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List of Citations from Science Citation Index for
S. Osher and J. A. Sethian, "Fronts Propagating with CurvatureDependent
Speed: Algorithms Based on HamiltonJacobi Formulations," Journal of
Computational Physics, 79(1): 1249, 1988.
1988: 1 1989: 2 1990: 4 1991: 11 1992: 24 1993: 23 1994: 19 1995: 29 1996: 45 1997: 60 1998: 52 1999: 71 2000: 74 2001: 72
Total citations: 487
As of 28 Jan 2002
By Year  By Citations  By Year with Abstract


1988 
 ASHURST, WT, SIVASHINSKY, GI, and YAKHOT, V, "FLAME FRONT PROPAGATION IN NONSTEADY HYDRODYNAMIC FIELDS," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 62, pp. 273284, 1988.
Abstract:
Mullins, in a series of papers, developed a surface dynamics
for phase interfaces whose evolution is controlled by mass
diffusion within the interface. It is our purpose here to
embed Mullin's theory within a general framework based on
balance laws for mass and capillary forces in conjunction with
a version of the second law, appropriate to a purely mechanical
theory, which asserts that the rate at which the free energy
increases cannot be greater than the energy inflow plus the
power supplied. We develop an appropriate constitutive theory,
and deduce general and approximate equations for the evolution
of the interface.


1989 
 POPE, SB, YEUNG, PK, and GIRIMAJI, SS, "THE CURVATURE OF MATERIALSURFACES IN ISOTROPIC TURBULENCE," PHYSICS OF FLUIDS AFLUID DYNAMICS, vol. 1, pp. 20102018, 1989.
Abstract:
Mullins, in a series of papers, developed a surface dynamics
for phase interfaces whose evolution is controlled by mass
diffusion within the interface. It is our purpose here to
embed Mullin's theory within a general framework based on
balance laws for mass and capillary forces in conjunction with
a version of the second law, appropriate to a purely mechanical
theory, which asserts that the rate at which the free energy
increases cannot be greater than the energy inflow plus the
power supplied. We develop an appropriate constitutive theory,
and deduce general and approximate equations for the evolution
of the interface.
 MARCUS, DL, and BERGER, SA, "THE INTERACTION BETWEEN A COUNTERROTATING VORTEX PAIR IN VERTICAL ASCENT AND A FREESURFACE," PHYSICS OF FLUIDS AFLUID DYNAMICS, vol. 1, pp. 19882000, 1989.
Abstract:
Mullins, in a series of papers, developed a surface dynamics
for phase interfaces whose evolution is controlled by mass
diffusion within the interface. It is our purpose here to
embed Mullin's theory within a general framework based on
balance laws for mass and capillary forces in conjunction with
a version of the second law, appropriate to a purely mechanical
theory, which asserts that the rate at which the free energy
increases cannot be greater than the energy inflow plus the
power supplied. We develop an appropriate constitutive theory,
and deduce general and approximate equations for the evolution
of the interface.


1990 
 LIONS, PL, and SOUGANIDIS, P, "CONVERGENCE OF MUSCL TYPE METHODS FOR SCALAR CONSERVATIONLAWS," COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE IMATHEMATIQUE, vol. 311, pp. 259264, 1990.
Abstract:
Mullins, in a series of papers, developed a surface dynamics
for phase interfaces whose evolution is controlled by mass
diffusion within the interface. It is our purpose here to
embed Mullin's theory within a general framework based on
balance laws for mass and capillary forces in conjunction with
a version of the second law, appropriate to a purely mechanical
theory, which asserts that the rate at which the free energy
increases cannot be greater than the energy inflow plus the
power supplied. We develop an appropriate constitutive theory,
and deduce general and approximate equations for the evolution
of the interface.
 DAVI, F, and GURTIN, ME, "ON THE MOTION OF A PHASE INTERFACE BY SURFACEDIFFUSION," ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND PHYSIK, vol. 41, pp. 782811, 1990.
Abstract:
Mullins, in a series of papers, developed a surface dynamics
for phase interfaces whose evolution is controlled by mass
diffusion within the interface. It is our purpose here to
embed Mullin's theory within a general framework based on
balance laws for mass and capillary forces in conjunction with
a version of the second law, appropriate to a purely mechanical
theory, which asserts that the rate at which the free energy
increases cannot be greater than the energy inflow plus the
power supplied. We develop an appropriate constitutive theory,
and deduce general and approximate equations for the evolution
of the interface.
 SETHIAN, JA, "NUMERICAL ALGORITHMS FOR PROPAGATING INTERFACES  HAMILTON JACOBI EQUATIONS AND CONSERVATIONLAWS," JOURNAL OF DIFFERENTIAL GEOMETRY, vol. 31, pp. 131161, 1990.
Abstract:
Mullins, in a series of papers, developed a surface dynamics
for phase interfaces whose evolution is controlled by mass
diffusion within the interface. It is our purpose here to
embed Mullin's theory within a general framework based on
balance laws for mass and capillary forces in conjunction with
a version of the second law, appropriate to a purely mechanical
theory, which asserts that the rate at which the free energy
increases cannot be greater than the energy inflow plus the
power supplied. We develop an appropriate constitutive theory,
and deduce general and approximate equations for the evolution
of the interface.
 OSHER, S, and RUDIN, LI, "FEATUREORIENTED IMAGEENHANCEMENT USING SHOCK FILTERS," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 27, pp. 919940, 1990.
Abstract:
Mullins, in a series of papers, developed a surface dynamics
for phase interfaces whose evolution is controlled by mass
diffusion within the interface. It is our purpose here to
embed Mullin's theory within a general framework based on
balance laws for mass and capillary forces in conjunction with
a version of the second law, appropriate to a purely mechanical
theory, which asserts that the rate at which the free energy
increases cannot be greater than the energy inflow plus the
power supplied. We develop an appropriate constitutive theory,
and deduce general and approximate equations for the evolution
of the interface.


1991 
 LAFON, F, and OSHER, S, "HIGHORDER FILTERING METHODS FOR APPROXIMATING HYPERBOLIC SYSTEMS OF CONSERVATIONLAWS," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 96, pp. 110142, 1991.
Abstract:
Because the stress resulting from compositional inhomogeneities
are long range, the local stress, diffusional flux and
equilibrium conditions at a point depend on the entire
composition distribution in a specimen. For a thin plate with
a onedimensional composition profile, this dependence is
simple; the local stress depends on the local composition and
on both the average composition and the first moment of the
composition profile, neither of which are local. A theory of
diffusion and equilibrium in a thin plate is developed, based
on a free energy that depends on composition, its gradients and
strain, and has a term for chemical effects at the plate
boundary. Under certain assumptions, a standard diffusion
equation is derived, with all of the nonlocal stress effects
in the boundary conditions. Solutions are altered by these new
conditions. Spontaneous bending is often a natural result of
diffusion.
 ASHURST, WT, and SIVASHINSKY, GI, "ON FLAME PROPAGATION THROUGH PERIODICFLOW FIELDS," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 80, pp. 159164, 1991.
Abstract:
Because the stress resulting from compositional inhomogeneities
are long range, the local stress, diffusional flux and
equilibrium conditions at a point depend on the entire
composition distribution in a specimen. For a thin plate with
a onedimensional composition profile, this dependence is
simple; the local stress depends on the local composition and
on both the average composition and the first moment of the
composition profile, neither of which are local. A theory of
diffusion and equilibrium in a thin plate is developed, based
on a free energy that depends on composition, its gradients and
strain, and has a term for chemical effects at the plate
boundary. Under certain assumptions, a standard diffusion
equation is derived, with all of the nonlocal stress effects
in the boundary conditions. Solutions are altered by these new
conditions. Spontaneous bending is often a natural result of
diffusion.
 WIKSWO, JP, WISIALOWSKI, TA, ALTEMEIER, WA, BALSER, JR, KOPELMAN, HA, and RODEN, DM, "VIRTUAL CATHODE EFFECTS DURING STIMULATION OF CARDIACMUSCLE  2DIMENSIONAL INVIVO EXPERIMENTS," CIRCULATION RESEARCH, vol. 68, pp. 513530, 1991.
Abstract:
We have found that when suprathreshold cathodal stimuli were
applied to the epicardium of canine ventricle, impulse
propagation originated at a "virtual cathode" with dimensions
greater than those of the physical cathode. We report the two
dimensional geometry of the virtual cathode as a function of
stimulus strength; the results are compared with the
predictions of an anisotropic, bidomain model of cardiac
conduction recently developed in our laboratories. Data were
collected in six pentobarbitalanesthetized dogs by using a
small plaque electrode sewn to the left ventricular epicardium.
Arrival times at closely spaced bipolar electrodes oriented
radially around a central cathode were obtained as a function
of stimulus strength and fiber orientation. The dimensions of
the virtual cathode were determined by linear back
extrapolation of arrival times to the time of stimulation. The
directional dependence of the conduction velocity was
consistent with previous reports: at 1 mA, longitudinal (0
degrees) and transverse (90degrees) velocities were 0.60 +/
0.03 and 0.29 +/ 0.02 m/sec, respectively. At 7 mA, the
longitudinal velocity was 0.75 +/ 0.05 m/sec, whereas there
was no significant change in the transverse velocity. In
contrast to conduction velocity, the virtual cathode was
smallest in the longitudinal orientation and largest between
45degrees and 60degrees. Virtual cathode size was dependent
on both orientation and stimulus strength: at 0degrees, the
virtual cathode was small (approximately 1 mm) and relatively
constant over the range of 17 mA; at oblique orientations (45
degrees90degrees), it displayed a roughly logarithmic
dependence on stimulus strength, approximately 1 mm at 1 mA and
approximately 3 mm at 7 mA. The bidomain, anisotropic model
reproduced both the stimulus strength and the fiberorientation
dependence of the virtual cathode geometry when the
intracellular and extracellular anisotropies were 10:1 and 4:1,
respectively, but not when the two anisotropies were equal. We
suggest that the virtual cathode provides a direct measure of
the determinants of cardiac activation; its complex geometry
appears to reflect the bidomain, anisotropic nature of cardiac
muscle.
 BARDI, M, and OSHER, S, "THE NONCONVEX MULTIDIMENSIONAL RIEMANN PROBLEM FOR HAMILTON JACOBI EQUATIONS," SIAM JOURNAL ON MATHEMATICAL ANALYSIS, vol. 22, pp. 344351, 1991.
Abstract:
Simple inequalities are presented for the viscosity solution of
a HamiltonJacobi equation in N space dimension when neither
the initial data nor the Hamiltonian need be convex (or
concave). The initial data are uniformly Lipschitz and can be
written as the sum of a convex function in a group of variables
and a concave function in the remaining variables, therefore
including the nonconvex Riemann problem. The inequalities
become equalities wherever a "maxmin" equals a "minmax" and
thus a representation formula for this problem is then
obtained, generalizing the classical Hopf's formulas.
 YANG, WH, "A DUALITY THEOREM FOR PLASTIC TORSION," INTERNATIONAL JOURNAL OF SOLIDS AND STRUCTURES, vol. 27, pp. 19811989, 1991.
Abstract:
Limit analysis of prismatic torsion bars was the earliest
attempt to apply plasticity theory to a continuum. The
simplicity of the problem made it feasible to use the two
dimensional Prandtl stress function, defined for the elastic
torsion problems, for the plastic stress distributions as well.
The gradient of the stress functions for plastic torsion has a
constant magnitude, and hence a function of this type assumes
the profile of a sand hill. This sand hill analogy of Nadai
(1950, The Theory of Flow and Fracture of Solids, McGrawHill,
U.K.) gave a visual sense of possible nonsmoothness of such
stress functions and thus discontinuous stress fields. Many
stress functions of plastic torsion for relatively simple
crosssections have been constructed graphically. However,
collapse modes in terms of warping functions were much less
reported. In this paper, we shall establish a duality theorem
which relates the correct stress function to the correct
warping function, thus providing the means to obtain complete
static and kinematic solutions. This dual variational
principle leads naturally to a general numerical algorithm
which guarantees convergence and accuracy. In this paper, we
shall only present three exact solutions to verify the theorem,
to demonstrate the possible nonsmooth feature of the solutions
and to reiterate this effective dual variational approach to
limit analysis in general.
 CHEN, YG, GIGA, Y, and GOTO, SI, "UNIQUENESS AND EXISTENCE OF VISCOSITY SOLUTIONS OF GENERALIZED MEANCURVATURE FLOW EQUATIONS," JOURNAL OF DIFFERENTIAL GEOMETRY, vol. 33, pp. 749786, 1991.
Abstract:
We construct a unique weak solution of the nonlinear PDE which
asserts each level set evolves in time according to its mean
curvature. This weak solution allows us then to define for any
compact set GAMMA0 a unique generalized motion by mean
curvature, existing for all time. We investigate the various
geometric properties and pathologies of this evolution.
 EVANS, LC, and SPRUCK, J, "MOTION OF LEVEL SETS BY MEANCURVATURE .1.," JOURNAL OF DIFFERENTIAL GEOMETRY, vol. 33, pp. 635681, 1991.
Abstract:
We construct a unique weak solution of the nonlinear PDE which
asserts each level set evolves in time according to its mean
curvature. This weak solution allows us then to define for any
compact set GAMMA0 a unique generalized motion by mean
curvature, existing for all time. We investigate the various
geometric properties and pathologies of this evolution.
 BRONSARD, L, and KOHN, RV, "MOTION BY MEANCURVATURE AS THE SINGULAR LIMIT OF GINZBURG LANDAU DYNAMICS," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 90, pp. 211237, 1991.
Abstract:
Seismic traveltimes can be computed efficiently on a regular
grid by an upwind finitedifference method. The method solves
a conservation law that describes changes in the gradient
components of the traveltime field. The traveltime field
itself is easily obtained from the solution of the conservation
law by numerical integration. The conservation law derives
from the eikonal equation, and its solution depicts the first
arrivaltime field. The upwind finitedifference scheme can be
implemented in fully vectorized form, in contrast to a similar
scheme proposed recently by Vidale. The resulting traveltime
field is useful both in Kirchhoff migration and modeling and in
seismic tomography. Many reliable methods exist for the
numerical solution of conservation laws, which appear in fluid
mechanics as statements of the conservation of mass, momentum,
etc. A firstorder upwind finitedifference scheme proves
accurate enough for seismic applications. Upwind schemes are
stable because they mimic the behavior of fluid flow by using
only information taken from upstream in the fluid. Other
common difference schemes are unstable, or overly dissipative,
at shocks (discontinuities in flow variables), which are time
gradient discontinuities in our approach to solving the eikonal
equation.
 VANTRIER, J, and SYMES, WW, "UPWIND FINITEDIFFERENCE CALCULATION OF TRAVELTIMES," GEOPHYSICS, vol. 56, pp. 812821, 1991.
Abstract:
Seismic traveltimes can be computed efficiently on a regular
grid by an upwind finitedifference method. The method solves
a conservation law that describes changes in the gradient
components of the traveltime field. The traveltime field
itself is easily obtained from the solution of the conservation
law by numerical integration. The conservation law derives
from the eikonal equation, and its solution depicts the first
arrivaltime field. The upwind finitedifference scheme can be
implemented in fully vectorized form, in contrast to a similar
scheme proposed recently by Vidale. The resulting traveltime
field is useful both in Kirchhoff migration and modeling and in
seismic tomography. Many reliable methods exist for the
numerical solution of conservation laws, which appear in fluid
mechanics as statements of the conservation of mass, momentum,
etc. A firstorder upwind finitedifference scheme proves
accurate enough for seismic applications. Upwind schemes are
stable because they mimic the behavior of fluid flow by using
only information taken from upstream in the fluid. Other
common difference schemes are unstable, or overly dissipative,
at shocks (discontinuities in flow variables), which are time
gradient discontinuities in our approach to solving the eikonal
equation.
 KEENER, JP, "AN EIKONALCURVATURE EQUATION FOR ACTIONPOTENTIAL PROPAGATION IN MYOCARDIUM," JOURNAL OF MATHEMATICAL BIOLOGY, vol. 29, pp. 629651, 1991.
Abstract:
We derive an "eikonalcurvature" equation to describe the
propagation of action potential wavefronts in myocardium. This
equation is used to study the effects of fiber orientation on
propagation in the myocardial wall. There are significant
computational advantages to the use of an eikonalcurvature
equation over a full ionic model of action potential spread.
With this model, it is shown that the experimentally observed
misalignment of spreading action potential "ellipses" from
fiber orientation in level myocardial surfaces is adequately
explained by the rotation of fiber orientation through the
myocardial wall. Additionally, it is shown that apparently
high propagation velocities on the epicardial and endocardial
surfaces are the result of propagation into the midwall region
and acceleration along midwall fibers before reemergence at an
outer surface at a time preceding what could be accomplished
with propagation along the surface alone.
 OSHER, S, and SHU, CW, "HIGHORDER ESSENTIALLY NONOSCILLATORY SCHEMES FOR HAMILTON JACOBI EQUATIONS," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 28, pp. 907922, 1991.
Abstract:
HamiltonJacobi (HJ) equations are frequently encountered in
applications, e.g., in control theory and differential games.
HJ equations are closely related to hyperbolic conservation
lawsin one space dimension the former is simply the integrated
version of the latter. Similarity also exists for the
multidimensional case, and this is helpful in the design of
difference approximations. In this paper highorder
essentially nonoscillatory (ENO) schemes for HJ equations are
investigated, which yield uniform highorder accuracy in smooth
regions and sharply resolve discontinuities in the derivatives.
The ENO scheme construction procedure is adapted from that for
hyperbolic conservation laws. The schemes are numerically
tested on a variety of onedimensional and twodimensional
problems, including a problem related to control optimization,
and highorder accuracy in smooth regions, good resolution of
discontinuities in the derivatives, and convergence to
viscosity solutions are observed.


1992 
 BRIO, M, and HUNTER, JK, "MACH REFLECTION FOR THE 2DIMENSIONAL BURGERSEQUATION," PHYSICA D, vol. 60, pp. 194207, 1992.
Abstract:
We study shock reflection for the two 2D Burgers equation. This
model equation is an asymptotic limit of the Euler equations,
and retains many of the features of the full equations. A von
Neumann type analysis shows that the 2D Burgers equation has
detachment, sonic, and Crocco points in complete analogy with
gas dynamics. Numerical solutions support the detachment/sonic
criterion for transition from regular to Mach reflection. There
is also strong numerical evidence that the reflected shock in
the 2D Burgers Mach reflection forms a smooth wave near the
Mach stem, as proposed by Colella and Henderson in their study
of the Euler equations.
 CHEN, XF, "GENERATION AND PROPAGATION OF INTERFACES IN REACTIONDIFFUSION SYSTEMS," TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 334, pp. 877913, 1992.
Abstract:
This paper is concerned with the asymptotic behavior, as
epsilon arrow pointing down and to the right 0, of the solution
(u(epsilon), upsilon(epsilon)) Of the second initialboundary
value problem of the reactiondiffusion system: [GRAPHICS]
where gamma > 0 is a constant. When upsilon isanelementof (
2 squareroot 3/9, 2 squareroot 3/9), f is bistable in the
sense that the ordinary differential equation u(t) = f(u,
upsilon) has two stable solutions u = h(upsilon) and u =
h+(upsilon) and one unstable solution u = h0(upsilon), where h
(upsilon) , h0(upsilon) , and h+(upsilon) are the three
solutions of the algebraic equation f(u, upsilon) = 0 . We show
that, when the initial data of upsilon is in the interval (2
squareroot 3/9, 2 squareroot 3/9) , the solution (u(epsilon),
upsilon(epsilon)) of the system tends to a limit (u, upsilon)
which is a solution of a free boundary problem, as long as the
free boundary problem has a unique classical solution. The
function u is a ''phase'' function in the sense that it
coincides with h+(upsilon) in one region OMEGA+ and with h
(upsilon) in another region OMEGA . The common boundary (free
boundary or interface) of the two regions OMEGA and OMEGA+
moves with a normal velocity equal to V(upsilon), where V(.) is
a function that can be calculated. The local (in time)
existence of a unique classical solution to the free boundary
problem is also established. Further we show that if initially
u(., 0)  h0(upsilon(.,0)) takes both positive and negative
values, then an interface will develop in a short time
O(epsilon\ln epsilon\) near the hypersurface where u(x, 0) 
h0(upsilon(x, 0)) = 0.
 KIMIA, BB, TANNENBAUM, A, and ZUCKER, SW, "ON THE EVOLUTION OF CURVES VIA A FUNCTION OF CURVATURE .1. THE CLASSICAL CASE," JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS, vol. 163, pp. 438458, 1992.
Abstract:
Because the stress resulting from compositional inhomogeneities
are long range, the local stress, diffusional flux and
equilibrium conditions at a point depend on the entire
composition distribution in a specimen. For a thin plate with
a onedimensional composition profile, this dependence is
simple; the local stress depends on the local composition and
on both the average composition and the first moment of the
composition profile, neither of which are local. A theory of
diffusion and equilibrium in a thin plate is developed, based
on a free energy that depends on composition, its gradients and
strain, and has a term for chemical effects at the plate
boundary. Under certain assumptions, a standard diffusion
equation is derived, with all of the nonlocal stress effects
in the boundary conditions. Solutions are altered by these new
conditions. Spontaneous bending is often a natural result of
diffusion.
 SETHIAN, JA, and STRAIN, J, "CRYSTALGROWTH AND DENDRITIC SOLIDIFICATION," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 98, pp. 231253, 1992.
Abstract:
Because the stress resulting from compositional inhomogeneities
are long range, the local stress, diffusional flux and
equilibrium conditions at a point depend on the entire
composition distribution in a specimen. For a thin plate with
a onedimensional composition profile, this dependence is
simple; the local stress depends on the local composition and
on both the average composition and the first moment of the
composition profile, neither of which are local. A theory of
diffusion and equilibrium in a thin plate is developed, based
on a free energy that depends on composition, its gradients and
strain, and has a term for chemical effects at the plate
boundary. Under certain assumptions, a standard diffusion
equation is derived, with all of the nonlocal stress effects
in the boundary conditions. Solutions are altered by these new
conditions. Spontaneous bending is often a natural result of
diffusion.
 GIGA, Y, and GOTO, S, "MOTION OF HYPERSURFACES AND GEOMETRIC EQUATIONS," JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, vol. 44, pp. 99111, 1992.
Abstract:
Because the stress resulting from compositional inhomogeneities
are long range, the local stress, diffusional flux and
equilibrium conditions at a point depend on the entire
composition distribution in a specimen. For a thin plate with
a onedimensional composition profile, this dependence is
simple; the local stress depends on the local composition and
on both the average composition and the first moment of the
composition profile, neither of which are local. A theory of
diffusion and equilibrium in a thin plate is developed, based
on a free energy that depends on composition, its gradients and
strain, and has a term for chemical effects at the plate
boundary. Under certain assumptions, a standard diffusion
equation is derived, with all of the nonlocal stress effects
in the boundary conditions. Solutions are altered by these new
conditions. Spontaneous bending is often a natural result of
diffusion.
 LARCHE, FC, and CAHN, JW, "PHASECHANGES IN A THIN PLATE WITH NONLOCAL SELFSTRESS EFFECTS," ACTA METALLURGICA ET MATERIALIA, vol. 40, pp. 947955, 1992.
Abstract:
Because the stress resulting from compositional inhomogeneities
are long range, the local stress, diffusional flux and
equilibrium conditions at a point depend on the entire
composition distribution in a specimen. For a thin plate with
a onedimensional composition profile, this dependence is
simple; the local stress depends on the local composition and
on both the average composition and the first moment of the
composition profile, neither of which are local. A theory of
diffusion and equilibrium in a thin plate is developed, based
on a free energy that depends on composition, its gradients and
strain, and has a term for chemical effects at the plate
boundary. Under certain assumptions, a standard diffusion
equation is derived, with all of the nonlocal stress effects
in the boundary conditions. Solutions are altered by these new
conditions. Spontaneous bending is often a natural result of
diffusion.
 GURTIN, ME, and SONER, HM, "SOME REMARKS ON THE STEFAN PROBLEM WITH SURFACESTRUCTURE," QUARTERLY OF APPLIED MATHEMATICS, vol. 50, pp. 291303, 1992.
Abstract:
This paper discusses a generalized Stefan problem which allows
for supercooling and superheating and for capillarity in the
interface between phases. Simple solutions are obtained
indicating the chief differences between this problem and the
classical Stefan problem. A weak formulation of the general
problem is given.
 ALVAREZ, L, LIONS, PL, and MOREL, JM, "IMAGE SELECTIVE SMOOTHING AND EDGEDETECTION BY NONLINEAR DIFFUSION .2.," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 29, pp. 845866, 1992.
Abstract:
A stable algorithm is proposed for image restoration based on
the "mean curvature motion" equation. Existence and uniqueness
of the "viscosity" solution of the equation are proved, a
L(infinity) stable algorithm is given, experimental results are
shown, and the subjacent vision model is compared with those
introduced recently by several vision researchers. The
algorithm presented appears to be the sharpest possible among
the multiscale image smoothing methods preserving uniqueness
and stability.
 TAYLOR, JE, CAHN, JW, and HANDWERKER, CA, "GEOMETRIC .1. MODELS OF CRYSTALGROWTH," ACTA METALLURGICA ET MATERIALIA, vol. 40, pp. 14431474, 1992.
Abstract:
Recent theoretical advances in the mathematical treatment of
geometric interface motion make more tractable the theory of a
wide variety of materials science problems where the interface
velocity is not controlled by longrangediffusion. Among the
interface motion problems that can be modelled as geometric are
certain types of phase changes, crystal growth, domain growth,
grain growth. ion beam and chemical etching, and coherency
stress driven interface migration. We provide an introduction
to nine mathematical methods for solving such problems, give
the limits of applicability of the methods, and discuss the
relations among them theoretically and their uses in
computation. Comparisons of some of them are made by displaying
how the same physical problems are treated in the various
applicable methods.
 CRANDALL, MG, ISHII, H, and LIONS, PL, "USERS GUIDE TO VISCOSITY SOLUTIONS OF 2NDORDER PARTIAL DIFFERENTIAL EQUATIONS," BULLETIN OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 27, pp. 167, 1992.
Abstract:
The notion of viscosity solutions of scalar fully nonlinear
partial differential equations of second order provides a
framework in which startling comparison and uniqueness
theorems, existence theorems, and theorems about continuous
dependence may now be proved by very efficient and striking
arguments. The range of important applications of these results
is enormous. This article is a selfcontained exposition of the
basic theory of viscosity solutions.
 MULDER, W, OSHER, S, and SETHIAN, JA, "COMPUTING INTERFACE MOTION IN COMPRESSIBLE GASDYNAMICS," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 100, pp. 209228, 1992.
Abstract:
A fully nonlinear evolution equation governing the propagation
of a premixed flame through a largescale spatially periodic
shear flow is derived, and steadystate solutions are obtained
numerically. The gas density is assumed to be constant across
the flame, but the local normal burning speed is allowed to
vary with the local strain and curvature along the flame front
in order to investigate the influence of the length scale of
the external flow on the average propagation speed of the
wrinkled flame. At fixed values of the amplitude of the flow
field variations an increase in the length scale (relative to
the flame thickness) is found to result in an increase in the
average flame propagation speed, in accordance with the
predictions of earlier theoretical investigations and with
experimental observations for the regime of largescale
turbulence. The propagation speed of the wrinkled flame is
calculated to exhibit the experimentally observed bending
effect, the tendency of the rate of change of the burning
velocity to decrease with increasing turbulence intensity at
low fixed turbulence Reynolds numbers. It is shown also how the
average flame speed depends on the ratio of the transverse to
longitudinal length scale associated with the periodic flow.
 ALDREDGE, RC, "THE PROPAGATION OF WRINKLED PREMIXED FLAMES IN SPATIALLY PERIODIC SHEARFLOW," COMBUSTION AND FLAME, vol. 90, pp. 121133, 1992.
Abstract:
A fully nonlinear evolution equation governing the propagation
of a premixed flame through a largescale spatially periodic
shear flow is derived, and steadystate solutions are obtained
numerically. The gas density is assumed to be constant across
the flame, but the local normal burning speed is allowed to
vary with the local strain and curvature along the flame front
in order to investigate the influence of the length scale of
the external flow on the average propagation speed of the
wrinkled flame. At fixed values of the amplitude of the flow
field variations an increase in the length scale (relative to
the flame thickness) is found to result in an increase in the
average flame propagation speed, in accordance with the
predictions of earlier theoretical investigations and with
experimental observations for the regime of largescale
turbulence. The propagation speed of the wrinkled flame is
calculated to exhibit the experimentally observed bending
effect, the tendency of the rate of change of the burning
velocity to decrease with increasing turbulence intensity at
low fixed turbulence Reynolds numbers. It is shown also how the
average flame speed depends on the ratio of the transverse to
longitudinal length scale associated with the periodic flow.
 KANSA, EJ, "A STRICTLY CONSERVATIVE SPATIAL APPROXIMATION SCHEME FOR THE GOVERNING ENGINEERING AND PHYSICS EQUATIONS OVER IRREGULAR REGIONS AND INHOMOGENEOUSLY SCATTERED NODES," COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 24, pp. 169190, 1992.
Abstract:
This paper reports the progress made in multiquadrics (MQ) as a
spatial approximation scheme for systems of governing equations
of engineering and physics by minimizing the spatial truncation
errors without excessive refinement. Although MQ is defined
over the general ndimensional real space, this paper is
limited to two spatial dimensions defined over a general non
convex irregular region containing inhomogeneously scattered
nodes. We have developed a strictly conservative interpolation
scheme over such irregular regions from which the partial
derivative estimates are obtained. In addition, we developed a
noniterative scheme to be used with domain decomposition to
ensure derivative continuity over contiguous regions. Jump
discontinuities for shock and material interfaces are likewise
treated by appropriate modification of the algorithm. We have
compared the relative errors of the derivative estimates
defined over an irregular region consisting of inhomogeneously
scattered nodes obtained by the MQ and Voronoi mesh schemes.
The MQ relative errors of the derivative estimates are three
orders of magnitude better than those obtained from the Voronoi
mesh method. (In our previous papers, we have shown that MQ is
superior in its derivative estimates over regular gridded
regions.) We have also used MQ to estimate derivatives within a
very narrow "shock" region with similar excellent results.
While comparing spatial approximation schemes for PDE's, we
found the MQ results to be superior in accuracy and were
calculated by far fewer operations than standard finite
difference schemes. Other authors have likewise used MQ
successfully to solve integral equations.
 DICARLO, A, GURTIN, ME, and PODIOGUIDUGLI, P, "A REGULARIZED EQUATION FOR ANISOTROPIC MOTIONBYCURVATURE," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 52, pp. 11111119, 1992.
Abstract:
For realistic interfacial energies, the equations of
anisotropic motionbycurvature exhibit backwardparabolic
behavior over portions of their domain, thereby inducing
phenomena such as the formation of facets and wrinkles. In this
paper, a physically consistent regularized equation that may be
used to analyze such phenomena is derived.
 ALVAREZ, L, GUICHARD, F, LIONS, PL, and MOREL, JM, "AXIOMS AND NEW OPERATORS OF MATHEMATICAL MORPHOLOGY," COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE IMATHEMATIQUE, vol. 315, pp. 265268, 1992.
Abstract:
We describe all multiscale causal, local, stable and shape
preserving filterings. This classification contains the
classical "morphological" operators, and some new ones.
 ZHU, JY, and SETHIAN, J, "PROJECTION METHODS COUPLED TO LEVEL SET INTERFACE TECHNIQUES," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 102, pp. 128138, 1992.
Abstract:
Stationary premixed flames in dualsource flow are considered.
The significant features of the dualsource system are that the
sources are of finite strength, and that a stagnation point is
located between the sources. A new mathematical model for front
propagation and advection is introduced that tracks the front
along streamlines. The equations for the stationary fronts of
the dualsource system are solved numerically. The assumption
of constantdensity potential flow is made to simplify the
problem and to illustrate the effects of the geometry alone. It
is shown that for sufficiently slow burning velocity (or
equivalently, small source separation), three stationary states
exist for closed, free flames, but one of them is unstable. In
addition, several types of burnerattached flames are observed.
Quasistationary evolution of a closed, free flame exhibits a
change of topology and hysteresis. Nonclosed flames are
predicted if local extinction due to flow strain is allowed.
 BREWSTER, ME, "STATIONARY PREMIXED FLAMES IN A DUALSOURCE SYSTEM," COMBUSTION AND FLAME, vol. 91, pp. 99105, 1992.
Abstract:
Stationary premixed flames in dualsource flow are considered.
The significant features of the dualsource system are that the
sources are of finite strength, and that a stagnation point is
located between the sources. A new mathematical model for front
propagation and advection is introduced that tracks the front
along streamlines. The equations for the stationary fronts of
the dualsource system are solved numerically. The assumption
of constantdensity potential flow is made to simplify the
problem and to illustrate the effects of the geometry alone. It
is shown that for sufficiently slow burning velocity (or
equivalently, small source separation), three stationary states
exist for closed, free flames, but one of them is unstable. In
addition, several types of burnerattached flames are observed.
Quasistationary evolution of a closed, free flame exhibits a
change of topology and hysteresis. Nonclosed flames are
predicted if local extinction due to flow strain is allowed.
 EVANS, LC, SONER, HM, and SOUGANIDIS, PE, "PHASETRANSITIONS AND GENERALIZED MOTION BY MEANCURVATURE," COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, vol. 45, pp. 10971123, 1992.
Abstract:
We study the limiting behavior of solutions to appropriately
rescaled versions of the AllenCahn equation, a simplified
model for dynamic phase transitions. We rigorously establish
the existence in the limit of a phaseantiphase interface
evolving according to mean curvature motion. This assertion is
valid for all positive time, the motion interpreted in the
generalized sense of EvansSpruck and ChenGigaGoto after the
onset of geometric singularities.
 DAVIS, SF, "AN INTERFACE TRACKING METHOD FOR HYPERBOLIC SYSTEMS OF CONSERVATIONLAWS," APPLIED NUMERICAL MATHEMATICS, vol. 10, pp. 447472, 1992.
Abstract:
This paper describes a method for tracking contact
discontinuities and material interfaces that arise in the
solution of hyperbolic systems of conservation laws. Numerical
results arc presented to show that the fronts are resolved to
within a mesh interval and smooth portions of the solution are
computed to within the accuracy of the underlying numerical
scheme.
 WU, MS, and DRISCOLL, JF, "A NUMERICALSIMULATION OF A VORTEX CONVECTED THROUGH A LAMINAR PREMIXED FLAME," COMBUSTION AND FLAME, vol. 91, pp. 310322, 1992.
Abstract:
A numerical study was conducted to understand how a vortex,
when convected at moderate speeds across a premixed flame, can
induce velocities that pull the flame along with the vortex,
causing flame elongation and unsteady flame stretch. If the
vortexinduced velocity that opposes flame motion is
sufficiently large, the flame cannot propagate over the vortex
and thus temporarily remains attached to the moving vortex. A
flame attachment criterion is discussed; when the criterion is
met the vortex forms cusps and pockets in the flame structure
similar to those observed experimentally. The net result of
increasing the vortex convection velocity is to reduce the
residence time of the vortex in the flame, which reduces the
degree of flame wrinkling. Vortex pairs that exert an extensive
strain on the flame were found to have significantly longer
residence times of interaction than vortices that exert
compressive strain; this difference in residence time helps to
explain why extensive strain on a flame is more probable in
turbulent flames than compressive strain. The calculated images
of the laminar flame shape show encouraging agreement with
experiment, which is another indication that flameinterface
simulations are a promising way to represent very wrinkled
turbulent premixed flames in a numerically efficient manner.
 ILMANEN, T, "GENERALIZED FLOW OF SETS BY MEANCURVATURE ON A MANIFOLD," INDIANA UNIVERSITY MATHEMATICS JOURNAL, vol. 41, pp. 671705, 1992.
Abstract:
The levelset flow of EvansSpruck and ChenGigaGoto is
generalized to a Riemannian manifold, using recent techniques
of CrandallIshii for viscosity solutions. Generally speaking,
the motion is not unique for noncompact closed sets, but the
definition can be modified to make the motion unique. We give
examples to show: (1) a smooth set can develop an interior that
originates from infinity (2) in the case of a Grayson
neckpinch, the evolving function u(x,t) need not remain C2.
 FRANZONE, PC, and GUERRI, L, "MODELS OF THE SPREADING OF EXCITATION IN MYOCARDIAL TISSUE," CRITICAL REVIEWS IN BIOMEDICAL ENGINEERING, vol. 20, pp. 211253, 1992.
Abstract:
We consider a macroscopic model of the excitation process in
the anisotropic myocardium involving the transmembrane,
extracellular, and extracardiac potentials upsilon, u(e), and
u0. The model is described by a reactiondiffusion (RD)
system, and the component upsilon exhibits a frontlike
behavior reflecting the features of the excitation process. In
numerical simulations, the presence of a moving excitation
layer imposes severe constraints on the time and space steps to
achieve stability and accuracy; consequently, application of
the model is very costly in terms of computer time. An
approximate model has been derived from the RD system by means
of a singular perturbation technique, and it is described by an
eikonal equation, nonlinear and elliptic, in the activation
time psi(x). Larger space steps are possible with this
equation. From psi(x), we can derive, for a given instant t,
the transmembrane potential upsilon and subsequently, by
solving an elliptic problem, we can compute the corresponding
extracellular and extracardiac potentials u(e) and u0. The
results of the RD and the eikonal models applied to a portion
of the ventricular wall are in excellent agreement; moreover,
the eikonal model requires only a small fraction of the
computer time needed by the RD system. Therefore, for large
scale simulations of the excitation process, only the eikonal
model has been used, and we investigate its ability to cope
with complex situations such as frontfront collisions and
related potential patterns.
 ALVAREZ, L, GUICHARD, F, LIONS, PL, and MOREL, JM, "FUNDAMENTAL EQUATIONS OF MULTISCALE ANALYSIS OF MOVIES," COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE IMATHEMATIQUE, vol. 315, pp. 11451148, 1992.
Abstract:
We describe all multiscale movie filtering which are causal,
local, shape preserving and galilean invariant.
 RUDIN, LI, OSHER, S, and FATEMI, E, "NONLINEAR TOTAL VARIATION BASED NOISE REMOVAL ALGORITHMS," PHYSICA D, vol. 60, pp. 259268, 1992.
Abstract:
A constrained optimization type of numerical algorithm for
removing noise from images is presented. The total variation of
the image is minimized subject to constraints involving the
statistics of the noise. The constraints are imposed using
Lagrange multipliers. The solution is obtained using the
gradientprojection method. This amounts to solving a time
dependent partial differential equation on a manifold
determined by the constraints. As t > infinity the solution
converges to a steady state which is the denoised image. The
numerical algorithm is simple and relatively fast. The results
appear to be stateoftheart for very noisy images. The method
is noninvasive, yielding sharp edges in the image. The
technique could be interpreted as a first step of moving each
level set of the image normal to itself with velocity equal to
the curvature of the level set divided by the magnitude of the
gradient of the image, and a second step which projects the
image back onto the constraint set.


1993 
 ROBERTS, S, "A LINE ELEMENT ALGORITHM FOR CURVE FLOW PROBLEMS IN THE PLANE," JOURNAL OF THE AUSTRALIAN MATHEMATICAL SOCIETY SERIES BAPPLIED MATHEMATICS, vol. 35, pp. 244261, 1993.
Abstract:
In this paper we shall describe a numerical method for the
solution of curve flow problems in which the normal velocity of
the curve depends locally on the position, normal and curvature
of the curve. The method involves approximating the curve by a
number of line elements (segments) which are only allowed to
move in a direction normal to the element. Hence the normal of
each line element remains constant throughout the evolution. In
regions of high curvature elements naturally tend to
accumulate. The method easily deals with the formation of cusps
as found in flame propagation problems and is computationally
comparable to a naive marker particle method. As a test of the
method we present a number of numerical experiments related to
mean curvature flow and flows associated with flame propagation
and bushfires.
 SAPIRO, G, and TANNENBAUM, A, "ON INVARIANT CURVE EVOLUTION AND IMAGEANALYSIS," INDIANA UNIVERSITY MATHEMATICS JOURNAL, vol. 42, pp. 9851009, 1993.
Abstract:
This paper deals with the mathematical theory of invariant
curve evolution. We present a highlevel procedure for the
formulation of geometric heat flows which are invariant with
respect to a given Lie group. This approach is based on the
classical theory of differential invariants. The affine group
is then analyzed in detail. Indeed, we give a rather complete
description of the properties of the affine geometric heat
equation. We moreover extend the results of [38] from the
convex to the nonconvex case. The paper concludes with a
summary of recent applications of curve evolution theory to
image analysis.
 FRANZONE, PC, and GUERRI, L, "SPREADING OF EXCITATION IN 3D MODELS OF THE ANISOTROPIC CARDIAC TISSUE .1. VALIDATION OF THE EIKONAL MODEL," MATHEMATICAL BIOSCIENCES, vol. 113, pp. 145209, 1993.
Abstract:
In this work we investigate, by means of numerical simulations,
the performance of two mathematical models describing the
spread of excitation in a threedimensional block representing
anisotropic cardiac tissue. The first model is characterized by
a reactiondiffusion system in the transmembrane and
extracellular potentials v and u. The second model is derived
from the first by means of a perturbation technique. It is
characterized by an eikonal equation, nonlinear and elliptic in
the activation time psi(x). The level surfaces psi(x) = t
represent the wavefront positions. The numerical procedures
based on the two models were applied to test functions and to
excitation processes elicited by local stimulations in a
relatively small block. The results are in excellent agreement,
and for the same problem the computation time required by the
eikonal equation is a small fraction of that needed for the
reactiondiffusion system. Thus we have strong evidence that
the eikonal equation provides a reliable and numerically
efficient model of the excitation process. Moreover, numerical
simulations have been performed to validate an approximate
model for the extracellular potential based on knowledge of the
excitation sequence. The features of the extracellular
potential distribution affected by the anisotropic conductivity
of the medium were investigated.
 OLIKER, VI, and URALTSEVA, NN, "EVOLUTION OF NONPARAMETRIC SURFACES WITH SPEED DEPENDING ON CURVATURE .2. THE MEANCURVATURE CASE," COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, vol. 46, pp. 97135, 1993.
Abstract:
We consider an evolution which starts as a flow of smooth
surfaces in nonparametric form propagating in space with normal
speed equal to the mean curvature of the current surface The
boundaries of the surfaces are assumed to remain fixed. G.
Huisken has shown that if the boundary of the domain over which
this flow is considered satisfies the ''mean curvature''
condition of H. Jenkins and J. Serrin (that is, the boundary of
the domain is convex ''in the mean'') then the corresponding
initial boundary value problem with Dirichlet boundary data the
smooth initial data admits a smooth SolUtion for all time. In
this paper we consider the case of arbitrary domains with
smooth boundaries not necessarily satisfying the condition of
JenkinsSerrin. In this case, even if the flow starts with
smooth initial data and homogeneous Dirichlet boundary data,
singularities may develop in finite time at the boundary of the
domain and the solution will not satisfy the boundary
condition. We prove, however. existence of solutions that are
smooth inside the domain for all time and become smooth up to
the boundary after elapsing of a sufficiently long period of
time. From that moment on such solutions assume the boundary
values in the classical sense. We also give sufficient
conditions that guarantee the existence of classical solutions
for all time t greaterthanorequalto 0. In addition. we
establish estimates of the rate at which solutions tend to zero
as t > infinity.
 IKEDA, T, and MIMURA, M, "AN INTERFACIAL APPROACH TO REGIONAL SEGREGATION OF 2 COMPETING SPECIES MEDIATED BY A PREDATOR," JOURNAL OF MATHEMATICAL BIOLOGY, vol. 31, pp. 215240, 1993.
Abstract:
We consider the problem of coexistence of two competing species
mediated by the presence of a predator. We employ a reaction
diffusion model equation with LotkaVolterra interaction, and
speculate that the possibility of coexistence is enhanced by
differences in the diffusion rates of the prey and their
predator. In the limit where the diffusion rate of the prey
tends to zero, a new equation is derived and the dynamics of
spatial segregation is discussed by means of the interfacial
dynamics approach. Also, we show that spatial segregation
permits periodic and chaotic dynamics for certain parameter
ranges.
 SONER, HM, "MOTION OF A SET BY THE CURVATURE OF ITS BOUNDARY," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 101, pp. 313372, 1993.
Abstract:
The connection between the weak theories for a class of
geometric equations and the asymptotics of appropriately
rescaled reactiondiffusion equations is rigorously
established. Two different scalings are studied. In the first,
the limiting geometric equation is a firstorder equation; in
the second, it is a generalization of the mean curvature
equation. Intrinsic definitions for the geometric equations are
obtained, and uniqueness under a geometric condition on the
initial surface is proved. In particular, in the case of the
mean curvature equation, this condition is satisfied by
surfaces that are strictly starshaped, that have positive mean
curvature, or that satisfy a condition that interpolates
between the positive mean curvature and the starshape
conditions.
 BARLES, G, SONER, HM, and SOUGANIDIS, PE, "FRONT PROPAGATION AND PHASE FIELDTHEORY," SIAM JOURNAL ON CONTROL AND OPTIMIZATION, vol. 31, pp. 439469, 1993.
Abstract:
The connection between the weak theories for a class of
geometric equations and the asymptotics of appropriately
rescaled reactiondiffusion equations is rigorously
established. Two different scalings are studied. In the first,
the limiting geometric equation is a firstorder equation; in
the second, it is a generalization of the mean curvature
equation. Intrinsic definitions for the geometric equations are
obtained, and uniqueness under a geometric condition on the
initial surface is proved. In particular, in the case of the
mean curvature equation, this condition is satisfied by
surfaces that are strictly starshaped, that have positive mean
curvature, or that satisfy a condition that interpolates
between the positive mean curvature and the starshape
conditions.
 KIMMEL, R, and BRUCKSTEIN, AM, "SHAPE OFFSETS VIA LEVEL SETS," COMPUTERAIDED DESIGN, vol. 25, pp. 154162, 1993.
Abstract:
An algorithm for shape offsetting is presented that is based on
levelset propagation. This algorithm avoids the topological
problems encountered in traditional offsetting algorithms, and
it deals with curvature singularities by including an 'entropy
condition' in its numerical implementation.
 KOBAYASHI, R, "MODELING AND NUMERICAL SIMULATIONS OF DENDRITIC CRYSTALGROWTH," PHYSICA D, vol. 63, pp. 410423, 1993.
Abstract:
A simple phase field model for one component melt growth is
presented. which includes anisotropy in a certain form. The
formation of various dendritic patterns can be shown by a
series of numerical simulations of this model. Qualitative
relations between the shapes of crystals and some physical
parameters are discussed. Also it is shown that noises give a
crucial influence on the side branch structure of dendrites in
some situations.
 HARABETIAN, E, "PROPAGATION OF SINGULARITIES, HAMILTONJACOBI EQUATIONS AND NUMERICAL APPLICATIONS," TRANSACTIONS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 337, pp. 5971, 1993.
Abstract:
We consider applications of HamiltonJacobi equations for which
the initial data is only assumed to be in L(infinity). Such
problems arise for example when one attempts to describe
several characteristic singularities of the compressible Euler
equations such as contact and acoustic surfaces, propagating
from the same discontinuous initial front. These surfaces
represent the level sets of solutions to a HamiltonJacobi
equation which belongs to a special class. For such Hamilton
Jacobi equations we prove the existence and regularity of
solutions for any positive time and convergence to initial data
along rays of geometrical optics at any point where the
gradient of the initial data exists. Finally, we present
numerical algorithms for efficiently capturing singular fronts
with complicated topologies such as corners and cusps. The
approach of using HamiltonJacobi equations for capturing
fronts has been used in [14] for fronts propagating with
curvaturedependent speed.
 CHOPP, DL, "COMPUTING MINIMALSURFACES VIA LEVEL SET CURVATURE FLOW," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 106, pp. 7791, 1993.
Abstract:
The propagation of a twodimensional wave front in an excitable
medium is dependent on the curvature of the front; current
theories of excitable reactiondiffusion models predict that,
when reaction is much faster than diffusion, the normal wave
speed (N) is approximately related to the curvature of the wave
front (kappa), the plane wave speed (c), and the diffusion
coefficient of the propagator variable (D), by the ''eikonal''
equation, N = c  Dkappa. We show that a simple model for
intracellular calcium (Ca2+) wave propagation does not obey the
eikonal equation, and postulate an alternative curvature
equation that is dependent on the parameter values used in the
model. This new curvature relation is confirmed by numerical
simulations. We raise the possibility that different models for
Ca2+ wave propagation will have qualitatively different spiral
wave patterns, providing a new way of distinguishing between
proposed models. The theory developed here also necessitates a
reconsideration of methods previously used to measure the
intracellular diffusion coefficient of Ca2+.
 CLARKE, JF, KARNI, S, QUIRK, JJ, ROE, PL, SIMMONDS, LG, and TORO, EF, "NUMERICAL COMPUTATION OF 2DIMENSIONAL UNSTEADY DETONATION WAVES IN HIGHENERGY SOLIDS," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 106, pp. 215233, 1993.
Abstract:
The propagation of a twodimensional wave front in an excitable
medium is dependent on the curvature of the front; current
theories of excitable reactiondiffusion models predict that,
when reaction is much faster than diffusion, the normal wave
speed (N) is approximately related to the curvature of the wave
front (kappa), the plane wave speed (c), and the diffusion
coefficient of the propagator variable (D), by the ''eikonal''
equation, N = c  Dkappa. We show that a simple model for
intracellular calcium (Ca2+) wave propagation does not obey the
eikonal equation, and postulate an alternative curvature
equation that is dependent on the parameter values used in the
model. This new curvature relation is confirmed by numerical
simulations. We raise the possibility that different models for
Ca2+ wave propagation will have qualitatively different spiral
wave patterns, providing a new way of distinguishing between
proposed models. The theory developed here also necessitates a
reconsideration of methods previously used to measure the
intracellular diffusion coefficient of Ca2+.
 SNEYD, J, and ATRI, A, "CURVATURE DEPENDENCE OF A MODEL FOR CALCIUM WAVEPROPAGATION," PHYSICA D, vol. 65, pp. 365372, 1993.
Abstract:
The propagation of a twodimensional wave front in an excitable
medium is dependent on the curvature of the front; current
theories of excitable reactiondiffusion models predict that,
when reaction is much faster than diffusion, the normal wave
speed (N) is approximately related to the curvature of the wave
front (kappa), the plane wave speed (c), and the diffusion
coefficient of the propagator variable (D), by the ''eikonal''
equation, N = c  Dkappa. We show that a simple model for
intracellular calcium (Ca2+) wave propagation does not obey the
eikonal equation, and postulate an alternative curvature
equation that is dependent on the parameter values used in the
model. This new curvature relation is confirmed by numerical
simulations. We raise the possibility that different models for
Ca2+ wave propagation will have qualitatively different spiral
wave patterns, providing a new way of distinguishing between
proposed models. The theory developed here also necessitates a
reconsideration of methods previously used to measure the
intracellular diffusion coefficient of Ca2+.
 LI, XL, "STUDY OF 3DIMENSIONAL RAYLEIGHTAYLOR INSTABILITY IN COMPRESSIBLE FLUIDS THROUGH LEVEL SET METHOD AND PARALLEL COMPUTATION," PHYSICS OF FLUIDS AFLUID DYNAMICS, vol. 5, pp. 19041913, 1993.
Abstract:
Computation of threedimensional (3D) RayleighTaylor
instability in compressible fluids is performed on a MIMD
computer. A secondorder TVD scheme is applied with a fully
parallelized algorithm to the 3D Euler equations. The
computational program is implemented for a 3D study of bubble
evolution in the RayleighTaylor instability with varying
bubble aspect ratio and for largescale simulation of a 3D
random fluid interface. The numerical solution is compared with
the experimental results by Taylor.
 MOSCO, U, "SOME VARIATIONAL ASPECTS OF DISCONTINUOUS MEDIA," BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, vol. 7A, pp. 149198, 1993.
Abstract:
A level set formulation for the solution of the HamiltonJacobi
equation F(x, y, u, u(x), u(y)) = 0 is Presented, where u is
prescribed on a set of closed bounded noncharacteristic curves.
A time dependent HamiltonJacobi equation is derived such that
the zero level set at various time t of this solution is
precisely the set of points (x, y) for which u(x, y) = t. This
gives a fast and simple numerical method for generating the
viscosity solution to F = 0. The level set capturing idea was
first introduced by Osher and Sethian [J. Comput. Phys., 79
(1988), pp. 1249], and the observation that this is useful for
an important computer vision problem of this type was then made
by Kimmel and Bruckstein in [Technion (Israel) Computer Science
Report, CIS #9209, 1992] following Bruckstein [Comput. Vision
Graphics Image Process, 44 (1988), pp. 139154]. Finally, it is
noted that an extension to many space dimensions is immediate.
 OSHER, S, "A LEVEL SET FORMULATION FOR THE SOLUTION OF THE DIRICHLET PROBLEM FOR HAMILTONJACOBI EQUATIONS," SIAM JOURNAL ON MATHEMATICAL ANALYSIS, vol. 24, pp. 11451152, 1993.
Abstract:
A level set formulation for the solution of the HamiltonJacobi
equation F(x, y, u, u(x), u(y)) = 0 is Presented, where u is
prescribed on a set of closed bounded noncharacteristic curves.
A time dependent HamiltonJacobi equation is derived such that
the zero level set at various time t of this solution is
precisely the set of points (x, y) for which u(x, y) = t. This
gives a fast and simple numerical method for generating the
viscosity solution to F = 0. The level set capturing idea was
first introduced by Osher and Sethian [J. Comput. Phys., 79
(1988), pp. 1249], and the observation that this is useful for
an important computer vision problem of this type was then made
by Kimmel and Bruckstein in [Technion (Israel) Computer Science
Report, CIS #9209, 1992] following Bruckstein [Comput. Vision
Graphics Image Process, 44 (1988), pp. 139154]. Finally, it is
noted that an extension to many space dimensions is immediate.
 SAPIRO, G, and TANNENBAUM, A, "AFFINE INVARIANT SCALESPACE," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 11, pp. 2544, 1993.
Abstract:
A new affine invariant scalespace for planar curves is
presented in this work. The scalespace is obtained from the
solution of a novel nonlinear curve evolution equation which
admits affine invariant solutions. This flow was proved to be
the affine analogue of the well known Euclidean shortening
flow. The evolution also satisfies properties such as
causality, which makes it useful in defining a scalespace.
Using an efficient numerical algorithm for curve evolution,
this continuous affine flow is implemented, and examples are
presented. The affineinvariant progressive smoothing property
of die evolution equation is demonstrated as well.
 ALVAREZ, L, GUICHARD, F, LIONS, PL, and MOREL, JM, "AXIOMS AND FUNDAMENTAL EQUATIONS OF IMAGEPROCESSING," ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, vol. 123, pp. 199257, 1993.
Abstract:
Imageprocessing transforms must satisfy a list of formal
requirements. We discuss these requirements and classify them
into three categories: ''architectural requirements'' like
locality, recursivity and causality in the scale space,
''stability requirements'' like the comparison principle and
''morphological requirements'', which correspond to shape
preserving properties (rotation invariance, scale invariance,
etc.). A complete classification is given of all image
multiscale transforms satisfying these requirements. This
classification yields a characterization of all classical
models and includes new ones, which all are partial
differential equations. The new models we introduce have more
invariance properties than all the previously known models and
in particular have a projection invariance essential for shape
recognition. Numerical experiments are presented and compared.
The same method is applied to the multiscale analysis of
movies. By introducing a property of Galilean invariance, we
find a single multiscale morphological model for movie
analysis.
 VASSILICOS, JC, and HUNT, JCR, "TURBULENT FLAMELET PROPAGATION," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 87, pp. 291327, 1993.
Abstract:
A formalism for a flamelet's evolution of its spatial
distribution is derived from a field equation which is slightly
more general than Williams' field equation. Unlike Williams'
field equation, the field equation used here, though non
linear, has the property that an arbitrary linear combination
of interface solutions (Heavyside type of functions) is also a
solution. We therefore can describe the location of the
flamelet with two interfaces rather than one, both moving
relative to the flow in the same direction. The volume between
these two interfaces is on average conserved; this makes it
possible to define a probability density for the spatial
distribution of the flamelet, and thereby derive equations
describing the evolution of the spatial distribution of folds
and wrinkles of the flame front. Three main conclusions are
reached in this paper using this formalism, through the exact
analytical study of a flamelet in an arbitrary 1d velocity
field, and through the numerical study of a flamelet in a
simulated 2d turbulent velocity field. (1) The rate of
advancement u(M) of the average location of the flame front can
be smaller than the turbulent flame speed u(T) at short times,
and sometimes even smaller than the laminar flame speed u(L)
(at short times). It is shown, in the case of an arbitrary 1d
velocity field, that u(M) = u(T) only after cusps have formed
on the flamelet, and u(M) < u(L) < u(T) before. (2) If the
turbulence is too weak or too strong compared with the laminar
flame speed, the dispersion of the flame is, at short times,
increased by the turbulence and reduced by the laminar flame
speed. (3) The dispersion of the flame is skewed towards the
direction of the flame's propagation at all times, even before
cusp formation.
 EVANS, LC, "CONVERGENCE OF AN ALGORITHM FOR MEANCURVATURE MOTION," INDIANA UNIVERSITY MATHEMATICS JOURNAL, vol. 42, pp. 533557, 1993.
Abstract:
Bence, Merriman and Osher [BMO] have proposed a new numerical
algorithm for computing mean curvature flow, in terms of
solutions of the usual heat equation, continually reinitialized
after short time steps. This paper employs nonlinear semigroup
theory to reconcile their algorithm with the ''levelset''
approach to mean curvature flow of OsherSethian [OS], Evans
Spruck [ES], and ChenGigaGoto [CGG].
 HAMAGUCHI, S, DALVIE, M, FAROUKI, RT, and SETHURAMAN, S, "A SHOCKTRACKING ALGORITHM FOR SURFACE EVOLUTION UNDER REACTIVEION ETCHING," JOURNAL OF APPLIED PHYSICS, vol. 74, pp. 51725184, 1993.
Abstract:
A new algorithm that determines the evolution of a surface
eroding under reactiveion etching is presented. The surface
motion is governed by both the HamiltonJacobi equation and the
entropy condition for a given etch rate. The trajectories of
''shocks'' and ''rarefaction waves'' are then directly tracked,
and thus this method may be regarded as a generalization of the
method of characteristics. This allows slope discontinuities to
be accurately calculated without artificial diffusion. The
algorithm is compared with ''geometric'' surface evolution
methods, such as the linesegment method.
 CASELLES, V, CATTE, F, COLL, T, and DIBOS, F, "A GEOMETRIC MODEL FOR ACTIVE CONTOURS IN IMAGEPROCESSING," NUMERISCHE MATHEMATIK, vol. 66, pp. 131, 1993.
Abstract:
We propose a new model for active contours based on a geometric
partial differential equation. Our model is intrinsec, stable
(satisfies the maximum principle) and permits a rigorous
mathematical analysis. It enables us to extract smooth shapes
(we cannot retrieve angles) and it can be adapted to find
several contours simultaneously. Moreover, as a consequence of
the stability, we can design robust algorithms which can be
engineed with no parameters in applications. Numerical
experiments are presented.
 SAPIRO, G, KIMMEL, R, SHAKED, D, KIMIA, BB, and BRUCKSTEIN, AM, "IMPLEMENTING CONTINUOUSSCALE MORPHOLOGY VIA CURVE EVOLUTION," PATTERN RECOGNITION, vol. 26, pp. 13631372, 1993.
Abstract:
A new approach to digital implementation of continuousscale
mathematical morphology is presented. The approach is based on
discretization of evolution equations associated with
continuous multiscale morphological operations. Those
equations, and their corresponding numerical implementation,
can be derived either directly from mathematical morphology
definitions or from curve evolution theory. The advantages of
the proposed approach over the classical discrete morphology
are demonstrated.


1994 
 ZHU, J, and RONNEY, PD, "SIMULATION OF FRONT PROPAGATION AT LARGE NONDIMENSIONAL FLOW DISTURBANCE INTENSITIES," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 100, pp. 183201, 1994.
Abstract:
Numerical modeling of propagating fronts in nonuniform two
dimensional flow fields is performed in order to simulate the
effect of such flows on premixed flame fronts. In particular,
the influence of the flow disturbance intensity (u') on the
mean front propagation rate (ST) is examined. A secondorder
numerical technique is employed that combines the level set (G
equation) formulation to describe the selfpropagation of the
front and a multidimensional upwind technique to describe the
convection of the front by the flow field. In this way the
effect of the nondimensional disturbance intensity (u'/SL) on
the nondimensional propagation rate (ST/SL) at values of
u'/SL >> 1 is computed. The dependence of the laminar
propagation speed (SL) on the flame stretch (including both
the front curvature and the velocity strain effects) is
incorporated in this formulation. We focus on front propagation
in simulated TaylorCouette flows in the ''Taylor vortex''
regime and the results are found to compare favorably with
recent experiments on the propagation of isothermal chemical
fronts in this flow. The formation of ''islands'' of reactants
is observed and its relation to front propagation rates is
discussed.
 SOILLE, P, "GENERALIZED GEODESY VIA GEODESIC TIME," PATTERN RECOGNITION LETTERS, vol. 15, pp. 12351240, 1994.
Abstract:
The time necessary to cover a path on a greyscale image is the
sum of the greylevel values along the path. The geodesic time
between two points in a greyscale image is defined as the
smallest amount of time allowing to link these points. The
geodesic time allows the definition of generalized geodesic
distances, erosions, dilations, and skeletons by influence
zones. An application to minimal path extraction on greyscale
images is presented.
 SAPIRO, G, and TANNENBAUM, A, "ON AFFINE PLANE CURVE EVOLUTION," JOURNAL OF FUNCTIONAL ANALYSIS, vol. 119, pp. 79120, 1994.
Abstract:
A mathematical model is developed for melting of a multilayered
medium while a heat source traverses one boundary. The finite
element method uses moving meshes, fronttracking using spines,
an automatic timestep algorithm, and an efficient solution of
the linearized equations. A novel solution method allows the
fixedmesh code to work unchanged but allows a moving mesh in
other problems. The finiteelement method is applied when the
heater mesh moves with respect to the multilayered medium mesh.
The same technique allows parallel processing for finite
element codes. The model is applied to several test problems
and then to the title problem.
 WESTERBERG, KW, WIKLOF, C, and FINLAYSON, BA, "TIMEDEPENDENT FINITEELEMENT MODELS OF PHASECHANGE PROBLEMS WITH MOVING HEATSOURCES," NUMERICAL HEAT TRANSFER PART BFUNDAMENTALS, vol. 25, pp. 119143, 1994.
Abstract:
A mathematical model is developed for melting of a multilayered
medium while a heat source traverses one boundary. The finite
element method uses moving meshes, fronttracking using spines,
an automatic timestep algorithm, and an efficient solution of
the linearized equations. A novel solution method allows the
fixedmesh code to work unchanged but allows a moving mesh in
other problems. The finiteelement method is applied when the
heater mesh moves with respect to the multilayered medium mesh.
The same technique allows parallel processing for finite
element codes. The model is applied to several test problems
and then to the title problem.
 ALVAREZ, L, and MAZORRA, L, "SIGNAL AND IMAGERESTORATION USING SHOCK FILTERS AND ANISOTROPIC DIFFUSION," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 31, pp. 590605, 1994.
Abstract:
The authors define a new class of filters for noise elimination
and edge enhancement by using shock filters and anisotropic
diffusion. Some nonlinear partial differential equations used
as models for these filters are studied. The authors develop
recursive and unconditional stable schemes which drastically
reduce the computational effort of the algorithms. A new fast
recursive approach to linear Gaussian filters is also shown by
using the heat equation.
 KARNI, S, "MULTICOMPONENT FLOW CALCULATIONS BY A CONSISTENT PRIMITIVE ALGORITHM," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 112, pp. 3143, 1994.
Abstract:
The dynamics of inviscid multicomponent fluids may be modelled
by the Euler equations, augmented by one (or more) additional
species equation(s). Attempts to compute solutions for extended
Euler models in conservation form, show strong oscillations and
other computational inaccuracies near material interfaces.
These are due to erroneous pressure fluctuations generated by
the conservative wave model. This problem does not occur in
single component computations and arises only in the presence
of several species. A nonconservative (primitive) Euler
formulation is proposed, which results in complete elimination
of the oscillations. The numerical algorithm uses small viscous
perturbations to remove leading order conservation errors and
is conservative to the order of numerical approximation.
Numerical experiments show clean monotonic solution profiles,
with acceptably small conservation error for shocks of weak to
moderate strengths. (C) 1994 Academic Press, Inc.
 MERRIMAN, B, BENCE, JK, and OSHER, SJ, "MOTION OF MULTIPLE JUNCTIONS  A LEVEL SET APPROACH," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 112, pp. 334363, 1994.
Abstract:
A coupled level set method for the motion of multiple junctions
is proposed. The new method extends the ''HamiltonJacobi''
level set formulation of Osher and Sethian. It retains the
feature of tracking fronts by following level sets and allows
the specification of arbitrary velocities on each front, The
diffusion equation is shown to generate curvature dependent
motion and this is used to develop an algorithm to move
multiple junctions with curvaturedependent speed. Systems of
reactiondiffusion equations are shown to possess inherent
properties which prohibit efficient numerical solutions when
applied to curvaturedependent motion. (C) 1994 Academic
Press, Inc.
 YU, KM, SUNG, CJ, and LAW, CK, "SOME ASPECTS OF THE FREELY PROPAGATING PREMIXED FLAME IN A SPATIALLY PERIODICFLOW FIELD," COMBUSTION AND FLAME, vol. 97, pp. 375383, 1994.
Abstract:
The premixed flame situated in a spatially periodic flow field
is examined using the passive propagation model with the local
flame speed affected by stretch and nonequidiffusion. Numerical
solution shows that the average flame speed increases with
either increasing fluctuation amplitude or increasing
wavelength of the imposed flow field, and that the flame
surface can locally extinguish for sufficiently large
fluctuation amplitude of the imposed flow. Perturbation
solutions in the weakly wrinkled flame and the thin flame
limits are presented. The formation of comers on the flame
surface in the thin flame limit is illustrated, and the
structure of the comer is further found to resemble that of the
Bunsen flame. The premixed flame situated in a twodimensional
periodic flow field is also analyzed in the Huygens limit,
leading to the observation that flame surface discontinuities
exist in the form of cones.
 KIMURA, M, "ACCURATE NUMERICAL SCHEME FOR THE FLOW BY CURVATURE," APPLIED MATHEMATICS LETTERS, vol. 7, pp. 6973, 1994.
Abstract:
An accurate finite difference scheme for the flow by curvature
in R2 is presented, and its convergence theorem is stated. The
numerical scheme has a correction term which is effective in
locating points uniformly and the effect prevents the
computation from breaking down.
 SORAVIA, P, "GENERALIZED MOTION OF A FRONT PROPAGATING ALONG ITS NORMAL DIRECTION  A DIFFERENTIALGAMES APPROACH," NONLINEAR ANALYSISTHEORY METHODS & APPLICATIONS, vol. 22, pp. 12471262, 1994.
Abstract:
The nonlinear interfacial instability of a liquid jet in a
coflowing compressible airstream is studied numerically. A
highresolution scheme which has secondorder accuracy in space
and time is coupled with a Lagrangian marker particle algorithm
to visualize the largescale motion of the interfaces in
compressible flow. A numerical algorithm based on an
approximate equation of state of a compressible liquid is
developed to allow this twofluid system to be governed by the
nonlinear unsteady Euler equations in conservative form. The
initial growth of small disturbances given by the simulations
agrees well with linear theory. The process of jet disruption
in compressible flow is demonstrated to consist of the
formation of liquid spikes, interweaving of the gas and liquid
and stretching and detachment of the liquid main center core.
 LI, HS, "NUMERICALSIMULATION OF THE INSTABILITY OF AN INVISCID LIQUID JET IN A COFLOWING COMPRESSIBLE AIRSTREAM," COMPUTERS & FLUIDS, vol. 23, pp. 853880, 1994.
Abstract:
The nonlinear interfacial instability of a liquid jet in a
coflowing compressible airstream is studied numerically. A
highresolution scheme which has secondorder accuracy in space
and time is coupled with a Lagrangian marker particle algorithm
to visualize the largescale motion of the interfaces in
compressible flow. A numerical algorithm based on an
approximate equation of state of a compressible liquid is
developed to allow this twofluid system to be governed by the
nonlinear unsteady Euler equations in conservative form. The
initial growth of small disturbances given by the simulations
agrees well with linear theory. The process of jet disruption
in compressible flow is demonstrated to consist of the
formation of liquid spikes, interweaving of the gas and liquid
and stretching and detachment of the liquid main center core.
 SUSSMAN, M, SMEREKA, P, and OSHER, S, "A LEVEL SET APPROACH FOR COMPUTING SOLUTIONS TO INCOMPRESSIBLE 2PHASE FLOW," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 114, pp. 146159, 1994.
Abstract:
A level set approach for computing solutions to incompressible
twophase flow is presented. The interface between the two
fluids is considered to be sharp and is described as the zero
level set of a smooth function. We use a secondorder
projection method which implements a secondorder upwinded
procedure for differencing the convection terms. A new
treatment of the level set method allows us to include large
density and viscosity ratios as well as surface tension. We
consider the motion of air bubbles in water and falling water
drops in air. (C) 1994 Academic Press, Inc.
 HOPPE, J, "SURFACE MOTIONS AND FLUIDDYNAMICS," PHYSICS LETTERS B, vol. 335, pp. 4144, 1994.
Abstract:
A certain class of surface motions, including those of a
relativistic membrane minimizing the threedimensional volume
swept out in Minkowski space, is shown to be equivalent to
threedimensional steadystate irrotational inviscid isentropic
gas dynamics. The SU(infinity) Nahm equations turn out to
correspond to motions where the time t at which the surface
moves through the point r is a harmonic function of the three
space coordinates. The solution also implies the linearisation
of a nontriviallooking scalar field theory.
 ILMANEN, T, "ELLIPTIC REGULARIZATION AND PARTIAL REGULARITY FOR MOTION BY MEANCURVATURE," MEMOIRS OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 108, pp. R3&, 1994.
Abstract:
I. We study Brakke's motion of varifolds by mean curvature in
the special case that the initial surface is an integral cycle,
giving a new existence proof by mean of elliptic
regularization. Under a uniqueness hypothesis, we obtain a
weakly continuous family of currents solving Brakke's motion.
II. These currents remain within the corresponding levelset
motion by mean curvature, as defined by EvansSpruck and Chen
GigaGoto. Now let T0 be the reduced boundary of a bounded set
of finite perimeter in R(n). If the levelset motion of the
support of T0 does not develop positive Lebesgue measure, then
there corresponds a unique integral ncurrent T, partial
derivative = T0, whose timeslices form a unit density Brakke
motion. Using Brakke's Regularity Theorem, spt T is smooth
H(n)almost everywhere. In consequence, almost every levelset
of the levelset flow is smooth H(n)almost everywhere in
spacetime.
 FALCONE, M, GIORGI, T, and LORETI, P, "LEVEL SETS OF VISCOSITY SOLUTIONS  SOME APPLICATIONS TO FRONTS AND RENDEZVOUS PROBLEMS," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 54, pp. 13351354, 1994.
Abstract:
The authors treat some applications of HamiltonJacobi
equations to the study of a flame front propagation model and
the rendezvous problem. The solution of both problems requires
the determination of the level sets of the viscosity solution
for the corresponding equation. In the flame front propagation
model described here, it is assumed that the evolution is
driven by a vector field satisfying a transversality condition
at time t = 0. The evolution in the normal direction with
variable velocity c(x) greater than or equal to 0 is considered
as a special case. This approach is constructive, permitting
the numerical solution of such problems.
 MCELIGOT, J, and MCELIGOT, DM, "PERSPECTIVE  SOME RESEARCH NEEDS IN CONVECTIVE HEATTRANSFER FOR INDUSTRY," JOURNAL OF FLUIDS ENGINEERINGTRANSACTIONS OF THE ASME, vol. 116, pp. 398404, 1994.
Abstract:
We study the limiting behavior (the macroscopic limit) of an
appropriately scaled spin system with GlauberKawasaki
dynamics. We rigorously establish the existence in the limit
of an interface evolving according to motion by mean curvature.
This limit is valid for all positive times, past possible
geometric singularities of the motion, which is interpreted in
the viscosity sense.
 KATSOULAKIS, MA, and SOUGANIDIS, PE, "INTERACTING PARTICLESYSTEMS AND GENERALIZED EVOLUTION OF FRONTS," ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, vol. 127, pp. 133157, 1994.
Abstract:
We study the limiting behavior (the macroscopic limit) of an
appropriately scaled spin system with GlauberKawasaki
dynamics. We rigorously establish the existence in the limit
of an interface evolving according to motion by mean curvature.
This limit is valid for all positive times, past possible
geometric singularities of the motion, which is interpreted in
the viscosity sense.
 SETHIAN, JA, "CURVATURE FLOW AND ENTROPY CONDITIONS APPLIED TO GRID GENERATION," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 115, pp. 440454, 1994.
Abstract:
We describe a numerical technique to generate logically
rectangular bodyfitted interior and exterior grids. The
technique is based on solving a HamiltonJacobitype equation
for a propagating level set function, using techniques borrowed
from hyperbolic conservation laws. Coordinate grid lines are
kept smooth through curvature terms which regularize the
equation of motion, and upwind difference schemes which satisfy
the correct entropy conditions of front propagation. The
resulting algorithm can be used to generate two and three
dimensional interior and exterior grids around reasonably
complex bodies which may contain sharp corners and significant
variations in curvature. The technique may also be easily
extended to problems of boundaryfitted moving grids. (C) 1994
Academic Press, Inc.
 BREWSTER, ME, "STATIONARY SELFPROPAGATING FRONTS IN POTENTIAL FLOW," PHYSICA D, vol. 79, pp. 306319, 1994.
Abstract:
We analyze the problem of stationary selfpropagating fronts in
potential flow. The issues of local existence and uniqueness
for solutions of the ODE describing stationary fronts,
multiplicity of solutions and linearized stability of a
stationary front as a solution of the (hyperbolic) evolution
equation are addressed. The results are illustrated in the case
of the dualsource system, which is a simple model of a
combustion system in which local extinction may arise. Model
extensions for combustion applications are presented.


1995 
 Altschuler, S, Angenent, SB, and Giga, Y, "Mean curvature flow through singularities for surfaces of rotation," JOURNAL OF GEOMETRIC ANALYSIS, vol. 5, pp. 293358, 1995.
Abstract:
In this paper, we study generalized ''viscosity'' solutions of
the mean curvature evolution which were introduced by Chen,
Giga, and Goto and by Evans and Spruck. We devote much of our
attention to solutions whose initial value is a compact,
smooth, rotationally symmetric hypersurface given by rotating a
graph around an axis. Our main result is the regularity of the
solution except at isolated points in spacetime and estimates
on the number of such points.
 Bruckstein, AM, Sapiro, G, and Shaked, D, "Evolutions of planar polygons," INTERNATIONAL JOURNAL OF PATTERN RECOGNITION AND ARTIFICIAL INTELLIGENCE, vol. 9, pp. 9911014, 1995.
Abstract:
Evolutions of closed planar polygons are studied in this work.
In the first part of the paper, the general theory of linear
polygon evolutions is presented, and two specific problems are
analyzed. The first one is a polygonal analog of a novel
affineinvariant differential curve evolution, for which the
convergence of planar curves to ellipses was proved. In the
polygon case, convergence to polygonal approximation of
ellipses, polygonal ellipses, is proven. The second one is
related to cyclic pursuit problems, and convergence, either to
polygonal ellipses or to polygonal circles, is proven. In the
second part, two possible polygonal analogues of the wellknown
Euclidean curve shortening flow are presented. The models
follow from geometric considerations. Experimental results show
that an arbitrary initial polygon converges to either regular
or irregular polygonal approximations of circles when evolving
according to the proposed Euclidean flows.
 SAPIRO, G, and TANNENBAUM, A, "AREA AND LENGTH PRESERVING GEOMETRIC INVARIANT SCALESPACES," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 17, pp. 6772, 1995.
Abstract:
In this paper, area preserving multiscale representations of
planar curves are described. This allows smoothing without
shrinkage at the same time preserving all the scalespace
properties. The representations are obtained deforming the
curve via geometric heat flows while simultaneously magnifying
the plane by a homethety which keeps the enclosed area
constant. When the Euclidean geometric heat now is used, the
resulting representation is Euclidean invariant, and similarly
it is affine invariant when the affine one is used. The flows
are geometrically intrinsic to the curve, and exactly satisfy
all the basic requirements of scalespace representations. In
the case of the Euclidean heat flow, it is completely local as
well. The same approach is used to define length preserving
geometric flows. A similarity (scale) invariant geometric heat
flow is studied as well in this work.
 MALLADI, R, SETHIAN, JA, and VEMURI, BC, "SHAPE MODELING WITH FRONT PROPAGATION  A LEVEL SET APPROACH," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 17, pp. 158175, 1995.
Abstract:
Shape modeling is an important constituent of computer vision
as well as computer graphics research. Shape models aid the
tasks of object representation and recognition. This paper
presents a new approach to shape modeling which retains some of
the attractive features of existing methods and overcomes some
of their limitations. Our techniques can be applied to model
arbitrarily complex shapes, which include shapes with
significant protrusions, and to situations where no a priori
assumption about the object's topology is made. A single
instance of our model, when presented with an image having more
than one object of interest, has the ability to split freely to
represent each object. This method is based on the ideas
developed by Osher and Sethian to model propagating
solid/liquid interfaces with curvature dependent speeds. The
interface (front) is a closed, nonintersecting, hypersurface
flowing along its gradient field with constant speed or a speed
that depends on the curvature, It is moved by solving a
''HamiltonJacobi'' type equation written for a function in
which the interface is a particular level set. A speed term
synthesized from the image is used to stop the interface in the
vicinityof object boundaries. The resulting equation of motion
is solved by employing entropysatisfying upwind finite
difference schemes. We present a variety of ways of computing
evolving front, including narrow bands, reinitializations, and
different stopping criteria. The efficacy of the scheme is
demonstrated with numerical experiments on some synthesized
images and some low contrast medical images.
 LIONS, PL, and SOUGANIDIS, PE, "CONVERGENCE OF MUSCL AND FILTERED SCHEMES FOR SCALAR CONSERVATIONLAWS AND HAMILTONJACOBI EQUATIONS," NUMERISCHE MATHEMATIK, vol. 69, pp. 441470, 1995.
Abstract:
This paper considers the questions of convergence of: (i) MUSCL
type (i.e. secondorder, TVD) finitedifference approximations
towards the entropic weak solution of scalar, onedimensional
conservation laws with strictly convex flux and (ii) higher
order schemes (filtered to ''preserve'' an upperbound on some
weak secondorder finite differences) towards the viscosity
solution of scalar, multidimensional HamiltonJacobi equations
with convex Hamiltonians.
 NAKAYAMA, K, HOPPE, J, and WADATI, M, "ON THE LEVELSET FORMULATION OF GEOMETRICAL MODELS," JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN, vol. 64, pp. 403407, 1995.
Abstract:
Levelset approach to the motion of surfaces is presented.
Applications to geometrical models in condensed matter physics
are given. The finger solution and its generalizations, which
were reported very recently, are derived in a simple way.
 KIMMEL, R, and SAPIRO, G, "SHORTENING 3DIMENSIONAL CURVES VIA 2DIMENSIONAL FLOWS," COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 29, pp. 4962, 1995.
Abstract:
In this paper, a curve evolution approach for the computation
of geodesic curves on 3D surfaces is presented. The algorithm
is based on deforming, via the curve shortening flow, an
arbitrary initial curve ending at two given surface points.
The 3D curve shortening flow is first transformed into an
equivalent 2D one. This 2D flow is implemented, using an
efficient numerical algorithm for curve evolution with fixed
end points.
 BARLES, G, and GEORGELIN, C, "A SIMPLE PROOF OF CONVERGENCE FOR AN APPROXIMATION SCHEME FOR COMPUTING MOTIONS BY MEANCURVATURE," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 32, pp. 484500, 1995.
Abstract:
We prove the convergence of an approximation scheme recently
proposed by Bence, Merriman, and Osher for computing motions of
hypersurfaces by mean curvature. Our proof is based on
viscosity solutions methods.
 CORRIAS, L, FALCONE, M, and NATALINI, R, "NUMERICAL SCHEMES FOR CONSERVATION LAWS VIA HAMILTONJACOBI EQUATIONS," MATHEMATICS OF COMPUTATION, vol. 64, pp. 555580, 1995.
Abstract:
We present some difference approximation schemes which converge
to the entropy solution of a scalar conservation law having a
convex flux. The numerical methods described here take their
origin from approximation schemes for HamiltonJacobiBellman
equations related to optimal control problems and exhibit
several interesting features: the convergence result still
holds for quite arbitrary time steps, the main assumption for
convergence can be interpreted as a discrete analogue of
Oleinik's entropy condition, numerical diffusion around the
shocks is very limited. Some tests are included in order to
compare the performances of these methods with other classical
methods (Godunov, TVD).
 EVANS, LC, and SPRUCK, J, "MOTION OF LEVEL SETS BY MEANCURVATURE .4.," JOURNAL OF GEOMETRIC ANALYSIS, vol. 5, pp. 77114, 1995.
Abstract:
We continue Our investigation of the ''levelset'' technique
for describing the generalized evolution of hypersurfaces
moving according to their mean curvature. The principal
assertion of this paper is a kind of reconciliation with the
geometric measure theoretic approach pioneered by K. Brakke: we
prove that almost every level set of the solution to the mean
curvature evolution PDE is in fact a unitdensity varifold
moving according to its mean curvature. In particular, a.e.
level set is endowed with a kind of ''geometric structure.''
The proof utilizes compensated compactness methods to pass to
limits in various geometric expressions.
 SAPIRO, G, and BRUCKSTEIN, AM, "THE UBIQUITOUS ELLIPSE," ACTA APPLICANDAE MATHEMATICAE, vol. 38, pp. 149161, 1995.
Abstract:
We discuss three different affine invariant evolution processes
for smoothing planar curves. The first one is derived from a
geometric heattype flow, both the initial and the smoothed
curves being differentiable. The second smoothing process is
obtained from a discretization of this affine heat equation. In
this case, the curves are represented by planar polygons. The
third process is based on Bspline approximations. For this
process, the initial curve is a planar polygon, and the
smoothed curves are differentiable and even analytic. We show
that, in the limit, all three affine invariant smoothing
processes collapse any initial curve into an elliptic point.
 ADALSTEINSSON, D, and SETHIAN, JA, "A FAST LEVEL SET METHOD FOR PROPAGATING INTERFACES," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 118, pp. 269277, 1995.
Abstract:
A method is introduced to decrease the computational labor of
the standard level set method for propagating interfaces. The
fast approach uses only points close to the curve at every time
step. We describe this new algorithm and compare its efficiency
and accuracy with the standard level set approach. (c) 1995
Academic Press, Inc.
 KIMMEL, R, AMIR, A, and BRUCKSTEIN, AM, "FINDING SHORTEST PATHS ON SURFACES USING LEVEL SETS PROPAGATION," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 17, pp. 635640, 1995.
Abstract:
We present a nerv algorithm for determining minimal length
paths between two regions on a three dimensional surface, The
numerical implementation is based on finding equal geodesic
distance contours from a given area, These contours are
calculated as zero sets of a bivariate function designed to
evolve so as to track the equal distance curves on the given
surface, The algorithm produces all paths of minimal length
between the source and destination areas on the surface given
as height values on a rectangular grid.
 GURTIN, ME, SONER, HM, and SOUGANIDIS, PE, "ANISOTROPIC MOTION OF AN INTERFACE RELAXED BY THE FORMATION OF INFINITESIMAL WRINKLES," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 119, pp. 54108, 1995.
Abstract:
A new algorithm for recovering depth to a Lambertian C1 smooth
object given its graylevel image under uniform illumination
from the viewing direction is presented. To recover depth, an
almost arbitrarily initialized surface is numerically
propagated on a rectangular grid, so that a level set of this
surface tracks the height contours of the depth function. The
image shading controls the propagation of the surface. When the
light direction is tilted with respect to the viewing direction
the problem is solved by tracking the projection of equal
height contours defined with respect to the light source
direction. This projection approach provides a solution that
overcomes ambiguity problems encountered in previous work,
while the level set approach of implementing the contour
propagation overcomes numerical problems and some of the
topology problems of the evolving contours. (C) 1995 Academic
Press, Inc.
 KIMMEL, R, and BRUCKSTEIN, AM, "TRACKING LEVEL SETS BY LEVEL SETS  A METHOD FOR SOLVING THE SHAPE FROM SHADING PROBLEM," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 62, pp. 4758, 1995.
Abstract:
A new algorithm for recovering depth to a Lambertian C1 smooth
object given its graylevel image under uniform illumination
from the viewing direction is presented. To recover depth, an
almost arbitrarily initialized surface is numerically
propagated on a rectangular grid, so that a level set of this
surface tracks the height contours of the depth function. The
image shading controls the propagation of the surface. When the
light direction is tilted with respect to the viewing direction
the problem is solved by tracking the projection of equal
height contours defined with respect to the light source
direction. This projection approach provides a solution that
overcomes ambiguity problems encountered in previous work,
while the level set approach of implementing the contour
propagation overcomes numerical problems and some of the
topology problems of the evolving contours. (C) 1995 Academic
Press, Inc.
 MALLADI, R, and SETHIAN, JA, "IMAGEPROCESSING VIA LEVEL SET CURVATURE FLOW," PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, vol. 92, pp. 70467050, 1995.
Abstract:
We present a controlled image smoothing and enhancement method
based on a curvature flow interpretation of the geometric heat
equation. Compared to existing techniques, the model has
several distinct advantages. (i) It contains just one
enhancement parameter. (ii) The scheme naturally inherits a
stopping criterion from the image; continued application of the
scheme produces no further change. (iii) The method is one of
the fastest possible schemes based on a curvaturecontrolled
approach.
 KATSOULAKIS, M, KOSSIORIS, GT, and REITICH, F, "GENERALIZED MOTION BY MEANCURVATURE WITH NEUMANN CONDITIONS AND THE ALLENCAHN MODEL FOR PHASETRANSITIONS," JOURNAL OF GEOMETRIC ANALYSIS, vol. 5, pp. 255279, 1995.
Abstract:
We study a sharpinterface model for phase transitions which
incorporates the interaction of tile phase boundaries with the
walls of a container Omega. In this model, the interfaces move
by their mean curvature and are normal to partial derivative
Omega. We first establish localintime existence and
uniqueness of smooth solutions for the mean curvature equation
with a normal contact angle condition. We then discuss global
solutions by interpreting the equation and the boundary
condition in a weak (viscosity) sense. Finally, we investigate
the relation of the aforementioned model with a transition
layer model. We prove that if Omega is convex, the transition
layer solutions converge to the sharpinterface solutions as
the thickness of the layer tends to zero. We conclude with a
discussion of the difficulties that arise in establishing this
result in nonconvex domains.
 ADALSTEINSSON, D, and SETHIAN, JA, "A LEVEL SET APPROACH TO A UNIFIED MODEL FOR ETCHING, DEPOSITION, AND LITHOGRAPHY .1. ALGORITHMS AND 2DIMENSIONAL SIMULATIONS," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 120, pp. 128144, 1995.
Abstract:
We apply a level set formulation to the problem of surface
advancement in a twodimensional topography simulation of
deposition, etching, and lithography processes in integrated
circuit fabrication. The level set formulation is based on
solving a HamiltonJacobi type equation for a propagating level
set function, using techniques borrowed from hyperbolic
conservation laws. Topological changes, corner a nd cusp
development, a nd accurate determination of geometric
properties such as curvature and normal direction are naturally
obtained in this setting. The equations of motion of a unified
model, including the effects of isotropic and unidirectional
deposition and etching, visibility, surface diffusion,
reflection, and material dependent etch/deposition rates are
presented and adapted to a level set formulation. The
development of this model and algorithm naturally extends to
three dimensions in a straightforward manner and is described
in part II of this paper (in press). (C) 1995 Academic Press,
Inc.
 KIMIA, BB, TANNENBAUM, AR, and ZUCKER, SW, "SHAPES, SHOCKS, AND DEFORMATIONS .1. THE COMPONENTS OF 2 DIMENSIONAL SHAPE AND THE REACTIONDIFFUSION SPACE," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 15, pp. 189224, 1995.
Abstract:
We undertake to develop a general theory of twodimensional
shape by elucidating several principles which any such theory
should meet. The principles are organized around two basic
intuitions: first, if a boundary were changed only slightly,
then, in general, its shape would change only slightly. This
leads us to propose an operational theory of shape based on
incremental contour deformations. The second intuition is that
not all contours are shapes, but rather only those that can
enclose ''physical'' material. A theory of contour deformation
is derived from these principles, based on abstract
conservation principles and HamiltonJacobi theory. These
principles are based on the work of Sethian (1985a, c), the
OsherSethian (1988), level set formulation the classical shock
theory of Lax (1971; 1973), as well as curve evolution theory
for a curve evolving as a function of the curvature and the
relation to geometric smoothing of GageHamiltonGrayson (1986;
1989). The result is a characterization of the computational
elements of shape: deformations, parts, bends, and seeds, which
show where to place the components of a shape. The theory
unifies many of the diverse aspects of shapes, and leads to a
space of shapes (the reaction/diffusion space), which places
shapes within a neighborhood of ''similar'' ones. Such
similarity relationships underlie descriptions suitable for
recognition.
 COLLINS, LR, "SPECTRALANALYSIS OF A SIMULATED PREMIXED FLAME SURFACE IN 2 DIMENSIONS," COMPUTERS & FLUIDS, vol. 24, pp. 663683, 1995.
Abstract:
This paper presents twodimensional direct numerical
simulations of a passive flame surface passing through
homogeneous isotropic turbulence. The flame was represented by
a field variable, G(x, t), whose isocontours constitute flame
surfaces. One well known complication in analyzing premixed
combustion in a homogeneous environment is decoupling the
effect of the decaying turbulent velocity field from the
dynamics of the flame surface. To overcome this, the velocity
field was made stationary by introducing a random forcing term
into the Navier Stokes equations. Forcing was done over two
different ranges of wavenumbers (k(f) = 1014, and k(f) = 80
84) thus creating turbulence with different length scales and
inertial range power laws. By comparing the response of the
flame to the two types of turbulence it was possible to
determine the effect the spectral distribution energy has on
the surface topology and mean rate of propagation. Indeed, the
flames were found to be remarkably sensitive to the spectral
distribution of the turbulent energy, and not just its
magnitude. Furthermore, a k(5/3) inertial range was shown to
produce a flame surface that was preferentially wrinkled at
intermediate to small scales for purely geometric reasons. By
defining a surface area spectrum it was possible to rationalize
this result by recognizing that flame surface area is closely
related to the dissipation spectrum of the scalar field.
Collectively the results suggest that knowledge of the energy
spectrum al a minimum is required to predict a turbulent flame
speed under general circumstances.
 RICHARDS, JR, BARIS, AN, and LENHOFF, AM, "DROP FORMATION IN LIQUIDLIQUID SYSTEMS BEFORE AND AFTER JETTING," PHYSICS OF FLUIDS, vol. 7, pp. 26172630, 1995.
Abstract:
The formation of drops resulting from the breakup of an
axisymmetric Newtonian liquid jet injected vertically into
another immiscible Newtonian liquid at various Reynolds numbers
is investigated here. The full transient from startup to
breakup into drops was simulated numerically by solving the
timedependent axisymmetric equations of motion and continuity
using a combination of the volumeoffluid (VOF) and
continuoussurfaceforce (CSF) methods. The numerical
simulation results compare well with previous experimental data
and are significantly more accurate than previous simplified
analyses based on drop formation before and after jetting over
a wide range of conditions. (C) 1995 American Institute of
Physics.
 KIMMEL, R, SIDDIQI, K, KIMIA, BB, and BRUCKSTEIN, AM, "SHAPE FROM SHADING  LEVEL SET PROPAGATION AND VISCOSITY SOLUTIONS," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 16, pp. 107133, 1995.
Abstract:
We present a new implementation of an algorithm aimed at
recovering a 3D shape from its 2D graylevel picture. In order
to reconstruct the shape of the object, an almost arbitrarily
initialized 3D function is propagated on a rectangular grid, so
that a level set of this function tracks the height contours of
the shape. The method imports techniques from differential
geometry, fluid dynamics, and numerical analysis and provides
an accurate shape from shading algorithm. The method solves
some topological problems and gracefully handles cases of non
smooth surfaces that give rise to shocks in the propagating
contours. Real and synthetic images of 3D profiles were
submitted to the algorithm and the reconstructed surfaces are
presented, demonstrating the effectiveness of the proposed
method.
 RHEE, CW, TALBOT, L, and SETHIAN, JA, "DYNAMICAL BEHAVIOR OF A PREMIXED TURBULENT OPEN VFLAME," JOURNAL OF FLUID MECHANICS, vol. 300, pp. 87115, 1995.
Abstract:
The levelset approach of Osher & Sethian to tracking
interfaces is successfully adapted to the simulation of a
premixed turbulent open Vflame including the effects of
exothermicity and baroclinicity. In accord with experimental
observations this algorithm, along with a flame anchoring
scheme, predicts flame cusping for a case in which a strong
vortex pair interacts with the flame front. The computed
velocity and scalar statistics obtained for the turbulent V
flame compare reasonably well with experimental results by
Cheng & Shepherd, and demonstrate the importance of flame
generated vorticity in the determination of flame dynamics and
product velocity characteristics.
 KIMMEL, R, SHAKED, D, KIRYATI, N, and BRUCKSTEIN, AM, "SKELETONIZATION VIA DISTANCE MAPS AND LEVEL SETS," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 62, pp. 382391, 1995.
Abstract:
The medial axis transform (MAT) of a shape, better known as its
skeleton, is frequently used in shape analysis and related
areas. In this paper a new approach for determining the
skeleton of an object is presented. The boundary is segmented
at points of maximal positive curvature and a distance map from
each of the segments is calculated. The skeleton is then
located by applying simple rules to the zero sets of distance
map differences. A framework is proposed for numerical
approximation of distance maps that is consistent with the
continuous case and hence does nor suffer from digitization
bias due to metrication errors of the implementation on the
grid. Subpixel accuracy in distance map calculation is obtained
by using graylevel information along the boundary of the shape
in the numerical scheme. The accuracy of the resulting
efficient skeletonization algorithm is demonstrated by several
examples. (C) 1995 Academic Press, Inc.
 KIMMEL, R, and BRUCKSTEIN, AM, "GLOBAL SHAPE FROM SHADING," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 62, pp. 360369, 1995.
Abstract:
A new approach for the reconstruction of a smooth three
dimensional object from its twodimensional graylevel image is
presented. An algorithm based on topological properties of
simple smooth surfaces is provided to solve the problem of
global reconstruction. Classifying singular points in the
shading image as maxima, minima, and two kinds of saddle points
serves as the key to the solution of the problem. The global
reconstruction procedure, being deterministic and using
topological properties of the surface, performs better than
other approaches proposed so far that are based on
classification of singular points according to the behavior of
characteristics in their neighborhood. The proposed algorithm
is simple and easy to implement and lends itself to a parallel
implementation. (C) 1995 Academic Press, Inc.
 PAUWELS, EJ, FIDDELAERS, P, and VANGOOL, LJ, "ENHANCEMENT OF PLANAR SHAPE THROUGH OPTIMIZATION OF FUNCTIONALS FOR CURVES," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 17, pp. 11011105, 1995.
Abstract:
We show how optimization of the Nordstrom and MumfordShah
functionals can be used to develop a type of curveevolution
that is able to preserve salient features of closed curves
while simultaneously suppressing noise and irrelevant details.
The idea is to characterize a curve by means of its angle
function and apply the appropriate dynamics to this
representation. Upon convergence, the resulting form of the
contour is reconstructed from the representation.
 CATTE, F, DIBOS, F, and KOEPFLER, G, "A MORPHOLOGICAL SCHEME FOR MEANCURVATURE MOTION AND APPLICATIONS TO ANISOTROPIC DIFFUSION AND MOTION OF LEVEL SETS," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 32, pp. 18951909, 1995.
Abstract:
This paper introduces a discrete scheme for mean curvature
motion using a morphological image processing approach. An
axiomatic approach of image processing and the mean curvature
partial differential equation (PDE) are briefly presented, then
the properties of the proposed scheme are studied. In
particular, consistency and convergence are proved. The
applications of mean curvature motion in image denoising and
form evolution are developed and experiences are presented.
 ADALSTEINSSON, D, and SETHIAN, JA, "A LEVEL SET APPROACH TO A UNIFIED MODEL FOR ETCHING, DEPOSITION, AND LITHOGRAPHY .2. 3DIMENSIONAL SIMULATIONS," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 122, pp. 348366, 1995.
Abstract:
We apply a level set formulation to the problem of surface
advancement in threedimensional topography simulation of
deposition, etching, and lithography processes in integrated
circuit fabrication. The level set formulation is based on
solving a HamiltonJacobitype equation for a propagating level
set function, using techniques borrowed from hyperbolic
conservation laws. Topological changes, corner and cusp
development, and accurate determination of geometric properties
such as curvature and normal direction are naturally obtained
in this setting. The equations of motion of a unified model,
including the effects of isotropic and unidirectional
deposition and etching, visibility, surface diffusion,
reflection, and material dependent etch/deposition rates are
presented and adapted to a level set formulation. In Part I of
this paper, the basic equations and algorithms for two
dimensional simulations were developed. In this paper, the
extension to three dimensions is presented. We show a large
collection of simulations, including threedimensional etching
and deposition into cavities under the effects of visibility,
directional and source flux functions, evolution of
lithographic profiles, discontinuous etch rates through
multiple materials, and nonconvex sputter yield flux
functions. In Part III of this paper, effects of reflection and
reemission and surface diffusion Will be presented. (C) 1995
Academic Press, Inc.
 ANGENENT, S, ILMANEN, T, and CHOPP, DL, "A COMPUTED EXAMPLE OF NONUNIQUENESS OF MEANCURVATURE FLOW IN R(3)," COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, vol. 20, pp. 19371958, 1995.
Abstract:
In this paper, we study generalized ''viscosity'' solutions of
the mean curvature evolution which were introduced by Chen,
Giga, and Goto and by Evans and Spruck. We devote much of our
attention to solutions whose initial value is a compact,
smooth, rotationally symmetric hypersurface given by rotating a
graph around an axis. Our main result is the regularity of the
solution except at isolated points in spacetime and estimates
on the number of such points.


1996 
 Calabi, E, Olver, PJ, and Tannenbaum, A, "Affine geometry, curve flows, and invariant numerical approximations," ADVANCES IN MATHEMATICS, vol. 124, pp. 154196, 1996.
Abstract:
A new geometric approach to the affine geometry of curves in
the plane and to affineinvariant curve shortening is
presented. We describe methods of approximating the affine
curvature with discrete finite difference approximations, based
on a general theory of approximating differential invariants of
Lie group actions by joint invariants. Applications to computer
vision are indicated. (C) 1996 Academic Press, Inc.
 Baillot, F, Bourehla, A, and Durox, D, "The characteristics method and cusped flame fronts," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 112, pp. 327350, 1996.
Abstract:
The kinematic effects of a spacetime forced velocity held upon
a thin premixed flame, stabilized above a circular cross
section burner, are studied in order to point out the non
linearities due to a sufficiently high velocity perturbation
level whose RMS amplitudes remain nonetheless inferior to the
normal burning velocity. The present calculation proposes to
seek a solution using the characteristics method, without any
linearized calculation, to express these effects. A front
evolution equation is interpreted as the differentiated form of
a conservation equation of the radial distance between two
points of the front. These modelling results are used to
interpret experiments of a vibrating flame subjected to a
spacetime sinusoidal velocity held. In this last case, the
limit of cusps formation is represented as a similarity law
expressing the nondimensional perturbation amplitude versus a
Strouhal number of the aeroacoustic reactive medium.
 LeVeque, RJ, and Shyue, KM, "Twodimensional front tracking based on high resolution wave propagation methods," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 123, pp. 354368, 1996.
Abstract:
We present a fully conservative, high resolution approach to
front tracking for nonlinear systems of conservation laws in
two space dimensions. An underlying uniform Cartesian grid is
used, with some cells cut by the front into two subcells. The
front is moved by solving a Riemann problem normal to each
segment of the front and using the motion of the strongest wave
to give an approximate location of the front at the end of the
time step. A high resolution finite volume method is then
applied on the resulting slightly irregular grid to update all
cell values. A ''large time step'' wave propagation algorithm
is used that remains stable in the small cut cells with a time
step that is chosen with respect to the uniform grid cells.
Numerical results on a radially symmetric problem show that
pointwise convergence with order between 1 and 2 is obtained in
both the cell values and location of the front. Other
computations are also presented. (C) 1996 Academic Press, Inc.
 Nochetto, RH, Paolini, M, and Verdi, C, "A dynamic mesh algorithm for curvature dependent evolving interfaces," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 123, pp. 296310, 1996.
Abstract:
A new finite element method is discussed for approximating
evolving interfaces in R(n) whose normal velocity equals mean
curvature plus a forcing function. The method is insensitive to
singularity formation and retains the local structure of the
limit problem and, thus, exhibits a computational complexity
typical of R(n1) without having the drawbacks of front
tracking strategies. A graded dynamic mesh around the
propagating front is the sole partition present at any time
step and is significantly smaller than a full mesh. Time
stepping is explicit, but stability constraints force small
time steps only when singularities develop, whereas relatively
large time steps are allowed before or past singularities, when
the evolution is smooth. The explicit marching scheme also
guarantees that at most one layer of elements has to be added
or deleted per time step, thereby making mesh updating simple
and, thus, practical. Performance and potentials are fully
documented via a number of numerical simulations in 2D, 3D, 4D,
and 8D, with axial symmetries. They include tori and cones for
the mean curvature flow, minimal and prescribed mean curvature
surfaces with given boundary, fattening for smooth driving
force, and volume constraint. (C) 1996 Academic Press, Inc.
 Lafon, F, and Osher, S, "High order two dimensional nonoscillatory methods for solving HamiltonJacobi scalar equations," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 123, pp. 235253, 1996.
Abstract:
For the computation of nonlinear solutions of HamiltonJacobi
scalar equations in two space dimensions, we develop high order
accurate numerical schemes that can be applied to complicated
geometries. Previously, the recently developed essentially
nonoscillatory (ENO) technology has been applied in simple
domains like squares or rectangles using dimensionbydimension
algorithms. On arbitrary two dimensional closed or multiply
connected domains, first order monotone methods were used. In
this paper, we propose two different techniques to construct
high order accurate methods using the ENO philosophy. Namely,
any arbitrary domain is triangulated by finite elements into
which two dimensional ENO polynomials are constructed. These
polynomials are then differentiated to compute a high order
accurate numerical solution. These new techniques are shown to
be very useful in the computation of numerical solutions of
various applications without significantly increasing CPU
running times as compared to dimensionbydimension algorithms.
Furthermore, these methods are stable and no spurious
oscillations are detected near singular points or curves. (C)
1996 Academic Press, Inc.
 Mulholland, AJ, and Gomatam, J, "The eikonal approximation to excitable reactiondiffusion systems: Travelling nonplanar wave fronts on the plane," PHYSICA D, vol. 89, pp. 329345, 1996.
Abstract:
Exact, nonplanar travelling solutions of the eikonal equation
on an infinite plane are presented for the first time. These
solutions are matched to produce corrugated wave fronts and
patterns such as 'spot' solutions as well as extended parabolic
type wave fronts. The stability of these solutions is also
analysed. The variational equation which belongs to a
generalised Wangerin class of differential equations is solved,
first with the aid of the LiouvilleGreen approximation for the
estimated eigenvalues characterising stability and then by a
more elaborate shootingmatching method. All of the three types
of travelling solutions are found to be geometrically stable.
It is suggested that some of these predictions are
experimentally testable.
 Sethian, JA, "A fast marching level set method for monotonically advancing fronts," PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, vol. 93, pp. 15911595, 1996.
Abstract:
A fast marching level set method is presented for monotonically
advancing fronts, which leads to an extremely fast scheme for
solving the Eikonal equation. Level set methods are numerical
techniques for computing the position of propagating fronts.
They rely on an initial value partial differential equation for
a propagating level set function and use techniques borrowed
from hyperbolic conservation laws. Topological changes, corner
and cusp development, and accurate determination of geometric
properties such as curvature and normal direction are naturally
obtained in this setting. This paper describes a particular
case of such methods for interfaces whose speed depends only on
local position. The technique works by coupling work on entropy
conditions for interface motion, the theory of viscosity
solutions for HamiltonJacobi equations, and fast adaptive
narrow band level set methods. The technique is applicable to a
variety of problems, including shapefromshading problems,
lithographic development calculations in microchip
manufacturing, and arrival time problems in control theory.
 Pnueli, Y, and Bruckstein, AM, "Gridless halftoning: A reincarnation of the old method," GRAPHICAL MODELS AND IMAGE PROCESSING, vol. 58, pp. 3864, 1996.
Abstract:
Despite continuing research and steady progress in the field of
digital halftones, there is a feeling that a wide gap in
quality still remains between the best available and the best
achievable results. A glance at manmade halftones readily
confirms this feeling. To bridge this gap, we propose the use
of computergenerated, yet gridless halftones. This involves
solving the halftoning problem on the continuous 2D plane
rather than on the usual discrete grid of pixels. In this
article we outline this new approach, describe its expected
advantages over existing techniques and demonstrate some of
them via a prototype system, DigiDurer, developed for this
purpose. (C) 1996 Academic Press, Inc.
 Yao, J, and Stewart, DS, "On the dynamics of multidimensional detonation," JOURNAL OF FLUID MECHANICS, vol. 309, pp. 225275, 1996.
Abstract:
We present an asymptotic theory for the dynamics of detonation
when the radius of curvature of the detonation shock is large
compared to the onedimensional, steady, ChapmanJouguet (CJ)
detonation reactionzone thickness. The analysis considers
additional timedependence in the slowly varying reaction zone
to that considered in previous works. The detonation is assumed
to have a sonic point in the reactionzone structure behind the
shock, and is referred to as an eigenvalue detonation. A new,
iterative method is used to calculate the eigenvalue relation,
which ultimately is expressed as an intrinsic, partial
differential equation (PDE) for the motion of the shock
surface. Two cases are considered for an ideal equation of
state. The first corresponds to a model of a condensedphase
explosive, with modest reaction rate sensitivity, and the
intrinsic shock surface PDE is a relation between the normal
detonation shock velocity, Dn, the first normal time
derivative of the normal shock velocity, Dn, and the shock
curvature, kappa. The second case corresponds to a gaseous
explosive mixture, with the large reaction rate sensitivity of
Arrhenius kinetics, and the intrinsic shock surface PDE is a
relation between the normal detonation shock velocity, Dn, its
first and second normal time derivatives of the normal shock
velocity, Dn, Dn, and the shock curvature, kappa, and its
first normal time derivative of the curvature, kappa. For the
second case, one obtains a onedimensional theory of pulsations
of plane CJ detonation and a theory that predicts the evolution
of selfsustained cellular detonation. Versions of the theory
include the limits of nearCJ detonation, and when the normal
detonation velocity is significantly below its CJ value. The
curvature of the detonation can also be of either sign,
corresponding to both diverging and converging geometries.
 Helenbrook, BT, Sung, CJ, Law, CK, and Ashurst, WT, "On stretchaffected flame propagation in vortical flows," COMBUSTION AND FLAME, vol. 104, pp. 460468, 1996.
Abstract:
Flame propagation through an array of vortices was studied with
a model which incorporated the variation of the local burning
velocity with stretch. Assuming incompressible how, the mean
burning velocities were calculated and compared to those of the
Huygens limit. It was found that stretch causes a decrease in
the mean burning velocity, and a mechanism which explains this
trend was identified. The study also demonstrated that, as
expected, stretch only has a significant effect on the mean
burning velocity for vortices whose size is of the same order
as that of the flame thickness.
 Chang, YC, Hou, TY, Merriman, B, and Osher, S, "A level set formulation of eulerian interface capturing methods for incompressible fluid flows," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 124, pp. 449464, 1996.
Abstract:
A level set formulation is derived for incompressible,
immiscible NavierStokes equations separated by a free surface.
The interface is identified as the zero level set of a smooth
function. Eulerian finite difference methods based on this
level set formulation are proposed. These methods are robust
and efficient and are capable of computing interface
singularities such as merging and reconnection. Numerical
experiments are presented to demonstrate the effectiveness of
the methods. (C) 1996 Academic Press, Inc.
 Sackinger, PA, Schunk, PR, and Rao, RR, "A newtonraphson pseudosolid domain mapping technique for free and moving boundary problems: A finite element implementation," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 125, pp. 83103, 1996.
Abstract:
An implicit, pseudosolid domain mapping technique is described
that facilitates finite element analysis of free and moving
boundary problems. The technique is based on an implicit, full
Newton strategy,free of restrictions on mesh structure; this
leads to many advantages over existing domain mapping
techniques. The fully coupled approach using Newton's method is
particularly effective for problems with strong coupling
between the internal bulk physics and the governing physics at
unknown free boundary locations. It is also useful when the
distinguishing conditions which constrain the free boundary
shape provide only an implicit dependence on the boundary
location. Unstructured meshes allow for efficient resolution of
internal and boundary layers and other regions of strong local
variations in the solution and they also reduce the amount of
user interaction required to define a problem since the meshes
may be generated automatically. The technique is readily
applied to steady or transient problems in complex geometries
of two and three dimensions. Examples are shown that include
free and moving boundary problems from solidification and
capillary hydrodynamics. (C) 1996 Academic Press, Inc.
 Malladi, R, and Sethian, JA, "Image processing: Flows under min/max curvature and mean curvature," GRAPHICAL MODELS AND IMAGE PROCESSING, vol. 58, pp. 127141, 1996.
Abstract:
We present a class of PDEbased algorithms suitable for image
denoising and enhancement. The techniques are applicable to
both saltandpepper grayscale noise and fullimage continuous
noise present in black and white images, grayscale images,
texture images, and color images. At the core, the techniques
rely on two fundamental ideas. First, a level set formulation
is used for evolving curves; use of this technique to flow
isointensity contours under curvature is known to remove noise
and enhance images. Second, the particular form of the
curvature how is governed by a minimax switch which selects a
range of denoising dependent on the size of switching window.
Our approach has several virtues. First, it contains only one
enhancement parameter, which in most cases is automatically
chosen. Second, the scheme automatically stops smoothing at a
point which depends on the switching window size; continued
application of the scheme produces no further change. Third,
the method is one of the fastest possible schemes based on a
curvaturecontrolled approach. (C) 1996 Academic Press, Inc.
 Kumar, A, Tannenbaum, AR, and Balas, GJ, "Optical flow: A curve evolution approach," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 5, pp. 598610, 1996.
Abstract:
A novel approach for the computation of optical how based on an
L(1) type minimization is presented, It is shown that the
approach has inherent advantages since it does not smooth the
flowvelocity across the edges and hence preserves edge
information, A numerical approach based on computation of
evolving curves is proposed for computing the optical flow
field. Computations are carried out on a number of real image
sequences in order to illustrate the theory as well as the
numerical approach.
 Maragos, P, "Differential morphology and image processing," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 5, pp. 922937, 1996.
Abstract:
Image processing via mathematical morphology has traditionally
used geometry to intuitively understand morphological signal
operators and set or lattice algebra to analyze them in the
space domain, In this paper, we provide a unified view and
analytic tools for a recently growing part of morphological
image processing that is based on ideas from differential
calculus and dynamical systems, This part includes both recent
and some earlier ideas on using partial differential or
difference equations (PDEs) to model distance propagation or
nonlinear multiscale processes in images. We briefly review
some nonlinear difference equations that implement discrete
distance transforms and relate them to numerical solutions of
the eikonal equation of optics. We also review some nonlinear
PDEs that model the evolution of multiscale morphological
operators and use morphological derivatives. Among the new
ideas presented, we develop some general 2D max/minsum
difference equations that model the space dynamics of 2D
morphological systems (including the distance computations) and
some nonlinear signal transforms, called slope transforms, that
can analyze these systems in a transform domain in ways
conceptually similar to the application of Fourier transforms
to linear systems. Thus, distance transforms are shown to be
bandpass slope filters, We view the analysis of the multiscale
morphological PDEs and of the eikonal PDE solved via weighted
distance tranforms as a unified area in nonlinear image
processing, which we call differential morphology, and briefly
discuss its potential applications to image processing and
computer vision.
 Aldredge, RC, "Premixed flame propagation in a highintensity, largescale vortical flow," COMBUSTION AND FLAME, vol. 106, pp. 2940, 1996.
Abstract:
The propagation of a premixed flame through a largescale
vortical how field is studied numerically by solving a front
propagation equation governing the evolution of a scalar field
whose zerolevel surface defines the location of a self
propagating interface. The flame front is considered to
propagate normal to itself with a constant speed, and the
density variation across the flame is considered to be zero.
Average burning rates are calculated for large velocity
fluctuation intensities, as the formulation allows naturally
for the formation of pockets of unburned gas downstream from
the reaction front. The propagation rate of the corrugated
flame front is found to vary periodically with time, with an
average that varies linearly with velocity fluctuation
intensity at large intensities. A mechanism is identified for
the decreasing sensitivity of the average burning rate to
increases in the fluctuation intensity, over an intermediate
range of intensities, in the zeroviscosity limit.
 Evans, LC, "A geometric interpretation of the heat equation with multivalued initial data," SIAM JOURNAL ON MATHEMATICAL ANALYSIS, vol. 27, pp. 932958, 1996.
Abstract:
We utilize the levelset method to interpret geometrically what
it means to solve the heat equation with multivalued initial
data. We prove that in one space dimension, the limits of
''geometrically natural'' approximations instantly unfold
multivalued initial data, according to an equalarea rule. In.
higher dimensions, the limits of certain ''analytically
natural'' approximations display similar effects.
 Aslam, TD, Bdzil, JB, and Stewart, DS, "Level set methods applied to modeling detonation shock dynamics," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 126, pp. 390409, 1996.
Abstract:
We give an extension of the level set formulation of Osher and
Sethian, which describes the dynamics of surfaces that
propagate under the influence of their own curvature. We
consider an extension of their original algorithms for finite
domains that includes boundary conditions. We discuss this
extension in the context of a specific application that comes
from the theory of detonation shock dynamics (DSD). We give an
outline of the theory of DSD which includes the formulation of
the boundary conditions that comprise the engineering model. We
give the formulation of the level set method, as applied to our
application with finite boundary conditions. We develop a
numerical method to implement arbitrarily complex 2D boundary
conditions and give a few representative calculations. We also
discuss the dynamics of level curve motion and point out
restrictions that arise when applying boundary conditions. (C)
1996 Academic Press, Inc.
 Li, XL, Jin, BX, and Glimm, J, "Numerical study for the threedimensional RayleighTaylor instability through the TVD/AC scheme and parallel computation," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 126, pp. 343355, 1996.
Abstract:
The RayleighTaylor instability is a gravity driven instability
of a contact surface between fluids of different densities, The
growth of this instability is sensitive to numerical or
physical mass diffusion. For this reason, high resolution of
the contact discontinuity is particularly important, In this
paper, we address this problem using a secondorder TVD finite
difference scheme with artificial compression. We describe our
numerical simulations of the 3D RayleighTaylor instability
using this scheme. The numerical solutions are compared to (a)
the exact 2D solution in the linear regime and (b) numerical
solutions using the TVD scheme and the front tracking method.
The computational program is used to study the evolution of a
single bubble and 3D bubble merger, i.e., the nonlinear
evolution of a single mode and the process of nonlinear mode
mode interaction. (C) 1996 Academic Press, Inc.
 Caselles, V, and Sbert, C, "What is the best causal scale space for threedimensional images?," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 56, pp. 11991246, 1996.
Abstract:
We study the unique affine invariant morphological scale space
in three dimensions. We discuss its properties and show that it
improves the dynamic shape model. We explain the algorithms and
display the first numerical experiments.
 Nochetto, RH, and Verdi, C, "Combined effect of explicit timestepping and quadrature for curvature driven flows," NUMERISCHE MATHEMATIK, vol. 74, pp. 105136, 1996.
Abstract:
The flow of a closed surface of codimension 1 in R(R) driven by
curvature is first approximated by a singularly perturbed
parabolic double obstacle problem with small parameter epsilon
> 0. Conforming piecewise linear finite elements, with mass
lumping, over a quasiuniform and weakly acute mesh of size h
are further used for space discretization, and combined with
forward differences for time discretization with uniform time
step tau. The resulting explicit schemes are the basis for an
efficient algorithm, the socalled dynamic mesh algorithm, and
exhibit finite speed of propagation and discrete nondegeneracy.
No iteration is required, not even to handle the obstacle
constraints. The zero level set of the fully discrete solution
is shown to converge past singularities to the true interface,
provided tau, h(2) approximate to 0(epsilon(4)) and no
fattening occurs. If the more stringent relations tau, h(2)
approximate to 0(epsilon(6)) are enforced, then an interface
rate of convergence O(epsilon) is derived in the vicinity of
regular points, along with a companion O(epsilon(1/2)) for type
I singularities. For smooth flows, an interface rate of
convergence of O(epsilon(2)) is proven, provided tau, h(2)
approximate to O(epsilon(5)) and exact integration is used for
the potential term. The analysis is based on constructing fully
discrete barriers via an explicit parabolic projection with
quadrature, which bears some intrinsic interest, Lipschitz
properties of viscosity solutions of the level set approach,
and discrete nondegeneracy. These basic ingredients are also
discussed.
 Corrias, L, "Fast LegendreFenchel transform and applications to Hamilton Jacobi equations and conservation laws," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 33, pp. 15341558, 1996.
Abstract:
We are interested in the study of a fast algorithm introduced
by Brenier computing the discrete LegendreFenchel transform of
a real function. We present convergence results and show how
the order of convergence grows with the regularity of the
Function to be transformed. applications to HamiltonJacobi
equations for front propagation problems and conservation laws
are presented.
 Zhao, HK, Chan, T, Merriman, B, and Osher, S, "Variational level set approach to multiphase motion," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 127, pp. 179195, 1996.
Abstract:
A coupled level set method for the motion of multiple junctions
(of, e.g., solid, liquid, and grain boundaries), which follows
the gradient flow for an energy functional consisting of
surface tension (proportional to length) and bulk energies
(proportional to area), is developed. The approach combines the
level set method of S. Osher and J. A. Sethian with a
theoretical variational formulation of the motion by F. Reitich
and H. M. Soner. The resulting method uses as many level set
functions as there are regions and the energy functional is
evaluated entirely in terms of level set functions. The
gradient projection method leads to a coupled system of
perturbed (by curvature terms) HamiltonJacobi equations. The
coupling is enforced using a single Lagrange multiplier
associated with a constraint which essentially prevents (a)
regions from overlapping and (b) the development of a vacuum.
The numerical implementation is relatively simple and the
results agree with (and go beyond) the theory as given in [12].
Other applications of this methodology, including the
decomposition of a domain into subregions with minimal
interface length, are discussed. Finally, some new techniques
and results in level set methodology are presented. (C) 1996
Academic Press, Inc.
 Harabetian, E, Osher, S, and Shu, CW, "An eulerian approach for vortex motion using a level set regularization procedure," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 127, pp. 1526, 1996.
Abstract:
We present an Eulerian, fixed grid, approach to solve the
motion of an incompressible fluid, in two and three dimensions,
in which the vorticity is concentrated on a lower dimensional
set. Our approach uses a decomposition of the vorticity of the
form xi = P(phi) eta, in which both phi (the level set
function) and eta (the vorticity strength vector) are smooth.
We derive coupled equations for phi and eta which give a
regularization of the problem. The regularization is
topological and is automatically accomplished through the use
of numerical schemes whose viscosity shrinks to zero with grid
size. There is no need for explicit filtering, even when
singularities appear in the front, The method also has the
advantage of automatically allowing topological changes such as
merging of surfaces. Numerical examples, including two and
three dimensional vortex sheets, twodimensional vortex dipole
sheets, and point vortices, are given. To our knowledge, this
is the first threedimensional vortex sheet calculation in
which the sheet evolution feeds back to the calculation of the
fluid velocity. Vortex in cell calculations for three
dimensional vortex sheets were done earlier by Trygvasson et
al. (C) 1996 Academic Press, Inc.
 Malladi, R, Sethian, JA, and Vemuri, BC, "A fast level set based algorithm for topologyindependent shape modeling," JOURNAL OF MATHEMATICAL IMAGING AND VISION, vol. 6, pp. 269289, 1996.
Abstract:
Shape modeling is an important constituent of computer vision
as well as computer graphics research. Shape models aid the
tasks of object representation and recognition. This paper
presents a new approach to shape modeling which retains some of
the attractive features of existing methods, and overcomes some
of their limitations. Our technique can be applied to model
arbitrarily complex shapes, which include shapes with
significant protrusions, and to situations where no a priori
assumption about the object's topology is made. A single
instance of our model, when presented with an image having more
than one object of interest, has the ability to split freely to
represent each object. This method is based on the ideas
developed by Osher and Sethian to model propagating
solid/liquid interfaces with curvaturedependent speeds. The
interface (front) is a closed, nonintersecting, hypersurface
flowing along its gradient field with constant speed or a speed
that depends on the curvature. It is moved by solving a
''HamiltonJacobi'' type equation written for a function in
which the interface is a particular level set. A speed term
synthesized from the image is used to stop the interface in the
vicinity of object boundaries. The resulting equation of motion
is solved by employing entropysatisfying upwind finite
difference schemes. We also introduce a new algorithm for rapid
advancement of the front using what we call a narrowband
update scheme. The efficacy of the scheme is demonstrated with
numerical experiments on low contrast medical images.
 Kimmel, R, Kiryati, N, and Bruckstein, AM, "Subpixel distance maps and weighted distance transforms," JOURNAL OF MATHEMATICAL IMAGING AND VISION, vol. 6, pp. 223233, 1996.
Abstract:
A new framework for computing the Euclidean distance and
weighted distance from the boundary of a given digitized shape
is presented. The distance is calculated with subpixel
accuracy. The algorithm is based on an equal distance contour
evolution process. The moving contour is embedded as a level
set in a time varying function of higher dimension. This
representation of the evolving contour makes possible the use
of an accurate and stable numerical scheme, due to Osher and
Sethian [22]. The relation between the classical shape from
shading problem and the weighted distance transform is
presented, as well as an algorithm that calculates the geodesic
distance transform on surfaces.
 Soravia, P, and Souganidis, PE, "Phasefield theory for FitzhughNagumotype systems," SIAM JOURNAL ON MATHEMATICAL ANALYSIS, vol. 27, pp. 13411359, 1996.
Abstract:
In this paper, we study the asymptotics of FitzhughNagumotype
systems of reactiondiffusion equations with bistable
nonlinearity. In the limit, we obtain an interface moving with
normal velocity determined by the dynamics and the scaling.
 Malladi, R, and Sethian, JA, "An O(N log N) algorithm for shape modeling," PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, vol. 93, pp. 93899392, 1996.
Abstract:
We present a shaperecovery technique in two dimensions and
three dimensions with specific applications in modeling
anatomical shapes from medical images. This algorithm models
extremely corrugated structures like the brain, is
topologically adaptable, and runs in O(N log N) time, where N
is the total number of points in the domain. Our technique is
based on a level set shaperecovery scheme recently introduced
by the authors and the fast marching method for computing
solutions to static HamiltonJacobi equations.
 Fierro, F, and Paolini, M, "Numerical evidence of fattening for the mean curvature flow," MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES, vol. 6, pp. 793813, 1996.
Abstract:
In this paper we describe some numerical simulations in the
context of mean curvature flow. We recover a few different
approaches in modeling the evolution of an interface Sigma
which evolves according to the law: V = k + g where V is the
velocity in the inward normal direction, k is the sum of the
principal curvatures and g is a given forcing term. We will
discuss about the phenomenon of fattening or nonuniqueness of
the solution, recalling what is known about this subject.
Finally we show some interesting numerical simulations that
suggest evidence of fattening starting from different initial
interfaces. Of particular interest is the result obtained for a
torus in R(4) which would be a first example of a regular and
compact surface showing evidence of fattening in the case of
pure motion by mean curvature (no forcing term).
 Sung, CJ, Sun, CJ, and Law, CK, "Analytic description of the evolution of twodimensional flame surfaces," COMBUSTION AND FLAME, vol. 107, pp. 114124, 1996.
Abstract:
The passive propagation of wrinkled, nonfolding, premixed
flames in quiescent and spatially periodic how fields is
investigated by employing the scalar held, Gequation
formulation. Rather than solving the Gequation directly, we
transform it into a gequation, which is a differential
equation governing the evolution of the slope of the flame
shape in twodimensional flows. For the Landau limit of flame
propagation with constant flame speed, the resulting gequation
degenerates to a quasilinear wave equation in a quiescent
flow. For the stretchaffected propagation mode in which the
flame propagation speed is curvaturedependent, the resulting
gequation is in the general form of the Burgers' equation.
Analytical solutions were obtained for several flame and flow
types, revealing some interesting characteristics of the
geometry and propagation of the flame, including the formation
of cusps and their inner structure, and the augmentation of the
average burning velocity through flame wrinkling.
 Tannenbaum, A, "Three snippets of curve evolution theory in computer vision," MATHEMATICAL AND COMPUTER MODELLING, vol. 24, pp. 103119, 1996.
Abstract:
In this paper, we discuss some uses of curve evolution theory
for problems in computer vision. We concentrate on three
problem areas: shape theory, active contours, and geometric
invariant scale spaces. The solutions to these key problems
will all be based on flows which are obtained in a completely
natural manner from geometric and physical principles.
 Kichenassamy, S, Kumar, A, Olver, P, Tannenbaum, A, and Yezzi, A, "Conformal curvature flows: From phase transitions to active vision," ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, vol. 134, pp. 275301, 1996.
Abstract:
In this paper, we analyze geometric active contour models from
a curve evolution point of view and propose some modifications
based on gradient flows relative to certain new featurebased
Riemannian metrics. This leads to a novel edgedetection
paradigm in which the feature of interest may be considered to
lie at the bottom of a potential well. Thus an edgeseeking
curve is attracted very naturally and efficiently to the
desired feature. Comparison with the AllenCahn model clarifies
some of the choices made in these models, and suggests
inhomogeneous models which may in return be useful in phase
transitions. We also consider some 3dimensional active surface
models based on these ideas. The justification of this model
rests on the careful study of the viscosity solutions of
evolution equations derived from a levelset approach.
 Ambrosio, L, and Soner, HM, "Level set approach to mean curvature flow in arbitrary codimension," JOURNAL OF DIFFERENTIAL GEOMETRY, vol. 43, pp. 693737, 1996.
Abstract:
We develop a level set theory for the mean curvature evolution
of surfaces with arbitrary codimension, thus generalizing the
previous work [8, 15] on hypersurfaces. The main idea is to
surround the evolving surface of codimensionk in R(d) by a
family of hypersurfaces (the level sets of a function) evolving
with normal velocity equal to the sum of the (d  k) smallest
principal curvatures. The existence and the uniqueness of a
weak (levelset) solution is easily established by using mainly
the results of [8] and the theory of viscosity solutions for
second order nonlinear parabolic equations. The level set
solutions coincide with the classical solutions whenever the
latter exist. The proof of this connection uses a careful
analysis of the squared distance from the surfaces. It is also
shown that varifold solutions constructed by Brakke [7] are
included in the levelset solutions. The idea of surrounding
the evolving surface by a family of hypersurfaces with a
certain property is related to the barriers of De Giorgi. An
introduction to the theory of barriers and its connection to
the level set solutions is also provided.
 Malladi, R, and Sethian, JA, "A unified approach to noise removal, image enhancement, and shape recovery," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 5, pp. 15541568, 1996.
Abstract:
We present a unified approach to noise removal, image
enhancement, and shape recovery in images. The underlying
approach relies on the level set formulation of curve and
surface motion, which leads to a class of PDEbased algorithms.
Beginning with an image, the first stage of this approach
removes noise and enhances the image by evolving the image
under flow controlled by min/max curvature and by the mean
curvature. This stage is applicable to both saltandpepper
greyscale noise and fullimage continuous noise present in
black and white images, greyscale images, texture images, and
color images. The noise removal/enhancement schemes applied in
this stage contain only one enhancement parameter, which in
most cases is automatically chosen. The other key advantage of
our approach is that a stopping criteria is automatically
picked from the image; continued application of the scheme
produces no further change. The second stage of our approach is
the shape recovery of a desired object; we again exploit the
level set approach to evolve an initial curve/surface toward
the desired boundary, driven by an imagedependent speed
function that automatically stops at the desired boundary.
 You, YL, Xu, WY, Tannenbaum, A, and Kaveh, M, "Behavioral analysis of anisotropic diffusion in image processing," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 5, pp. 15391553, 1996.
Abstract:
In this paper, we analyze the behavior of the anisotropic
diffusion model of Perona and Malik. The main idea is to
express the anisotropic diffusion equation as coming from a
certain optimization problem, so its behavior can be analyzed
based on the shape of the corresponding energy surface. We show
that anisotropic diffusion is the steepest descent method for
solving an energy minimization problem. It is demonstrated that
an anisotropic diffusion is well posed when there exists a
unique global minimum for the energy functional and that the
ill posedness of a certain anisotropic diffusion is caused by
the fact that its energy functional has an infinite number of
global minima that are dense in the image space. We give a
sufficient condition for an anisotropic diffusion to be well
posed and a sufficient and necessary condition for it to be ill
posed due to the dense global minima. The mechanism of
smoothing and edge enhancement of anisotropic diffusion is
illustrated through a particular orthogonal decomposition of
the diffusion operator into two parts: one that diffuses
tangentially to the edges and therefore acts as an anisotropic
smoothing operator, and the other that flows normally to the
edges and thus acts as an enhancement operator.
 Falcone, M, and Lanucara, P, "Parallel algorithms for HamiltonJacobi equations," ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK, vol. 76, pp. 355358, 1996.
Abstract:
We discuss the parallel implemention of some algorithms for
HamiltonJacobi equations based on dynamic programming. Our
model problem is the first order partial differential equation
related to the isotropic motion of fronts in the normal
direction with velocity c(x). We present a local version of the
serial algorithm which is well suited for the parallelisation
via a domain decomposition strategy and we discuss the
performance obtained using PVM and Linda on a cluster of UNIX
workstations.
 Schwendeman, DW, "A front dynamics approach to curvaturedependent flow," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 56, pp. 15231538, 1996.
Abstract:
A front dynamics approach is developed to study the evolution
of planar curves whose normal speed depends on curvature. The
formulation is similar to Whitham's shock dynamics theory for
the propagation of shock Nave in gases but assumes a different
propagation rule. Equations that describe the motion of the
front are obtained, and these are evolution equations for the
normal direction and local are length of the front. The
solution of these equations leads to the front positions using
an appropriate integration along rays. A similarity solution of
the equations is found for the evolution of an initial corner.
Freeboundary problems for the motion of a junction connecting
front segments are discussed. A numerical method is presented
to calculate the evolution of any number of front segments. The
segments can be closed or open, connected to Nail boundaries or
not, or connected to other segments at 3segment junctions.
Several sample problems are considered to illustrate the
method. An extension of the method for curvaturedependent
motion under a constant area constraint is also discussed.
 Kimia, BB, and Siddiqi, K, "Geometric heat equation and nonlinear diffusion of shapes and images," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 64, pp. 305322, 1996.
Abstract:
Visual tasks often require a hierarchical representation of
shapes and images in scales ranging from coarse to fine. A
variety of linear and nonlinear smoothing techniques, such as
Gaussian smoothing, anisotropic diffusion, regularization,
etc., have been proposed, leading to scalespace
representations. We propose a geometric smoothing method based
on local curvature for shapes and images. The deformation by
curvature, or the geometric heat equation, is a special case of
the reactiondiffusion framework proposed in [41]. For shapes,
the approach is analogous to the classical heat equation
smoothing, but with a renormalization by arelength at each
infinitesimal step. For images, the smoothing is similar to
anisotropic diffusion in that, since the component of diffusion
in the direction of the brightness gradient is nil, edge
location is left intact. Curvature deformation smoothing for
shape has a number of desirable properties: it preserves
inclusion order, annihilates extrema and inflection points
without creating new ones, decreases total curvature, satisfies
the semigroup property allowing for local iterative
computations, etc. Curvature deformation smoothing of an image
is based on viewing it as a collection of isointensity level
sets, each of which is smoothed by curvature. The reassembly of
these smoothed level sets into a smoothed image follows a
number of mathematical properties; it is shown that the
extension from smoothing shapes to smoothing images is
mathematically sound due to a number of recent results [21]. A
generalization of these results [14] justifies the extension of
the entire entropy scale space for shapes [42] to one for
images, where each isointensity level curve is deformed by a
combination of constant and curvature deformation. The scheme
has been implemented and is illustrated for several medical,
aerial, and range images. (C) 1996 Academic Press, Inc.
 Chakraborty, A, Staib, LH, and Duncan, JS, "Deformable boundary finding in medical images by integrating gradient and region information," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 15, pp. 859870, 1996.
Abstract:
Accurately segmenting and quantifying structures is a key issue
in biomedical image analysis. The two conventional methods of
image segmentation, regionbased segmentation, and boundary
finding, often suffer from a variety of limitations. Here we
propose a method which endeavors to integrate the two
approaches in an effort to form a unified approach that is
robust to noise and poor initialization. Our approach uses
Green's theorem to derive the boundary of a homogeneous region
classified area in the image and integrates this with a gray
level gradientbased boundary finder. This combines the
perceptual notions of edge/shape information with gray level
homogeneity. A number of experiments were performed both on
synthetic and real medical images of the brain and heart to
evaluate the new approach, and it is shown that the integrated
method typically performs better when compared to conventional
gradientbased deformable boundary finding. Further, this
method yields these improvements with little increase in
computational overhead, an advantage derived from the
application of the Green's theorem.
 Crandall, MG, and Lions, PL, "Convergent difference schemes for nonlinear parabolic equations and mean curvature motion," NUMERISCHE MATHEMATIK, vol. 75, pp. 1741, 1996.
Abstract:
Explicit finite difference schemes are given for a collection
of parabolic equations which may have all of the following
complex features: degeneracy, quasilinearity, full
nonlinearity, and singularities. In particular, the equation of
''motion by mean curvature'' is included. The schemes are
monotone and consistent, so that convergence is guaranteed by
the general theory of approximation of viscosity solutions of
fully nonlinear problems. In addition, an intriguing new type
of nonlocal problem is analyzed which is related to the
schemes, and another very different sort of approximation is
presented as well.
 Caselles, V, and Coll, B, "Snakes in movement," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 33, pp. 24452456, 1996.
Abstract:
In this paper, we propose a geometric partial differential
equation (PDE) for tracking one or several moving objects from
a sequence of images, which is based on a geometric model for
active contours. The active contour approach permits us to
simultaneously handle both aspects: finding the boundaries and
tracking them. We also describe a numerical scheme to solve the
geometric equation and we present some numerical experiments.
 Walkington, NJ, "Algorithms for computing motion by mean curvature," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 33, pp. 22152238, 1996.
Abstract:
We propose a finite element algorithm for computing the motion
of a surface moving by mean curvature. The algorithm uses the
level set formulation so that changes in topology of the
surface can be accommodated. Stability is deduced by showing
that the discrete solutions satisfy both L(infinity) and W1,W
1 bounds. Existence of discrete solutions and connections with
Brakke flows are established. Some numerical examples and
application to related problems, such as the phase field
equations, are also presented.
 Kimmel, R, and Kiryati, N, "Finding the shortest paths on surfaces by fast global approximation and precise local refinement," INTERNATIONAL JOURNAL OF PATTERN RECOGNITION AND ARTIFICIAL INTELLIGENCE, vol. 10, pp. 643656, 1996.
Abstract:
Finding the shortest path between points on a surface is a
challenging global optimization problem. It is difficult to
devise an algorithm that is computationally efficient, locally
accurate and guarantees to converge to the globally shortest
path. In this paper a two stage coarsetofine approach for
finding the shortest paths is suggested. In the first stage the
algorithm of Ref. 10 that combines a 3D length estimator with
graph search is used to rapidly obtain an approximation to the
globally shortest path. In the second stage the approximation
is refined to become a shorter geodesic curve, i.e., a locally
optimal path. This is achieved by using an algorithm that
deforms an arbitrary initial curve ending at two given surface
points via geodesic curvature shortening flow. The 3D curve
shortening how is transformed into an equivalent 2D one that is
implemented using an efficient numerical algorithm for curve
evolution with fixed end points, introduced in Ref. 9.
 Alikakos, ND, Fusco, G, and Kowalczyk, M, "Finite dimensional dynamics and interfaces intersecting the boundary: Equilibria and quasiinvariant manifold," INDIANA UNIVERSITY MATHEMATICS JOURNAL, vol. 45, pp. 11191155, 1996.
Abstract:
In the present paper we consider the AllenCahn equation in a
class of domains consisting of a rectangular part with two
attachments on its sides. We establish the existence of
stationary solutions with nearly flat interfaces intersecting
orthogonally the boundary of the domain at its rectangular
part. We also show that the stability of these equilibria
depends on the geometry of the domain. Finally we obtain some
results regarding the dynamics of the AllenCahn equation,
namely we construct an approximation of the invariant manifold
associated with the equilibria (quasiinvariant manifold).
Analysis of the vector field near this manifold suggests that
the normal velocity of the flat interfaces is exponentially
small in epsilon.
 McAuliffe, MJ, Eberly, D, Fritsch, DS, Chaney, EL, and Pizer, SM, "Scalespace boundary evolution initialized by cores," VISUALIZATION IN BIOMEDICAL COMPUTING, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1131, pp. 173182, 1996.
Abstract:
A novel interactive segmentation method has been developed
which uses estimated boundaries, generated from cores, to
initialize a scalespace boundary evolution process in
greyscale medical images. Presented is an important addition to
core extraction methodology that improves core generation for
objects that are in the presence of interfering objects. The
boundary at the scale of the core (BASOC) and its associated
width information, both derived from the core, are used to
initialize the second stage of the segmentation process. In
this automatic refinement stage, the BASOC is allowed to evolve
in a splinesnakelike manner that makes use of objectrelevant
width information to make robust measurements of local edge
positions.


1997 
 Sethian, JA, and Adalsteinsson, D, "An overview of level set methods for etching, deposition, and lithography development," IEEE TRANSACTIONS ON SEMICONDUCTOR MANUFACTURING, vol. 10, pp. 167184, 1997.
Abstract:
The range of surface evolution problems in etching, deposition,
and lithography development offers significant challenge for
numerical methods in front tracking. Level set methods for
evolving interfaces are specifically designed for profiles
which can develop sharp corners, change topology, and undergo
orders of magnitude changes in speed, They are based on solving
a HamiltonJacobi type equation for a level set function, using
techniques borrowed from hyperbolic conservation laws. Over the
past few years, a body of level set methods have been developed
with application to microfabrication problems, In this paper,
we give an overview of these techniques, describe the
implementation in etching, deposition, and lithography
simulations, and present a collection of fast level set
methods, each aimed at a particular application, In the case of
photoresist development and isotropic etching/deposition, the
fast marching level set method, introduced by Sethian in [39],
[40], can track the threedimensional photoresist process
through a 200x200x 200 rate function grid in under 55 s on a
Sparc10. In the case of more complex etching and deposition,
the narrow band level set method, introduced in Adalsteinsson
and Sethian in [2], can be used to handle problems in which the
speed of the interface delicately depends on the orientation of
the interface versus an incoming beam, the effects of
visibility, surface tension, reflection and reemission, and
complex threedimensional effects, Our applications include
photoresist development, etching/deposition problems under the
effects of masking, visibility, complex flux integrations over
sources, nonconvex sputter deposition problems, and
simultaneous deposition and etch phenomena.
 Sapiro, G, Cohen, A, and Bruckstein, AM, "A subdivision scheme for continuousscale Bsplines and affine invariant progressive smoothing," JOURNAL OF MATHEMATICAL IMAGING AND VISION, vol. 7, pp. 2340, 1997.
Abstract:
Multiscale representations and progressive smoothing constitute
an important topic in different fields as computer vision,
CAGD, and image processing. In this work, a multiscale
representation of planar shapes is first described. The
approach is based on computing classical Bsplines of
increasing orders, and therefore is automatically affine
invariant. The resulting representation satisfies basic scale
space properties at least in a qualitative form, and is simple
to implement. The representation obtained in this way is
discrete in scale, since classical Bsplines are functions in
Ck2, where k is an integer bigger or equal than two. We
present a subdivision scheme for the computation of Bsplines
of finite support at continuous scales. With this scheme, B
splines representations in Cr are obtained for any real r in
[0, infinity), and the multiscale representation is extended to
continuous scale. The proposed progressive smoothing receives a
discrete set of points as initial shape, while the smoothed
curves are represented by continuous (analytical) functions,
allowing a straightforward computation of geometric
characteristics of the shape.
 Miller, K, "A geometricalmechanical interpretation of gradientweighted moving finite elements," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 34, pp. 6790, 1997.
Abstract:
The usual explanation of the gradientweighted moving finite
element (GWMFE) method has been in terms of its variational
interpretation. This paper presents a more intuitive
geometricalmechanical interpretation of GWMFE as a balance of
forces on the nodes, forces concentrated onto the nodes by the
laws of leverage. It also presents significant simplifications
in the ''internodal viscosity'' terms for regularization of the
nodal movements, plus some simple ''linear internodal
tensions'' for regularization of the longterm nodal
positioning. These simplifications of the regularizations are
especially important in two and three space dimensions. One of
the generalizations which follows from the geometrical
mechanical interpretation is a promising but still untested
second GWMFE formulation for systems of PDEs. The original MFE
method is seen to be the smallslope limit of GWMFE under
''vertical rescaling.'' Reporting on the design and extensive
numerical trials of robust and versatile GWMFE systems codes in
one and two dimensions is deferred to two forthcoming papers by
Carlson and the author [Design and application of a gradient
weighted moving finite element code, Part I, in 1D, SIAM J.
Sci. Comput., to appear] and [Design and application of a
gradientweighted moving finite element code, Part II, in 2D,
SIAM J. Sci. Comput., to appear]. Here only a few illustrative
examples are presented involving motion of surfaces by mean
curvature, i.e., by surface tension.
 Shu, CW, "Uniformly high order essentially nonoscillatory schemes .3. Preface," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 131, pp. 12, 1997.
Abstract:
We present a numerical method for computing solutions of the
incompressible Euler or NavierStokes equations when a
principal feature of the flow is the presence of an interface
between two fluids with different fluid properties. The method
is based on a secondorder projection method for variable
density flows using an ''approximate projection'' formulation.
The boundary between the fluids is tracked with a secondorder,
volumeoffluid interface tracking algorithm. We present
results for viscious RayleighTaylor problems at early time
with equal and unequal viscosities to demonstrate the
convergence of the algorithm. We also present computational
results for the RayleighTaylor instability in airhelium and
for bubbles and drops in an airwater system without surface
tension to demonstrate the behavior of the algorithm on
problems with large density and viscosity contrasts. (C) 1997
Academic Press.
 Puckett, EG, Almgren, AS, Bell, JB, Marcus, DL, and Rider, WJ, "A highorder projection method for tracking fluid interfaces in variable density incompressible flows," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 130, pp. 269282, 1997.
Abstract:
We present a numerical method for computing solutions of the
incompressible Euler or NavierStokes equations when a
principal feature of the flow is the presence of an interface
between two fluids with different fluid properties. The method
is based on a secondorder projection method for variable
density flows using an ''approximate projection'' formulation.
The boundary between the fluids is tracked with a secondorder,
volumeoffluid interface tracking algorithm. We present
results for viscious RayleighTaylor problems at early time
with equal and unequal viscosities to demonstrate the
convergence of the algorithm. We also present computational
results for the RayleighTaylor instability in airhelium and
for bubbles and drops in an airwater system without surface
tension to demonstrate the behavior of the algorithm on
problems with large density and viscosity contrasts. (C) 1997
Academic Press.
 Catte, F, "Convergence of iterated affine and morphological filters by nonlinear semigroup theory," NONLINEAR ANALYSISTHEORY METHODS & APPLICATIONS, vol. 28, pp. 19351942, 1997.
Abstract:
This paper presents a general framework to generate multiscale
representations of image data. The process is considered as an
initial value problem with an acquired image as initial
condition and a geometrical invariant as ''driving force'' of
an evolutionary process. The geometrical invariants are
extracted using the family of Gaussian derivative operators.
These operators naturally deal with scale as a free parameter
and solve the illposedness problem of differentiation.
Stability requirements for numerical approximation of evolution
schemes using Gaussian derivative operators are derived and
establish an intuitive connection between the allowed timestep
and scale. This approach has been used to generalize and
implement a variety of nonlinear diffusion schemes. Results on
test images and medical images are shown.
 Niessen, WJ, Romeny, BMT, Florack, LMJ, and Viergever, MA, "A general framework for geometrydriven evolution equations," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 21, pp. 187205, 1997.
Abstract:
This paper presents a general framework to generate multiscale
representations of image data. The process is considered as an
initial value problem with an acquired image as initial
condition and a geometrical invariant as ''driving force'' of
an evolutionary process. The geometrical invariants are
extracted using the family of Gaussian derivative operators.
These operators naturally deal with scale as a free parameter
and solve the illposedness problem of differentiation.
Stability requirements for numerical approximation of evolution
schemes using Gaussian derivative operators are derived and
establish an intuitive connection between the allowed timestep
and scale. This approach has been used to generalize and
implement a variety of nonlinear diffusion schemes. Results on
test images and medical images are shown.
 HajHariri, H, Shi, Q, and Borhan, A, "Thermocapillary motion of deformable drops at finite Reynolds and Marangoni numbers," PHYSICS OF FLUIDS, vol. 9, pp. 845855, 1997.
Abstract:
We present the results of numerical simulations of the three
dimensional thermocapillary motion of deformable viscous drops
under the influence of a constant temperature gradient within a
second liquid medium. In particular, we examine the effects of
shape deformations and convective transport of momentum and
energy on the migration velocity of the drop. A numerical
method based on a continuum model for the fluidfluid interface
is used to account for finite drop deformations. An octtree
adaptive grid refinement scheme is integrated into the
numerical method in order to track the interface without the
need for interface reconstruction. Interface deformations
arising from the convection of energy at small Reynolds numbers
are found to be negligible. On the other hand, deformations of
the drop shape due to inertial effects? though small in
magnitude, are found to retard the motion of the drop. The
steady drop shapes are found to resemble oblate or prolate
spheroids without fore and aft symmetry, with the direction of
elongation of the drop depending on the value of the density
ratio between the two phases. As in the case of a gas bubble,
convection of energy is shown to retard the thermocapillary
motion of a viscous drop, as the isotherms get wrapped around
the front surface of the drop and effectively reduce the
surface temperature gradient which drives the motion. The
effect of inertia on the mobility of viscous drops is found to
be weaker than that in the case of gas bubbles. (C) 1997
American Institute of Physics.
 Tek, H, and Kimia, BB, "Volumetric segmentation of medical images by threedimensional bubbles," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 65, pp. 246258, 1997.
Abstract:
The segmentation of structure from images is an inherently
difficult problem in computer vision and a bottleneck to its
widespread application, e.g., in medical imaging, This paper
presents an approach for integrating local evidence such as
regional homogeneity and edge response to form global structure
for figureground segmentation. This approach is motivated by a
shockbased morphogenetic language, where the growth of four
types of shocks results in a complete description of shape,
Specifically, objects are randomly hypothesized in the form of
fourthorder shocks (seeds) which then grow, merge, split,
shrink, and, in general, deform under physically motivated
''forces,'' but slow down and come to a halt near differential
structures. Two major issues arise in the segmentation of 3D
images using this approach. First, it is shown that the
segmentation of 3D images by 3D bubbles is superior to a slice
byslice segmentation by 2D bubbles or by ''21/2D bubbles''
which are inherently 2D but use 3D information for their
deformation. Specifically, the advantages lie in an intrinsic
treatment of the underlying geometry and accuracy of
reconstruction. Second, gaps and weak edges, which frequently
present a significant problem for 2D and 3D segmentation, are
regularized by curvaturedependent curve and surface
deformations which constitute diffusion processes, The 3D
bubbles evolving in the 3D reactiondiffusion space are a
powerful tool in the segmentation of medical and other images,
as illustrated for several realistic examples. (C) 1997
Academic Press.
 Faugeras, O, and Keriven, R, "Level set methods and the stereo problem," SCALESPACE THEORY IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1252, pp. 272283, 1997.
Abstract:
We present a novel geometric approach for solving the stereo
problem for an arbitrary number of images (greater than or
equal to 2). It is based upon the definition of a variational
principle that must be satisfied by the surfaces of the objects
in the scene and their images. The EulerLagrange equations
which are deduced from the variational principle provide a set
of PDE's which are used to deform an initial set of surfaces
which then move towards the objects to be detected. The level
set implementation of these PDE's potentially provides an
efficient and robust way of achieving the surface evolution and
to deal automatically with changes in the surface topology
during the deformation, i.e. to deal with multiple objects.
Results of a two dimensional implementation of our theory are
presented on synthetic and real images.
 Yezzi, A, Kichenassamy, S, Kumar, A, Olver, P, and Tannenbaum, A, "A geometric snake model for segmentation of medical imagery," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 16, pp. 199209, 1997.
Abstract:
In this note, we employ the new geometric active contour models
formulated in [25] and [26] for edge detection and segmentation
of magnetic resonance imaging (MRI), computed tomography (CT),
and ultrasound medical imagery, Our method is based on defining
featurebased metrics on a given image which in turn leads to a
novel snake paradigm in which the feature of interest mag be
considered to lie at the bottom of a potential well, Thus, the
snake is attracted very quickly and efficiently to the desired
feature.
 Caselles, V, Kimmel, R, and Sapiro, G, "Geodesic active contours," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 22, pp. 6179, 1997.
Abstract:
A novel scheme for the detection of object boundaries is
presented. The technique is based on active contours evolving
in time according to intrinsic geometric measures of the image.
The evolving contours naturally split and merge, allowing the
simultaneous detection of several objects and both interior and
exterior boundaries. The proposed approach is based on the
relation between active contours and the computation of
geodesics or minimal distance curves. The minimal distance
curve lays in a Riemannian space whose metric is defined by the
image content. This geodesic approach for object segmentation
allows to connect classical ''snakes'' based on energy
minimization and geometric active contours based on the theory
of curve evolution. Previous models of geometric active
contours are improved, allowing stable boundary detection when
their gradients suffer from large variations, including gaps.
Formal results concerning existence, uniqueness, stability, and
correctness of the evolution are presented as well. The scheme
was implemented using an efficient algorithm for curve
evolution. Experimental results of applying the scheme to real
images including objects with holes and medical data imagery
demonstrate its power. The results may be extended to 3D object
segmentation as well.
 Sethian, JA, "Tracking interfaces with level sets," AMERICAN SCIENTIST, vol. 85, pp. 254263, 1997.
Abstract:
Consider a closed surface in R(n) of codimension 1 which
propagates in the normal direction with velocity proportional
to its mean curvature plus a forcing term. This geometric
problem is first approximated by a singularly perturbed
parabolic double obstacle problem with small parameter epsilon
> 0. Conforming piecewise linear finite elements over a quasi
uniform and strongly acute mesh of size h are further used for
space discretization and combined with backward differences for
time discretization with uniform timestep tau. It is shown
that the zero level set of the fully discrete solution
converges past singularities to the true interface, provided
tau, h(2) approximate to o(epsilon(3)) and no fattening occurs.
If the more stringent relations tau, h(2) approximate to
O(epsilon(4)) are enforced, then a linear rate of convergence
O(epsilon) for interfaces is derived in the vicinity of regular
points, namely those for which the underlying viscosity
solution is nondegenerate. Singularities and their smearing
effect are also studied. The analysis is based on constructing
discrete barriers via a parabolic projection, Lipschitz
dependence of viscosity solutions with respect to perturbations
of data, and discrete nondegeneracy. These issues are proven,
along with quasi optimality in two dimensions of the parabolic
projection in L(infinity) with respect to both order and
regularity requirements for functions in Wp(2,1).
 Nochetto, RH, and Verdi, C, "Convergence past singularities for a fully discrete approximation of curvaturedriven interfaces," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 34, pp. 490512, 1997.
Abstract:
Consider a closed surface in R(n) of codimension 1 which
propagates in the normal direction with velocity proportional
to its mean curvature plus a forcing term. This geometric
problem is first approximated by a singularly perturbed
parabolic double obstacle problem with small parameter epsilon
> 0. Conforming piecewise linear finite elements over a quasi
uniform and strongly acute mesh of size h are further used for
space discretization and combined with backward differences for
time discretization with uniform timestep tau. It is shown
that the zero level set of the fully discrete solution
converges past singularities to the true interface, provided
tau, h(2) approximate to o(epsilon(3)) and no fattening occurs.
If the more stringent relations tau, h(2) approximate to
O(epsilon(4)) are enforced, then a linear rate of convergence
O(epsilon) for interfaces is derived in the vicinity of regular
points, namely those for which the underlying viscosity
solution is nondegenerate. Singularities and their smearing
effect are also studied. The analysis is based on constructing
discrete barriers via a parabolic projection, Lipschitz
dependence of viscosity solutions with respect to perturbations
of data, and discrete nondegeneracy. These issues are proven,
along with quasi optimality in two dimensions of the parabolic
projection in L(infinity) with respect to both order and
regularity requirements for functions in Wp(2,1).
 Sapiro, G, and Caselles, V, "Histogram modification via differential equations," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 135, pp. 238268, 1997.
Abstract:
The explicit use of partial differential equations (PDEs) in
image processing became a major research topic in the past
years. In this work we present a framework for histogram
(pixelvalue distribution) modification via ordinary and
partial differential equations. In this way, the image contrast
is improved. We show that the histogram can be modified to
achieve any given distribution as the steady state solution of
an image now. The contrast modification can be performed while
simultaneously reducing noise in a unique PDE, avoiding noise
sharpening effects of classical algorithms. The approach is
extended to local contrast enhancement as well. A variational
interpretation of the flow is presented and theoretical results
on the existence of solutions are given. (C) 1997 Academic
Press.
 Coward, AV, Renardy, YY, Renardy, M, and Richards, JR, "Temporal evolution of periodic disturbances in twolayer Couette flow," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 132, pp. 346361, 1997.
Abstract:
The timedependent motion for a twolayer Couette flow
consisting of fluids of different viscosities is simulated
numerically by using an algorithm based on the Volume of Fluid
(VOF) method. Interfacial tension is included via a continuous
surface force (CSF) algorithm. The algorithm is finetuned to
handle the motion which is driven by a shearinduced
interfacial instability due to the viscosity stratification.
The code is validated against linear theory. Two prototypical
situations are presented, one at a moderately high Reynolds
number and the other at a lower Reynolds number. The initial
condition is seeded with the eigenmode of largest growth rate,
with amplitudes that are varied from those that capture the
linear regime to larger values for nonlinear regimes. Issues of
free surface advection and viscosity interpolation are
discussed. The onset of nonlinearity occurs at the interface
and is quadratic, followed by wave steepening. (C) 1997
Academic Press.
 Caselles, V, Kimmel, R, and Sapiro, G, "Minimal surfaces based object segmentation," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 19, pp. 394398, 1997.
Abstract:
A geometric approach for 3D object segmentation and
representation is presented. The segmentation is obtained by
deformable surfaces moving towards the objects to be detected
in the 3D image. The model is based on curvature motion and the
computation of surfaces with minimal areas, better known as
minimal surfaces. The space where the surfaces are computed is
induced from the 3D image (volumetric data) in which the
objects are to be detected. The model links between classical
deformable surfaces obtained via energy minimization, and
intrinsic ones derived from curvature based flows. The new
approach is stable, robust, and automatically handles changes
in the surface topology during the deformation.
 Fejes, S, and Rosenfeld, A, "Discrete active models and applications," PATTERN RECOGNITION, vol. 30, pp. 817835, 1997.
Abstract:
Optimization processes based on ''active models'' play central
roles in many areas of computational vision as well as
computational geometry. Unfortunately, current models usually
require highly complex and sophisticated mathematical machinery
and at the same time they suffer from a number of limitations
which impose restrictions on their applicability. In this paper
a simple class of discrete active models, called migration
processes (MPs), is presented. The processes are based on
iterated averaging over neighborhoods defined by constant
geodesic distance. It is demonstrated that the MP modela
system of selforganizing active particleshas a number of
advantages over previous models, both parametric active models
(''snakes'') and implicit (contour evolution) models. Due to
the generality of the MP model, the process can be applied to
derive natural solutions to a variety of optimization
problems,including defining (minimal) surface patches given
their boundary curves; finding shortest paths joining sets of
points; and decomposing objects into ''primitive'' parts. (C)
1997 Pattern Recognition Society.
 Leveque, RJ, and Li, ZL, "Immersed interface methods for Stokes flow with elastic boundaries or surface tension," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 18, pp. 709735, 1997.
Abstract:
A secondorder accurate interface tracking method for the
solution of incompressible Stokes flow problems with moving
interfaces on a uniform Cartesian grid is presented. The
interface may consist of an elastic boundary immersed in the
fluid or san interface between two different fluids. The
interface is represented by a cubic spline along which the
singularly supported elastic or surface tension force can be
computed. The Stokes equations are then discretized using the
secondorder accurate finite difference methods for elliptic
equations with singular sources developed in our previous paper
[SIAM J. Numer. Anal., 31(1994), pp. 10191044]. The resulting
velocities are interpolated to the interface to determine the
motion of the interface. An implicit quasiNewton method is
developed that allows reasonable time steps to be used.
 Caselles, V, Coll, B, and Morel, JM, "Scale space versus topographic map for natural images," SCALESPACE THEORY IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1252, pp. 2949, 1997.
Abstract:
We call "natural" image any photograph of an outdoor or indoor
scene taken by a standard camera. In such images, most observed
objects undergo occlusions and the illumination condition and
contrast response of the camera are unknown. Actual Scale Space
theories do not incorporate obvious restrictions imposed by the
physics of image generation. The heat equation (linear scale
space) is not contrast invariant and destroys Tjunctions. The
same is true for the recently proposed curvature equations
(mean curvature motion and affine shortening): They break the
symmetry of junctions. To apply directly these models to
natural world images, With occlusions, is irrevelant. Returning
to the edge detection problem, in which scale space theory
originates, we show how level lines can be found in an image
without smoothing. As an alternative to edge detection/scale
space, we propose to define the line structure in a natural
image by its topographic map (set of all level lines). We also
show that a modification of morphological scale space can help
to the visualization of the topographic map.
 Denet, B, "A Lagrangian method to simulate turbulent flames with reconnections," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 123, pp. 247260, 1997.
Abstract:
A 1D lagrangian formulation, equivalent to the 2D G equation of
2D propagating fronts in the geometrical optics approximation,
is introduced. Fractal flames are obtained numerically by this
method for flow fields containing a large number of scales.
 Olver, PJ, Sapiro, G, and Tannenbaum, A, "Invariant geometric evolutions of surfaces and volumetric smoothing," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 57, pp. 176194, 1997.
Abstract:
The study of geometric flows for smoothing, multiscale
representation, and analysis of two and threedimensional
objects has received much attention in the past few years. In
this paper, we first survey the geometric smoothing of curves
and surfaces via geometric heattype flows, which are invariant
under the groups of Euclidean and affine motions. Second, using
the general theory of differential invariants, we determine the
general formula for a geometric hypersurface evolution which is
invariant under a prescribed symmetry group. As an application,
we present the simplest affine invariant flow for (convex)
surfaces in threedimensional space, which, like the affine
invariant curve shortening flow, will be of fundamental
importance in the processing of threedimensional images.
 Niessen, WJ, Vincken, KL, Weickert, JA, and Viergever, MA, "Nonlinear multiscale representations for image segmentation," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 66, pp. 233245, 1997.
Abstract:
In order to segment an image the use of information at multiple
scales is invaluable. The hyperstack, a linkingmodelbased
segmentation technique, uses intensity to link points in
adjacent levels of a scale space stack. This approach has been
successfully applied to linear multiscale representations.
Multiscale representions which satisfy two scale space
properties, viz. a causality criterion and a semigroup property
in differential form, are valid inputs as well. In this paper
we consider linear scale space, gradientdependent diffusion,
and the Euclidean shortening flow. Since no global scale
parameter is available in the latter two approaches we compare
scale levels based on evolution time, information theoretic
measures, and by counting the number of objects. The multiscale
representations are compared with respect to their performance
in image segmentation tasks on test and MR images. The
hyperstack proves to be rather insensitive to the underlying
multiscale representation although the nonlinear
representations reduced the number of post processing steps.
(C) 1997 Academic Press.
 Okatani, T, and Deguchi, K, "Shape reconstruction from an endoscope image by shape from shading technique for a point light source at the projection center," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 66, pp. 119131, 1997.
Abstract:
This paper presents a method for reconstructing the 3D shape of
an object from its endoscope image based on image shading. The
primary problem is that the endoscope has a light source near
the object surface. Most of the conventional shape from shading
methods assumed that the light source was distant from the
object surface and simplified the analysis. To deal with the
near light source, we use the configuration of the endoscope
that the light source of the endoscope is well approximated by
an imaginary point source at the projection center. In
addition, we introduce a notion of equal distance contours of
the object surface; by propagating the contours using the image
shading, we reconstruct the object shape. This is an extension
of the KimmelBruckstein algorithm of shape from shading to the
endoscope images. Experimental results for real medical
endoscope images of the stomach wall show the feasibility of
this method and also show its promising availability for
morphological analyses of tumors on human inner organs. (C)
1997 Academic Press.
 Frankel, ML, "Turbulent fronts and selffractalizing ornaments generated by an interface dynamics equation," INTERNATIONAL JOURNAL OF BIFURCATION AND CHAOS, vol. 7, pp. 239252, 1997.
Abstract:
We present results of numerical experimentation with a 2D
version of an equation of surface dynamics that has been
derived earlier in the context of flame fronts [Fankel &
Sivashinsky, 1987, 1988] and solidliquid interfaces [Frankel,
1988]. Our observations confirm qualitative predictions of
Frankel & Sivashinsky [1987, 1988]: the curves develop chaotic
cellular pattern and accelerate while imbedding is sustained.
However, if we allow selfintersections, in a different range
of parameters the equation gives birth to remarkably complex
and beautiful fractallike structures either entirely chaotic
or preserving any symmetry if inherited from the initial
configuration. This accumulation of complexity is also
manifested in exponential growth of the length while diameter
of the set increases linearly which results in increasingly
dense covering of the plane. Based on these observations we
introduce concepts of selffractalizing family and asymptotic
fractal dimension, which turns out to be equal to two.
 Kimmel, R, "Intrinsic scale space for images on surfaces: The geodesic curvature flow," SCALESPACE THEORY IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1252, pp. 212223, 1997.
Abstract:
A scale space for images painted on surfaces is introduced.
Based on the geodesic curvature pow of the isogray level
contours of an image painted on the given surface, the image is
evolved and forms the natural geometric scale space. Its
geometrical properties are discussed as well as the intrinsic
nature of the proposed flow. I.e. the flow is invariant to the
bending of the surface.
 Xu, K, "BGKbased scheme for multicomponent flow calculations," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 134, pp. 122133, 1997.
Abstract:
This paper concerns the extension of the gaskinetic BGKtype
scheme to multicomponent flow calculations. In this new scheme,
each component satisfies its individual gaskinetic BGK
equation and the equilibrium states for each component are
coupled in space and time to have common temperature and
velocity. The particle diffusion in gas mixtures is included
naturally in the gaskinetic model. The current scheme can
handle strong shocks and be oscillationfree through the
material interface. The scheme guarantees the exact mass
conservation for each component and the exact conservation of
total momentum and energy in the whole particle system. As a
special application, the current scheme is applied to gas
vacuum interaction case, where the mass densities for other
components are set to zero in the whole domain. The extension
of the current approach to three dimensions is straightforward.
With the definition of phi = rho((1))  rho((2)) in the two
component gas flow, similar to the level set method we can
follow explicitly the time evolution of the material interface
(phi = 0). The numerical results confirm the accuracy and
robustness of the BGKtype scheme. (C) 1997 Academic Press.
 Papalexandris, MV, Leonard, A, and Dimotakis, PE, "Unsplit schemes for hyperbolic conservation laws with source terms in one space dimension," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 134, pp. 3161, 1997.
Abstract:
The present work is concerned with an application of the theory
of characteristics to conservation laws with source terms in
one space dimension, such as the Euler equations for reacting
flows. Spacetime paths are introduced on which the
flow/chemistry equations decouple to a characteristic set of
ODE's for the corresponding homogeneous laws, thus allowing the
introduction of functions analogous to the Riemann invariants
in classical theory. The geometry of these paths depends on the
spatial gradients of the solution. This particular
decomposition can be used in the design of efficient unsplit
algorithms for the numerical integration of the equations. As a
first step, these ideas are implemented for the case of a
scalar conservation law with a nonlinear source term. The
resulting algorithm belongs to the class of MUSCLtype, shock
capturing schemes. Its accuracy and robustness are checked
through a series of tests. The stiffness of the source term is
also studied. Then, the algorithm is generalized for a system
of hyperbolic equations, namely the Euler equations for
reacting flows. A numerical study of unstable detonations is
performed. (C) 1997 Academic Press.
 Granier, B, Lerat, A, and Wu, ZN, "An implicit centered scheme for steady and unsteady incompressible one and twophase flows," COMPUTERS & FLUIDS, vol. 26, pp. 373393, 1997.
Abstract:
Based on artificial compressibility and dual timestepping, an
implicit scheme is developed for solving the steady and
unsteady incompressible NavierStokes equations for one and
twophase flows. The scheme is centered but, due to its
internal dissipation, it needs no staggered grid or upwinding
to be stable. Its stability with respect to pseudo and physical
times and convergence to a steady state are analyzed for a
scalar model equation and partially for the full 2D Navier
Stokes equations. The scheme is applied to the calculation of
flow over a flat plate and steady or unsteady flows in lid
driven cavities. Thanks to a levelset interface tracking
method, the method is also applied to model the impingement of
a liquid drop on a solid wail. Accurate solutions are obtained
compared to analytical, numerical and experimental published
results. (C) 1997 Elsevier Science Ltd.
 Katsoulakis, MA, and Souganidis, PE, "Stochastic Ising models and anisotropic front propagation," JOURNAL OF STATISTICAL PHYSICS, vol. 87, pp. 6389, 1997.
Abstract:
We study Ising models with general spinflip dynamics obeying
the detailed balance law. After passing to suitable macroscopic
limits, we obtain interfaces moving with normal velocity
depending anisotropically on their principal curvatures and
direction. In addition we deduce ( directiondependent) Kubo
Greentype formulas for the mobility and the Hessian of the
surface tension, thus obtaining an explicit description of
anisotropy in terms of microscopic quantities. The choice of
dynamics affects only the mobility, a scalar function of the
direction.
 Xiao, F, Yabe, T, Ito, T, and Tajima, M, "An algorithm for simulating solid objects suspended in stratified flow," COMPUTER PHYSICS COMMUNICATIONS, vol. 102, pp. 147160, 1997.
Abstract:
An efficient difference algorithm for computing directly
deformation less solid objects suspended in stratified flow in
2D has been developed. The objects are represented by colour
functions (or density functions) and predicted by a sharpness
preserving scheme that is able to prevent the numerical
diffusion across the sharp interface, Pressure distribution is
then calculated by a unified solver and the solid object is
treated as a mass of material of high sound speed, The motion
of the solid object is decomposed into translation and
rotation, and the force as well as the torque that cause change
in the motion of the solid body are evaluated by an averaging
calculation over the region occupied by the solid body.
Calculations are conducted on a fixed grid system, Operations
for reconstructing moving interfaces or dealing with inner
boundary conditions are not necessary.
 Li, ZL, "Immersed interface methods for moving interface problems," NUMERICAL ALGORITHMS, vol. 14, pp. 269293, 1997.
Abstract:
A second order difference method is developed for the nonlinear
moving interface problem of the form u(t) + lambda uu(x) =
(beta u(x))(x)  f(x, t), x is an element of [0,alpha) boolean
OR (alpha, 1], d alpha/dt = w(t, alpha; u, u(x)), where
alpha(t) is the moving interface. The coefficient beta(x, t)
and the source term f(x, t) can be discontinuous across
alpha(t) and moreover, f(x, t) may have a delta or/and delta
prime function singularity there. As a result, although the
equation is parabolic, the solution u and its derivatives may
be discontinuous across alpha(t). Two typical interface
conditions are considered. One condition occurs in Stefanlike
problems in which the solution is known on the interface. A new
stable interpolation strategy is proposed. The other type
occurs in a onedimensional model of Peskin's immersed boundary
method in which only jump conditions are given across the
interface. The CrankNicolson difference scheme with
modifications near the interface is used to solve for the
solution u(x, t) and the interface alpha(t) simultaneously.
Several numerical examples, including models of icemelting and
glaciation, are presented. Second order accuracy on uniform
grids is confirmed both for the solution and the position of
the interface.
 Lindeberg, T, and Garding, J, "Shapeadapted smoothing in estimation of 3D shape cues from affine deformations of local 2D brightness structure," IMAGE AND VISION COMPUTING, vol. 15, pp. 415434, 1997.
Abstract:
This article describes a method for reducing the shape
distortions due to scale space smoothing that arise in the
computation of 3D shape cues using operators (derivatives)
defined from scalespace representation. More precisely, we are
concerned with a general class of methods for deriving 3D
shape cues from a 2D image data based on the estimation of
locally linearized deformations of brightness patterns. This
class constitutes a common framework for describing several
problems in computer vision (such as shapefromtexture, shape
from disparitygradients, and motion estimation) and for
expressing different algorithms in terms of similar types of
visual frontendoperations. It is explained how surface
orientation estimates will be biased due to the use of
rotationally symmetric smoothing in the image domain. These
effects can be reduced by extending the linear scalespace
concept into an affine Gaussian scalespace representation and
by performing affine shape adaptation of the smoothing kernels.
This improves the accuracy of the surface orientation
estimates, since the image descriptors, on which the methods
are based, will be relative invariant under affine
transformations, and the error thus confined to the higher
order terms in the locally linearized perspective
transformation. A straightforward algorithm is presented for
performing shape adaptation in practice. Experiments on real
and synthetic images with known orientation demonstrate that in
the presence of moderately high noise levels the accuracy is
improved by typically one order of magnitude.
 Hou, TY, Li, ZL, Osher, S, and Zhao, HK, "A hybrid method for moving interface problems with application to the HeleShaw flow," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 134, pp. 236252, 1997.
Abstract:
In this paper, a hybrid approach which combines the immersed
interface method with the level set approach is presented. The
fast version of the immersed interface method is used to solve
the differential equations whose solutions and their
derivatives may be discontinuous across the interfaces due to
the discontinuity of the coefficients or/and singular sources
along the interfaces. The moving interfaces then are updated
using the newly developed fast level set formulation which
involves computation only inside some small tubes containing
the interfaces. This method combines the advantage of the two
approaches and gives a secondorder Eulerian discretization for
interface problems. Several key steps in the implementation are
addressed in detail. This new approach is then applied to Hele
Shaw flow, an unstable flow involving two fluids with very
different viscosity. (C) 1997 Academic Press.
 Sussman, M, and Smereka, P, "Axisymmetric free boundary problems," JOURNAL OF FLUID MECHANICS, vol. 341, pp. 269294, 1997.
Abstract:
We present a number of threedimensional axisymmetric free
boundary problems for two immiscible fluids, such as air and
water. A level set method is used where the interface is the
zero level set of a continuous function while the two fluids
are solutions of the incompressible NavierStokes equation. We
examine the rise and distortion of an initially spherical
bubble into cap bubbles and toroidal bubbles. Steady solutions
for gas bubbles rising in a liquid are computed, with
favourable comparisons to experimental data. We also study the
inviscid limit and compare our results with a boundary integral
method. The problems of an air bubble bursting at a free
surface and a liquid drop hitting a free surface are also
computed.
 Chen, S, Merriman, B, Osher, S, and Smereka, P, "A simple level set method for solving Stefan problems," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 135, pp. 829, 1997.
Abstract:
A simple level set method for solving Stefan problems is
presented. This method can be applied to problems involving
dendritic solidification. Our method consists of an implicit
finite difference scheme for solving the heat equation and a
level set approach for capturing the front between solid and
liquid phases of a pure substance. Our method is accurate with
respect to some exact solutions of the Stefan problem. Results
indicate that this method can handle topology changes and
complicated interfacial shapes and that it can numerically
simulate many of the physical features of dendritic
solidification. (C) 1997 Academic Press.
 Kowalczyk, M, "Exponentially stow dynamics and interfaces intersecting the boundary," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 138, pp. 5585, 1997.
Abstract:
We introduce a relaxation model for front propagation problems.
Our proposed relaxation approximation is a semilinear
hyperbolic system without singularities. It yields a direction
dependent normal velocity at the leading term and captures, in
the ChapmanEnskog expansion, the higher order curvature
dependent corrections, including possible anisotropies. (C)
1997 Academic Press.
 Jin, S, and Katsoulakis, MA, "Relaxation approximations to front propagation," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 138, pp. 380387, 1997.
Abstract:
We introduce a relaxation model for front propagation problems.
Our proposed relaxation approximation is a semilinear
hyperbolic system without singularities. It yields a direction
dependent normal velocity at the leading term and captures, in
the ChapmanEnskog expansion, the higher order curvature
dependent corrections, including possible anisotropies. (C)
1997 Academic Press.
 Hsiau, ZK, Kan, EC, McVittie, JP, and Dutton, RW, "Robust, stable, and accurate boundary movement for physical etching and deposition simulation," IEEE TRANSACTIONS ON ELECTRON DEVICES, vol. 44, pp. 13751385, 1997.
Abstract:
The increasing complexity of VLSI device interconnect features
and fabrication technologies encountered by semiconductor
etching and deposition simulation necessitates improvements in
the robustness, numerical stability, and physical accuracy of
the boundary movement method, The volumemeshbased level set
method, integrated with the physical models in SPEEDIE,
demonstrates accuracy and robustness for simulations on a wide
range of etching/deposition processes The surface profile is
reconstructed from the wellbehaved level set function without
rulebased algorithms, Adaptive gridding is used to accelerate
the computation, Our algorithm can be easily extended from two
dimensional (2D) to threedimensional (3D), and applied to
model microstructures consisting of multiple materials,
Efficiency benchmarks show that this boundary movement method
is practical in 2D, and competitive for larger scale or 3D
modeling applications.
 Glimm, J, Kranzer, HC, Tan, D, and Tangerman, FM, "Wave fronts for HamiltonJacobi equations: The general theory for Riemann solutions in Rn," COMMUNICATIONS IN MATHEMATICAL PHYSICS, vol. 187, pp. 647677, 1997.
Abstract:
The HamiltonJacobi equation describes the dynamics of a
hypersurface in Rn. This equation is a nonlinear conservation
law and thus has discontinuous solutions. The dependent
variable is a surface gradient and the discontinuity is a
surface cusp. Here we investigate the intersection of cusp
hypersurfaces. These intersections define (n1)dimensional
Riemann problems for the HamiltonJacobi equation. We propose
the class of HamiltonJacobi equations as a natural higher
dimensional generalization of scalar equations which allow a
satisfactory theory of higherdimensional Riemann problems, The
first main result of this paper is a general framework for the
study of higherdimensional Riemann problems for Hamilton
Jacobi equations. The purpose of the framework is to understand
the structure of HamiltonJacobi wave interactions in an
explicit and constructive manner. Specialized to two
dimensional Riemann problems (i.e., the intersection of cusp
curves on surfaces embedded in R3), this framework provides
explicit solutions to a number of cases of interest. We are
specifically interested in models of deposition and etching,
important processes for the manufacture of semiconductor chips.
We also define elementary waves as Riemann solutions which
possess a common group velocity, Our second main result, for
elementary waves, is a complete characterization in terms of
algebraic constraints on the data. When satisfied, these
constraints allow a consistently defined closed form expression
for the solution. We also give a computable characterization
for the admissibility of an elementary wave which is inductive
in the codimension of the wave, and which generalizes the
classical Oleinik condition for scalar conservation laws in one
dimension.
 Cohen, LD, and Kimmel, R, "Global minimum for active contour models: A minimal path approach," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 24, pp. 5778, 1997.
Abstract:
A new boundary detection approach for shape modeling is
presented. It detects the global minimum of an active contour
model's energy between two end points. Initialization is made
easier and the curve is not trapped at a local minimum by
spurious edges. We modify the ''snake'' energy by including the
internal regularization term in the external potential term.
Our method is based on finding a path of minimal length in a
Riemannian metric. We then make use of a new efficient
numerical method to find this shortest path. It is shown that
the proposed energy, though based only on a potential
integrated along the curve, imposes a regularization effect
like snakes. We explore the relation between the maximum
curvature along the resulting contour and the potential
generated from the image. The method is capable to close
contours, given only one point on the objects' boundary by
using a topologybased saddle search routine. We show examples
of our method applied to real aerial and medical images.
 Kimmel, R, Kiryati, N, and Bruckstein, AM, "Analyzing and synthesizing images by evolving curves with the OsherSethian method," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 24, pp. 3755, 1997.
Abstract:
Numerical analysis of conservation laws plays an important role
in the implementation of curve evolution equations. This paper
reviews the relevant concepts in numerical analysis and the
relation between curve evolution, HamiltonJacobi partial
differential equations, and differential conservation laws.
This close relation enables us to introduce finite difference
approximations, based on the theory of conservation laws, into
curve evolution. It is shown how curve evolution serves as a
powerful tool for image analysis, and how these mathematical
relations enable us to construct efficient and accurate
numerical schemes. Some examples demonstrate the importance of
the CFL condition as a necessary condition for the stability of
the numerical schemes.
 Bellettini, G, "Some results on minimal barriers for geometric movements of sets," BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, vol. 11A, pp. 485512, 1997.
Abstract:
We discuss some properties of barriers and minimal barriers in
the sense of De Giorgi for geometric evolutions of subsets of
Rn We point out the role played by the regularizations
M*(E,f(H)), M*(E, f(H)) in the comparison with others
generalized geometric flows, and in the fattening phenomenon.
 Li, XL, and Zhang, Q, "A comparative numerical study of the RichtmyerMeshkov instability with nonlinear analysis in two and three dimensions," PHYSICS OF FLUIDS, vol. 9, pp. 30693077, 1997.
Abstract:
A shock driven interfacial instability, known as the
RichtmyerMeshkov instability, is studied numerically in two
and three dimensions and in the nonlinear regime. The numerical
solution is tested for convergence under computational mesh
refinement and is compared with the predictions of a recently
developed nonlinear theory based on the Pade approximation and
asymptotic matching, Good agreement has been found between
numerical solutions and predictions of the nonlinear theory in
both two and three dimensions and for both the reflected shock
and the reflected rarefaction wave cases. The numerical study
is extended to the reshock experiment in which the fluid
interface interacts initially with the incident shock. Later,
as the transmitted shock bounces back from the wall, the fluid
interface is reshocked. (C) 1997 American Institute of
Physics.
 Caselles, V, Kimmel, R, Sapiro, G, and Sbert, C, "Minimal surfaces: a geometric three dimensional segmentation approach," NUMERISCHE MATHEMATIK, vol. 77, pp. 423451, 1997.
Abstract:
A novel geometric approach for three dimensional object
segmentation is presented. The scheme is based on geometric
deformable surfaces moving towards the objects to be detected,
We show that this model is related to the computation of
surfaces of minimal area (local minimal surfaces). The space
where these surfaces are computed is induced from the three
dimensional image in which the objects are to be detected. The
general approach also shows the relation between classical
deformable surfaces obtained via energy minimization and
geometric ones derived from curvature flows in the surface
evolution framework. The scheme is stable, robust, and
automatically handles changes in the surface topology during
the deformation. Results related to existence, uniqueness,
stability, and correctness of the solution to this geometric
deformable model are presented as well. Based on an efficient
numerical algorithm for surface evolution, we present a number
of examples of object detection in real and synthetic images.
 Jayaraman, V, Udaykumar, HS, and Shyy, WS, "Adaptive unstructured grid for threedimensional interface representation," NUMERICAL HEAT TRANSFER PART BFUNDAMENTALS, vol. 32, pp. 247265, 1997.
Abstract:
Movingboundary problems arise in numerous important physical
phenomena, and often form complex shapes during their
evolution. The ability to track the interface in such cases in
two dimensions is well established. However, modifying the grid
representing the interface as it evolves in threedimensional
space introduces additional issues. In the current work, three
dimensional interfaces are represented by adaptive unstructured
grids. The grids are restructured and refined based on the
shape and size of the triangular elements in the grid that
forms the interfaces. As the interface deforms, points are
automatically added to ensure that the accuracy of interface
representation remains consistent. Results are presented to
show how complex interface features, including surface
curvatures and normals, can be captured by modifying an
existing method that uses an approximation to the Dupin
indicatrix.
 Ulitsky, M, and Collins, LR, "Application of the eddy damped quasinormal Markovian spectral transport theory to premixed turbulent flame propagation," PHYSICS OF FLUIDS, vol. 9, pp. 34103430, 1997.
Abstract:
The eddy damped quasinormal Markovian (EDQNM) turbulence
theory was applied to a modified KuramotoSivashinsky field
equation to develop a spectral model for investigating the
single and twopoint scalar statistics associated with a flame
front (treated as a passive scalar interface) propagating
through isotropic turbulence. As a result of the presence of a
uniform mean gradient in the scalar field, all correlations
involving the scalar were found to be functions of both the
wave number, k, and mu, the cosine of the angle between the ik
vector and the mean gradient vector. An infinite Legendre
expansion separated out the wave number and angle dependencies,
where the first term in each series accounted for the isotropic
contribution to the correlations and the higher order terms
accounted for the anisotropy introduced as a result of the mean
gradient. It was found that while strong anisotropy existed in
the scalar field at short times, at steady state the scalar
field became nearly isotropic. A parameter study was then
conducted to ascertain the effect of independently varying
u'/s(L) and Relambda (where u' is the rms velocity, s(L) is
the laminar burning velocity, and Relambda is the Reynolds
number based on the Taylor microscale). The turbulent burning
velocity increased with increases in either u'/s(L) or Re
lambda, however, the model predicted a finite turbulent burning
velocity as u'/s(L) > infinity, even though flame quenching
was not accounted for. This finite asymptote for the burning
velocity was traced to tile constitutive relationship used for
the flame thickness and the ratio of the Markstein length to
the flame thickness. It was also shown that the dominant
wrinkling of the flame surface and subsequent contribution to
the turbulent burning velocity occurred at smaller and smaller
length scales as the inertial range of the scalar spectrum
increased. Single point models will therefore have great
difficulty reproducing this significant result. Scalar spectra
exhibited changes over all wave numbers as either u'/s(L) or
Relambda, was modified. Transfer spectra, which arose in the
form of convolution integrals as a result of the advection and
propagation processes, were also analyzed and separated into
their pairwise spectral interactions to determine which
nonlinear terms in each integral were dominant. (C) 1997
American Institute of Physics.
 Udaykumar, HS, Kan, HC, Shyy, W, and TranSonTay, R, "Multiphase dynamics in arbitrary geometries on fixed Cartesian grids," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 137, pp. 366405, 1997.
Abstract:
in this work. a mixed EulerianLagrangian algorithm, called
ELAFINT (Eulerian Lagrangian algorithm for interface tracking
is developed further and applied to compute hows with solid
fluid and fluidfluid interfaces. The method is capable of
handling fluid Bows in the presence of both irregularly shaped
solid boundaries and moving boundaries on a fixed Cartesian
grid. The held equations are solved on the underlying fixed
grid using a collocated variable, pressurebased formulation.
The moving boundary is tracked explicitly the Lagrangian
translation of marker particles. The moving boundary passes
through the grid and the immersed boundary technique is used to
handle its interaction with the underlying grid. The internal
solid boundaries are dealt with by using a cutcell technique.
Particular attention is directed toward conservation and
consistency in the vicinity of both phase boundaries. The
complex geometry feature has been tested for a variety of Bow
problems. The performance of the immersed boundary
representation is demonstrated in the simulation of Newtonian
Liquid drops. The combination of the two features is then
employed in the simulation of motion of drops through
constricted tubes. The capabilities developed here can be
useful for solving Bow problems involving moving and stationary
complex boundaries. (C) 1997 Academic Press.
 Alvarez, L, and Morales, F, "Affine morphological multiscale analysis of corners and multiple junctions," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 25, pp. 95107, 1997.
Abstract:
In this paper we study the application of the Affine
Morphological Scale Space (AMSS) to the analysis of
singularities (corners or multiple junctions) of the shapes
present in a 2D image. We introduce a new family of travelling
wave solutions of AMSS which determines the evolution of the
initial shapes given by conics. We characterize the evolution
of corners accross the scales according to their angle, We
develop a numerical algorithm to compute AMSS accross the
scales and we present some experimental results about corners
and multiple junction detection.
 Sapiro, G, "Color snakes," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 68, pp. 247253, 1997.
Abstract:
A framework for object segmentation in vectorvalued images is
presented in this paper. The first scheme proposed is based on
geometric active contours moving toward the objects to be
detected in the vectorvalued image. Object boundaries are
obtained as geodesics or minimal weighteddistance curves,
where the metric is given by a definition of edges in vector
valued data. The curve flow corresponding to the proposed
active contours holds formal existence, uniqueness, stability,
and correctness results. The scheme automatically handles
changes in the deforming curve topology. The technique is
applicable, for example, to color and texture images as well as
multiscale representations. We then present an extension of
these vector active contours, proposing a possible image flow
for vectorvalued image segmentation. The algorithm is based on
moving each one of the image level sets according to the
proposed vector active contours. This extension also shows the
relation between active contours and a number of partial
differentialequationsbased image processing algorithms as
anisotropic diffusion and shock filters. (C) 1997 Academic
Press.
 Kimmel, R, "Intrinsic scale space for images on surfaces: The geodesic curvature flow," GRAPHICAL MODELS AND IMAGE PROCESSING, vol. 59, pp. 365372, 1997.
Abstract:
A scale space for images painted on surfaces is introduced.
Based on the geodesic curvature flow of the isogray level
contours of an image painted on the given surface, the image is
evolved and forms the natural geometric scale space. Its
geometrical properties are discussed as well as the intrinsic
nature of the proposed flow; i.e., the flow is invariant to the
bending of the surface. (C) 1997 Academic Press.
 Siddiqi, K, Kimia, BB, and Shu, CW, "Geometric shockcapturing ENO schemes for subpixel interpolation, computation and curve evolution," GRAPHICAL MODELS AND IMAGE PROCESSING, vol. 59, pp. 278301, 1997.
Abstract:
Subpixel methods that locate curves and their singularities,
and that accurately measure geometric quantities, such as
orientation and curvature, are of significant importance in
computer vision and graphics. Such methods often use local
surface fits or structural models for a local neighborhood of
the curve to obtain the interpolated curve. Whereas their
performance is good in smooth regions of the curve, it is
typically poor in the vicinity of singularities. Similarly, the
computation of geometric quantities is often regularized to
deal with noise present in discrete data. However, in the
process, discontinuities are blurred over, leading to poor
estimates at them and in their vicinity. In this paper we
propose a geometric interpolation technique to overcome these
limitations by locating curves and obtaining geometric
estimates while (1) not blurring across discontinuities and (2)
explicitly and accurately placing them, The essential idea is
to avoid the propagation of information across singularities.
This is accomplished by a onesided smoothing technique, where
information is propagated from the direction of the side with
the ''smoother'' neighborhood. When both sides are nonsmooth,
the two existing discontinuities are relieved by placing a
single discontinuity, or shock. The placement of shacks is
guided by geometric continuity constraints, resulting in
subpixel interpolation with accurate geometric estimates. Since
the technique was originally motivated by curve evolution
applications, we demonstrate its usefulness in capturing not
only smooth evolving curves, but also ones with orientation
discontinuities. In particular, the technique is shown to be
far better than traditional methods when multiple or entire
curves are present in a very small neighborhood. (C) 1997
Academic Press.
 Adalsteinsson, D, and Sethian, JA, "A level set approach to a unified model for etching, deposition, and lithography .3. Redeposition, reemission, surface diffusion, and complex simulations," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 138, pp. 193223, 1997.
Abstract:
Previously, Adalsteinsson and Sethian have applied the level
set formulation to the problem of surface advancement in two
and threedimensional topography simulation of deposition,
etching, and lithography processes in integrated circuit
fabrication. The level set formulation is based on solving a
HamiltonJacobi type equation for a propagating level set
function, using techniques borrowed from hyperbolic
conservation laws. Topological changes, corner, and cusp
development, and accurate determination of geometric properties
such as curvature and normal direction are naturally obtained
in this setting. Part I presented the basic equations and
algorithms for two dimensional simulations, including the
effects of isotropic and unidirectional deposition and
etching, visibility, reflection, and material dependent
etch/deposition rates. Part II focused on the extension to
three dimensions. This paper completes the series, and add the
effects of redeposition, reemission, and surface diffusion.
This requires the solution of the transport equations for
arbitrary geometries, and leads to simulations that contain
multiple simultaneous competing effects of visibility,
directional and source flux functions, complex sputter yield
flux functions, wide ranges of sticking coefficients for the
reemission and redeposition functions, multilayered fronts and
thin film layers. (C) 1997 Academic Press.
 Ansorge, R, and Sonar, T, "Information loss, abstract entropy and mathematical description of the second law of thermodynamics," ZEITSCHRIFT FUR ANGEWANDTE MATHEMATIK UND MECHANIK, vol. 77, pp. 803821, 1997.
Abstract:
We intend to discuss the relations between the snore abstract
notion of entropy and the ideas of information using nonlinear
partial differential equations as an example. Hyperbolic
equations of the first order where vanishing of the
characteristics in a discontinuity may be interpreted as loss
of initial information, will serve as illustration. The problem
of nonphysical rarefaction shocks in discussed. and with Lax'
shock condition a first entropy condition is introduced. The
approximation of a physical problem with friction by an
inviscid problem leads to a demand for an entropy inequality
characterizing the solutions of the model with friction in the
limit, when friction vanishes. This demand for an additional
condition, namely for an entropy inequality as mathematical
pendant to the second fundamental law also dominates the
numerics of the presented equations. A is only the fulfilment
of a discrete entropy condition  i.e. of a discretized form of
the second fundamental law  that provides for the convergence
of finite difference procedures.
 Aldredge, RC, "A fluxlimiting scheme for solution of the eikonal evolution equation," INTERNATIONAL COMMUNICATIONS IN HEAT AND MASS TRANSFER, vol. 24, pp. 11711176, 1997.
Abstract:
A finitedifference scheme is derived for numerical solution of
the eikonal scalarfield front propagation equation. The
numerical scheme is implemented for the description of the
dynamics of a flame surface propagating through a vortical flow
field. Attributes of the numerical scheme are its accuracy and
numerical stability when large velocity fluctuations in the
flow are present, and the absence of explicit artificial
viscosity. (C) 1997 Elsevier Science Ltd.
 Aldredge, RC, "An analytical model for flame propagation in lowMachnumber, variabledensity flow," INTERNATIONAL COMMUNICATIONS IN HEAT AND MASS TRANSFER, vol. 24, pp. 11631169, 1997.
Abstract:
A simple model problem is formulated to describe the coupling
between premixedflame and flowfield dynamics resulting from
gas expansion within the flame. The energy conservation
equation is integrated analytically across the flame in order
to reduce the number of governing equations for the
computational problem. A system of six equations and associated
boundary conditions are formulated for computation of the time
evolution of an initially prescribed threedimensional velocity
field and the flame surface. (C) 1997 Elsevier Science Ltd.
 Carmel, E, and CohenOr, D, "Warpguided objectspace morphing," VISUAL COMPUTER, vol. 13, pp. 465478, 1997.
Abstract:
We present an algorithm that builds a correspondence between
two arbitrary genus0 objects and generates a sequence of
inbetween objects. A warp function deforms the source object
and aligns it with the target object. An iterative polygon
evolution algorithm blurs the details of the warped source and
target objects into two convex objects with similar shapes that
are projected to two identical circles. Merging the topologies
of the projected objects and reconstructing the original
objects results in two objects with identical topologies. A
twopart transformation produces the morph sequence. The rigid
part moves and rotates the objects to their relative positions.
The elastic part establishes the position of each of the
vertices forming the inbetween object.
 McLaughlin, RM, and Zhu, JY, "The effect of finite front thickness on the enhanced speed of propagation," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 129, pp. 89112, 1997.
Abstract:
Recently, Majda and Souganidis have presented a rigorous
asymptotic theory governing the largescale renormalized flame
front dynamics for a reactiondiffusionadvection system
involving KPP type chemistries and small scale turbulence. This
theory is valid in the context of an infinitely thin reaction
layer. Embid, Majda and Souganidis have explored this rigorous
theory within the context of a shear layer flow geometry, and
demonstrate that the enhanced burning speed is sensitively
dependent upon the presence of a mean wind transverse to the
direction of the flame propagation. Here, we address the effect
that a thin reaction layer may have on the enhanced flame
propagation for the case of a small scale shear layer with and
without a transverse mean wind. We show through high resolution
numerical simulations that the enhanced burning speed is
sensitively dependent on the presence of a transverse mean wind
in a qualitatively similar fashion to the asymptotic theory;
but further, we exhibit that a finite reaction zone yields
effective burning speeds which are smaller than the theoretical
predictions for infinitely thin flame fronts and that the decay
of these corrections depends upon the relative scale separation
between the reaction layer scale, the turbulence scale, and the
integral scale.
 Souganidis, PE, "Front propagation: Theory and applications," VISCOSITY SOLUTIONS AND APPLICATIONS, LECTURE NOTES IN MATHEMATICS, vol. 1660, pp. 186242, 1997.
Abstract:
A level set method for capturing the interface between two
fluids is combined with a variable density projection method to
allow for computation of a twophase flow where the interface
can merge/break and the flow can have a high Reynolds number. A
distance function formulation of the level set method enables
us to compute flows with large density ratios (1000/1) and
flows that are surface tension driven, with no emotional
involvement. Recent work has improved the accuracy of the
distance function formulation and the accuracy of the advection
scheme. We compute flows involving air bubbles and water drops,
among others. We validate our code against experiments and
theory. (C) 1998 Elsevier Science Ltd. All rights reserved.
 Sapiro, G, and Caselles, V, "Contrast enhancement via image evolution flows," GRAPHICAL MODELS AND IMAGE PROCESSING, vol. 59, pp. 407416, 1997.
Abstract:
A framework for contrast enhancement via image evolution hows
and variational formulations is introduced in this paper.
First, an algorithm for histogram modification via image
evolution equations is presented. We show that the image
histogram can be modified to achieve any given distribution as
the steady state solution of this differential equation. We
then prove that the proposed evolution equation solves an
energy minimization problem. This gives a new interpretation to
histogram modification and contrast enhancement in general.
This interpretation is completely formulated in the image
domain, in contrast with classical techniques for histogram
modification which are formulated in a probabilistic domain.
From this, new algorithms for contrast enhancement, including,
for example, image and perception models, can be derived, Based
on the energy formulation and its corresponding differential
form, we show that the proposed histogram modification
algorithm can be combined with image regularization schemes,
This allows us to perform simulations contrast enhancement and
denoising, avoiding common noise sharpening effects in
classical schemes, Theoretical results regarding the existence
of solutions to the proposed equations are presented. (C) 1997
Academic Press.


1998 
 Gundlach, C, "Pseudospectral apparent horizon finders: An efficient new algorithm," PHYSICAL REVIEW D, vol. 57, pp. 863875, 1998.
Abstract:
We review the problem of finding an apparent horizon in Cauchy
data (Sigma,g(ab),Kab) in three space dimensions without
symmetries. We describe a family of algorithms which includes
the pseudospectral apparent horizon finder of Nakamura et al.
and the curvature flow method proposed by Tod as special cases.
We suggest that other algorithms in the family may combine the
speed of the former with the robustness of the latter. A
numerical implementation for Cauchy data given on a grid in
Cartesian coordinates is described, and tested on Brill
Lindquist and Kerr initial data. The new algorithm appears
faster and more robust than previous ones.
 Franzone, PC, Guerri, L, Pennacchio, M, and Taccardi, B, "Spread of excitation in 3D models of the anisotropic cardiac tissue. II. Effects of fiber architecture and ventricular geometry," MATHEMATICAL BIOSCIENCES, vol. 147, pp. 131171, 1998.
Abstract:
We investigate a threedimensional macroscopic model of wave
front propagation related to the excitation process in the left
ventricular wall represented by an anisotropic bidomain. The
whole left ventricle is modeled, whereas, in a previous paper,
only a flat slab of myocardial tissue was considered. The
direction of cardiac fibers, which affects the anisotropic
conductivity of the myocardium, rotates from the epito the
endocardium. If the ventricular wall is conceived as a set of
packed surfaces, the fibers may be tangent to them or more
generally may cross them obliquely; the latter case is
described by an "imbrication angle." The effect of a simplified
Purkinje network also is investigated. The cardiac excitation
process, more particularly the depolarization phase, is modeled
by a nonlinear elliptic equation, called an eikonal equation,
in the activation time. The numerical solution of this equation
is obtained by means of the finite element method, which
includes an upwind treatment of the Hamiltonian part of the
equation. By means of numerical simulations in an idealized
model of the left ventricle, we try to establish whether the
eikonal approach contains the essential basic elements for
predicting the features of the activation patterns
experimentally observed. We discuss and compare these results
with those obtained in our previous papers [1,2] for a flat
part of myocardium. The general rules governing the spread of
excitation after local stimulations, previously delineated for
the flat geometry, are extended to the present, more realistic
monoventricular model. (C) 1998 Elsevier Science Inc.
 Lorigo, LM, Faugeras, O, Grimson, WEL, Keriven, R, and Kikinis, R, "Segmentation of bone in clinical knee MRI using texturebased geodesic active contours," MEDICAL IMAGE COMPUTING AND COMPUTERASSISTED INTERVENTION  MICCAI'98, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1496, pp. 11951204, 1998.
Abstract:
This paper presents a method for automatic segmentation of the
tibia and femur in clinical magnetic resonance images of knees.
Texture information is incorporated into an active contours
framework through the use of vectorvalued geodesic snakes with
local variance as a second value at each pixel, in addition to
intensity. This additional information enables the system to
better handle noise and the nonuniform intensities found
within the structures to be segmented. It currently operates
independently on 2D images (slices of a volumetric image) where
the initial contour must be within the structure but not
necessarily near the boundary. These separate segmentations are
stacked to display the performance on the entire 3D structure.
 Jain, AK, Zhong, Y, and DubuissonJolly, MP, "Deformable template models: A review," SIGNAL PROCESSING, vol. 71, pp. 109129, 1998.
Abstract:
In this paper, we review the recently published work on
deformable models. We have chosen to concentrate on 2D
deformable models and relate the energy minimization approaches
to the Bayesian formulations. We categorize the various active
contour systems according to the definition of the deformable
model. We also present in detail one particular formulation for
deformable templates which combines edge, texture, color and
region information for the external energy and model
deformations using wavelets, splines or Fourier descriptors. We
explain how these models can be used for segmentation, image
retrieval in a large database and object tracking in a video
sequence. (C) 1998 Elsevier Science B.V. All rights reserved.
 Gyure, MF, Ratsch, C, Merriman, B, Caflisch, RE, Osher, S, Zinck, JJ, and Vvedensky, DD, "Levelset methods for the simulation of epitaxial phenomena," PHYSICAL REVIEW E, vol. 58, pp. R6927R6930, 1998.
Abstract:
We introduce a model for epitaxial phenomena based on the
motion of island boundaries, which is described by the level
set method. Our model treats the growing film as a continuum in
the lateral direction, but retains atomistic discreteness in
the growth direction. An example of such an "island dynamics"
model using the levelset method is presented and compared with
the corresponding rate equation description. Extensions of our
methodology to more general settings are then discussed.
[S1063651X(98)502127].
 Li, J, Renardy, YY, and Renardy, M, "A numerical study of periodic disturbances on twolayer Couette flaw," PHYSICS OF FLUIDS, vol. 10, pp. 30563071, 1998.
Abstract:
The flow of two viscous liquids is investigated numerically
with a volume of fluid scheme. The scheme incorporates a semi
implicit Stokes solver to enable computations at low Reynolds
numbers, and a secondorder velocity interpolation. The code is
validated against linear theory for the stability of twolayer
Couette flow, and weakly nonlinear theory for a Hopf
bifurcation. Examples of longtime wave saturation are shown.
The formation of fingers for relatively small initial
amplitudes as well as larger amplitudes are presented in two
and three dimensions as initialvalue problems. Fluids of
different viscosity and density are considered, with an
emphasis on the effect of the viscosity difference. Results at
low Reynolds numbers show elongated fingers in two dimensions
that break in three dimensions to form drops, while different
topological changes take place at higher Reynolds numbers. (C)
1998 American Institute of Physics. [S10706631(98)006126].
 Jia, W, "An accurate semiLagrangian scheme designed for incompressible NavierStokes equations written in generalized coordinates," TRANSACTIONS OF THE JAPAN SOCIETY FOR AERONAUTICAL AND SPACE SCIENCES, vol. 41, pp. 105117, 1998.
Abstract:
An accurate SemiLagrangian (SL) scheme for incompressible
NavierStokes equations written in generalized coordinates was
developed. The scheme explicitly calculates the advection phase
of the governing equations by a newly developed SL transport
model and the viscous term by a modified CrankNicholson
method. Several numerical schemes approximating the trajectory
and the unknowns at the upstream departure point have been
verified to construct a wellbalanced, totally accurate
numerical solver. Employing the contravariant velocity, the
scheme directly predicates the generalized coordinates of the
upstream departure point by the fourstage fourth order Runge
Kutta method. The velocity at the departure point is
interpolated with the third order accuracy. Unlike the
traditional Eulerian schemes, the scheme allows a large time
step length free from the CFL condition while keeping accuracy.
The unsteady flows around a 2D circular cylinder are accurately
predicted in 1/6 to 1/3 of the CPU times required for the
traditional Eulerian solvers. The proposed scheme is expected
to replace the traditional Eulerian solvers to quickly
investigate the practical flow problems.
 Niessen, WJ, Romeny, BMT, and Viergever, MA, "Geodesic deformable models for medical image analysis," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 17, pp. 634641, 1998.
Abstract:
In this paper implicit representations of deformable models for
medical image enhancement and segmentation are considered. The
advantage of implicit models over classical explicit models is
that their topology can he naturally adapted to objects in the
scene, A geodesic formulation of implicit deformable models is
especially attractive since it has the energy minimizing
properties of classical models, The aim of this pager is
twofold, First, a modification to the customary geodesic
deformable model approach is introduced by considering all the
level sets in the image as energy minimizing contours. This
approach is used to segment multiple objects simultaneously and
for enhancing and segmenting cardiac computed tomography (CT)
and magnetic resonance images. Second, the approach is used to
effectively compare implicit and explicit models for specific
tasks. This shows the complementary character of implicit
models since in case of poor contrast boundaries or gaps in
boundaries e.g. due to partial volume effects, noise, or motion
artifacts, they do not perform well, since the approach is
completely datadriven.
 Whitaker, RT, "A levelset approach to 3D reconstruction from range data," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 29, pp. 203231, 1998.
Abstract:
This paper presents a method that uses the level sets of
volumes to reconstruct the shapes of 3D objects from range
data. The strategy is to formulate 3D reconstruction as a
statistical problem: find that surface which is mostly likely,
given the data and some prior knowledge about the application
domain. The resulting optimization problem is solved by an
incremental process of deformation. We represent a deformable
surface as the level set of a discretely sampled scalar
function of three dimensions, i.e., a volume. Such levelset
models have been shown to mimic conventional deformable surface
models by encoding surface movements as changes in the
greyscale values of the volume. The result is a voxelbased
modeling technology that offers several advantages over
conventional parametric models, including flexible topology, no
need for reparameterization, concise descriptions of
differential structure, and a natural scale space for
hierarchical representations. This paper builds on previous
work in both 3D reconstruction and levelset modeling. It
presents a fundamental result in surface estimation from range
data: an analytical characterization of the surface that
maximizes the posterior probability. It also presents a novel
computational technique for levelset modeling, called the
sparsefield algorithm, which combines the advantages of a
levelset approach with the computational efficiency and
accuracy of a parametric representation. The sparsefield
algorithm is more efficient than other approaches, and because
it assigns the level set to a specific set of grid points, it
positions the levelset model more accurately than the grid
itself. These properties, computational efficiency and subcell
accuracy, are essential when trying to reconstruct the shapes
of 3D objects. Results are shown for the reconstruction objects
from sets of noisy and overlapping range maps.
 Lowengrub, J, and Truskinovsky, L, "Quasiincompressible CahnHilliard fluids and topological transitions," PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON SERIES A MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES, vol. 454, pp. 26172654, 1998.
Abstract:
One of the fundamental problems in simulating the motion of
sharp interfaces between immiscible fluids is a description of
the transition that occurs when the interfaces merge and
reconnect. It is well known that classical methods involving
sharp interfaces fail to describe this type of phenomena.
Following some previous work in this area, we suggest a
physically motivated regularization of the Euler equations
which allows topological transitions to occur smoothly. In this
model, the sharp interface is replaced by a narrow transition
layer across which the fluids may mix. The model describes a
flow of a binary mixture, and the internal structure of the
interface is determined by both diffusion and motion. An
advantage of our regularization is that it automatically yields
a continuous description of surface tension, which can play an
important role in topological transitions. An additional scalar
held is introduced to describe the concentration of one of the
fluid components and the resulting system of equations couples
the Euler (or NavierStokes) and the CahnHilliard equations.
The model takes into account weak nonlocality (dispersion)
associated with an internal length scale and localized
dissipation due to mixing. The nonlocality introduces a
dimensional surface energy; dissipation is added to handle the
loss of regularity of solutions to the sharp interface
equations and to provide a mechanism for topological changes.
In particular, we study a nontrivial limit when both components
are incompressible, the pressure is kinematic but the velocity
field is nonsolenoidal (quasiincompressibility). To
demonstrate the effects of quasiincompressibility, we analyse
the linear stage of spinodal decomposition in one dimension. We
show that when the densities of the fluids are not perfectly
matched, the evolution of the concentration field causes fluid
motion even if the fluids are inviscid. In the limit of
infinitely thin and wellseparated interfacial layers, an
appropriately scaled quasiincompressible EulerCahnHilliard
system converges to the classical sharp interface model. In
order to investigate the behaviour of the model outside the
range of parameters where the sharp interface approximation is
sufficient, we consider a simple example of a change of
topology and show that the model permits the transition to
occur without an associated singularity.
 Ishii, K, and Soner, HM, "Regularity and convergence of crystalline motion," SIAM JOURNAL ON MATHEMATICAL ANALYSIS, vol. 30, pp. 1937, 1998.
Abstract:
We consider the motion of polygons by crystalline curvature. We
show that "smooth" polygon evolves by crystalline curvature
"smoothly" and that it shrinks to a point in finite time. We
also establish the convergence of crystalline motion to the
motion by mean curvature.
 Li, CL, and Cheng, YY, "Inversion of a homogeneous cylinder of arbitrary shape by genetic algorithm and shape mutation," MICROWAVE AND OPTICAL TECHNOLOGY LETTERS, vol. 19, pp. 188192, 1998.
Abstract:
A computational method combining the genetic algorithm (GA) and
shape mutation is reported for electromagnetic imaging of a
homogeneous cylinder of arbitrary shape. By measuring the
scattered field, the shape location, size, and permittivity of
the object are retrieved quite successfully. The forward
problem is solved based on the equivalent source current and
the method of moments (MoM) while the inverse problem is
reformulated into an optimization one, and is solved by the
proposed scheme. Numerical simulation shows that, even with a
bad initial guess, good reconstruction can be obtained for a
hollow object or multiplecylinder object as long as the noise
level is less than or equal to 20 dB. (C) 1998 John Wiley &
Sons, Inc.
 Lock, N, Jaeger, M, Medale, M, and Occelli, R, "Local mesh adaptation technique for front tracking problems," INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, vol. 28, pp. 719736, 1998.
Abstract:
A numerical model is developed for the simulation of moving
interfaces in viscous incompressible flows. The model is based
on the finite element method with a pseudoconcentration
technique to track the front. Since a Eulerian approach is
chosen, the interface is advected by the flow through a fixed
mesh. Therefore, material discontinuity across the interface
cannot be described accurately. To remedy this problem, the
model has been supplemented with a local mesh adaptation
technique. This latter consists in updating the mesh at each
time step to the interface position, such that element
boundaries lie along the front. It has been implemented for
unstructured triangular finite element meshes. The outcome of
this technique is that it allows an accurate treatment of
material discontinuity across the interface and, if necessary,
a modelling of interface phenomena such as surface tension by
using specific boundary elements. For illustration, two
examples are computed and presented in this paper: the broken
dam problem and the RayleighTaylor instability. Good agreement
has been obtained in the comparison of the numerical results
with theory or available experimental data. (C) 1998 John Wiley
& Sons, Ltd.
 Barth, TJ, and Sethian, JA, "Numerical schemes for the HamiltonJacobi and level set equations on triangulated domains," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 145, pp. 140, 1998.
Abstract:
Borrowing from techniques developed for conservation law
equations, numerical schemes which discretize the Hamilton
Jacobi (HJ), level set, and Eikonal equations on triangulated
domains are presented. The first scheme is a provably monotone
discretization for the HJ equations. Unfortunately, the basic
scheme lacks Lipschitz continuity of the numerical Hamiltonian.
By employing a "virtual" edge flipping technique local
Lipschitz continuity of the numerical flux is restored on acute
triangulations. Next, schemes are introduced and developed
based on the weaker concept of positive coefficient
approximations for homogeneous Hamiltonians. These schemes
possess a discrete maximum principle on arbitrary
triangulations and exhibit local Lipschitz continuity of the
numerical Hamiltonian. Finally, a class of PetrovGalerkin
approximations is considered. These schemes are stabilized via
a leastsquares bilinear form. The PetrovGalerkin schemes do
not possess a discrete maximum principle but generalize to high
order accuracy. Discretization of the level set equation also
requires the numerical approximation of a mean curvature term.
A simple masslumped Galerkin approximation is presented in
Section 6 and analyzed using maximum principle analysis. The
use of unstructured meshes permits several forms of mesh
adaptation which have been incorporated into numerical
examples. These numerical examples include discretizations of
convex and nonconvex forms of the HJ equation, the Eikonal
equation, and the level set equation. (C) 1998 Academic Press.
 Harabetian, E, and Osher, S, "Regularization of illposed problems via the level set approach," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 58, pp. 16891706, 1998.
Abstract:
We introduce a new formulation for the motion of curves in R2
(easily extendable to the motion of surfaces in R3), when the
original motion generally corresponds to an illposed problem
such as the CauchyRiemann equations. This is, in part, a
generalization of our earlier work in [6], where we applied
similar ideas to compute flows with highly concentrated
vorticity, such as vortex sheets or dipoles, for incompressible
Euler equations. Our new formulation involves extending the
level set method of [12] to problems in which the normal
velocity is not intrinsic. We obtain a coupled system of two
equations, one of which is a level surface equation. This
yields a fixedgrid, Eulerian method which regularizes the ill
posed problem in a topological fashion. We also present an
analysis of curvature regularizations and some other
theoretical justification. Finally, we present numerical
results showing the stability properties of our approach and
the novel nature of the regularization, including the
development of bubbles for curves evolving under CauchyRiemann
flow.
 Zhai, J, "Heat flow with tangent penalisation converging to mean curvature motion," PROCEEDINGS OF THE ROYAL SOCIETY OF EDINBURGH SECTION A MATHEMATICS, vol. 128, pp. 875894, 1998.
Abstract:
In this paper, we prove that mean curvature motion can be
regarded as the singular limit of the following model:
[GRAPHICS] where epsilon > 0 is a Small parameter and W(u) =(1
 u(1)(2))(2)(1u(2)(2))(2) + 1. This model is related to the
LandauLifshitz equation of ferromagnetism. Local existence of
classical solutions of the Dirichlet problem and global
existence of the travelling wave solutions are also obtained.
 Xiao, F, and Ebisuzaki, T, "An efficient numerical model for multiphase fluid dynamics," ADVANCES IN ENGINEERING SOFTWARE, vol. 29, pp. 345352, 1998.
Abstract:
This work presents some newly developed efficient numerical
schemes and techniques for the dynamics of multiphase flows.
In the first part of this paper a semiLagrangian method for an
advection equation, as an important part of our numerical
model, is introduced. The scheme is constructed from a rational
function and proved to be convexity preserving. It has third
order accuracy in the smooth region and possesses an
oscillation suppressing property near discontinuities or steep
gradients. There is no need to calculate the slope limiter to
eliminate numerical oscillation as other high resolution
schemes do. The paper also discusses in the second part some
numerical algorithms that prove important to the establishment
of an efficient and robust code for simulating the dynamics of
multimaterial flows, such as the calculation of a moving
boundary, the unified procedure for evaluating pressure over
different materials and the equivalent volume force
formulations for different types of forces. (C) 1998 Elsevier
Science Ltd.
 Ruuth, SJ, "Efficient algorithms for diffusiongenerated motion by mean curvature," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 144, pp. 603625, 1998.
Abstract:
The problem of simulating the motion of evolving surfaces with
junctions according to some curvaturedependent speed arises in
a number of applications. By alternately diffusing and
sharpening characteristic functions for each region, a variety
of motions have been obtained which allow for topological
mergings and breakings and produce no overlapping regions or
vacuums. However, the usual finite difference discretization of
these methods is often excessively slow when accurate solutions
are sought, even in two dimensions. Vie propose a new, spectral
discretization of these diffusiongenerated methods which
obtains greatly improved efficiency over the usual finite
difference approach. These efficiency gains are obtained, in
part through the use of a quadraturebased refinement
technique, by integrating Fourier modes exactly and by
neglecting the contributions of rapidly decaying solution
transients. Indeed, numerical studies demonstrate that the new
algorithm is often more than 1000 times faster than the usual
finite difference discretization. Our findings are demonstrated
on several examples. (C) 1998 Academic Press.
 Kawata, Y, Niki, N, Ohmatsu, H, Kakinuma, R, Eguchi, K, Kaneko, M, and Moriyama, N, "Quantitative surface characterization of pulmonary nodules based on thinsection CT images," IEEE TRANSACTIONS ON NUCLEAR SCIENCE, vol. 45, pp. 21322138, 1998.
Abstract:
Characterization of pulmonary nodules plays a significant role
in the differential diagnosis of lung cancer. This paper
presents a method to quantify surface characteristics of small
pulmonary nodules with welldefined surface based on thin
section CT images. The segmentation of the threedimensional
(3D) nodule images are obtained by a 3D deformable surfaces
approach. The feature extraction algorithms are designed to
quantify the surface characteristic parameters from 3D nodule
images by using surface curvatures and ridge lines.
Experimental results of our method, applied to patients 3D
nodule images, demonstrate it performance.
 Qian, J, Tryggvason, G, and Law, CK, "A front tracking method for the motion of premixed flames," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 144, pp. 5269, 1998.
Abstract:
A front tracking method to study multifluid flows in which a
sharp interface separates incompressible fluids of different
densities and viscosities is adopted to simulate the unsteady
motion of an infinitely thin premixed name characterized by
significant chemical heat release and hence thermal expansion.
The How field is discretized by a conservative finite
difference approximation on a stationary grid, and the flame
surface is explicitly represented by connected marker points
that move with the local flame speed, relative to the flow
field. The performance of the method is tested by applying it
to a steady planar name and the DarrieusLandau instability.
The numerical results are in good agreement with analytical
results. The method is also applied to the interaction between
a name and a vortex array. The results show that the name can
destroy the vorticity originally in the unburnt gas and
generate vorticity of opposite sign in the burnt gas. (C) 1998
Academic Press.
 Canic, S, and Mirkovic, D, "A numerical study of Riemann problems for the twodimensional unsteady transonic small disturbance equation," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 58, pp. 13651393, 1998.
Abstract:
We study a twoparameter family of Riemann problems for the
unsteady transonic small disturbance (UTSD) equation, also
called the twodimensional Burgers equation. The two
parameters, a and b, which define oblique shock initial data,
correspond to the slopes of the initial shock waves in the
upper halfplane. For each a and b, the three constant states
in the upper halfplane satisfy the RankineHugoniot conditions
across the shocks. This leads to a twoparameter family of
oblique shock interaction problems. In this paper we present a
numerical study of global solution behavior for the values of a
and b in a previously obtained bifurcation diagram. Our study
supplements the related theoretical results and conjectures
recently obtained by S. Canic and B. L. Keyfitz. We employ a
high resolution numerical method which reveals fine solution
structures. Our findings confirm theoretical results and
conjectures about the solution patterns and deepen the
understanding of the structure of several intricate wave
interactions arising in this model.
 Gunther, J, Thomas, EL, Clingman, S, and Ober, CK, "Curvature driven relaxation of disclination loops in liquid crystals," POLYMER, vol. 39, pp. 44974503, 1998.
Abstract:
Relaxation of disclination loops created during shear flow of a
low molar mass and a polymer liquid crystal were monitored
using a special shear stage with a videomicroscope. Loops in
the polymer system generally displayed initially highly
distorted contours. In the small molecule liquid crystal, the
loop contours consistently exhibited very simple, generally
convex shapes. In the polymer system, the complex line shape
reflects the many prior looploop coalescence events due to the
greater density of loops than in the small molecule system.
Sequential images of loops were analysed to determine the
velocity of the disclination loops as a function of the local
curvature. Observations and simulations indicate that local
disclination line curvature is a driving force in loop
evolution. The reduction of regions of high loop curvature is
inherently slower in the polymer liquid crystal due to the
higher viscosity. In addition, the motion of the disclination
contour at very high values of curvature in the polymer liquid
crystal is slowed due to the presence of lower molecular weight
components at the defect core which themselves must diffuse
along with the line defect. (C) 1998 Elsevier Science Ltd. All
rights reserved.
 Sharp, NG, and Hancock, ER, "Density propagation for surface tracking," PATTERN RECOGNITION LETTERS, vol. 19, pp. 177188, 1998.
Abstract:
This paper describes a novel approach to surface tracking in
volumetric image stacks. It draws on a statistical model of the
uncertainties inherent in the characterisation of feature
contours to compute an evidential field for putative inter
frame displacements. This field is computed using Gaussian
density kernels which are parameterised in terms of the
variancecovariance matrices for contour displacement. The
underlying variance model accommodates the effects of raw image
noise on the estimated surface normals. The evidential field
effectively couples contour displacements to the intensity
features on successive frames through a statistical process of
contour tracking. Hard contours are extracted using a
dictionarybased relaxation process. The method is evaluated on
both MRI data and simulated data. (C) 1998 Elsevier Science
B.V. All rights reserved.
 Zhang, H, Zheng, LL, Prasad, Y, and Hou, TY, "A curvilinear level set formulation for highly deformable free surface problems with application to solidification," NUMERICAL HEAT TRANSFER PART BFUNDAMENTALS, vol. 34, pp. 120, 1998.
Abstract:
A curvilinear level set formulation has been developed for
highly deformable free surface problems. In this new scheme,
the grid lines follow the irregular domain generated by the
multizone adaptive grid generation (MAGG) scheme [1] and free
surfaces are captured by level set functions among the
curvilinear grids. NavierStokes equations are discretized and
solved based on a multiphase curvilinear finitevolume method
[2], and the level set function is solved based on a finite
difference method using the secondorder essentially
nonoscillatory (ENO) scheme [3]. The scheme is capable of
accurately and efficiently representing the deformation,
oscillation, merger, and separation of free surfaces. The
effectiveness and robustness of the algorithm are demonstrated
by using it for problems involving merger of bubbles, mold
filling, and the spreading and solidification of molten
droplets on a cold substrate where both free surface and
solidification interfaces move and the mass of the liquid phase
is continuously decreased.
 Rougon, N, and Preteux, F, "Directional adaptive deformable models for segmentation," JOURNAL OF ELECTRONIC IMAGING, vol. 7, pp. 231256, 1998.
Abstract:
We address the problem of adapting the functions controlling
the material properties of 2D snakes, and show how introducing
oriented smoothness constraints results in a novel class of
active contour models for segmentation, which extends standard
isotropic inhomogeneous membrane/thinplate stabilizers. These
constraints, expressed as adaptive L2 matrix norms, are
defined by two secondorder symmetric and positive definite
tensors that are invariant with respect to rigid motions in the
image plane. These tensors, equivalent to directional adaptive
stretching and bending densities, are quadratic with respect to
first and secondorder derivatives of the image luminance,
respectively. A representation theorem specifying their
canonical form is established and a geometrical interpretation
of their effects is developed. Within this framework, it is
shown that by achieving a directional control of regularization
such nonisotropic constraints consistently relate the
differential properties (metric and curvature) of the
deformable model with those of the underlying luminance
surface, yielding a satisfying preservation of image contour
characteristics. In particular, this model adapts to
nonstationary curvature variations along image contours to be
segmented, thus providing a consistent solution to curvature
underestimation problems encountered near high curvature
contour points by classical snakes evolving with constant
material parameters. Optimization of the model within
continuous and discrete frameworks is discussed in detail.
Finally, accuracy and robustness of the model are established
on synthetic images. Its efficacy is further demonstrated on 2
D MRI sequences for which comparisons with segmentations
obtained using classical snakes are provided. (C) 1998 SPIE and
IS&T. [S10179909(98)021011].
 Zhao, HK, Merriman, B, Osher, S, and Wang, L, "Capturing the behavior of bubbles and drops using the variational level set approach," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 143, pp. 495518, 1998.
Abstract:
We reproduce the general behavior of complicated bubble and
droplet motions using the variational level set formulation
introduced by the authors earlier. Our approach here ignores
inertial effects; thus the motion is only correct as an
approximation for very viscous problems. However, the steady
states are true equilibrium solutions. Inertial forces will be
added in future work. The problems include: soap bubbles
colliding and merging, drops falling or remaining attached to a
(generally irregular) ceiling, and liquid penetrating through a
funnel in both two and three dimensions. Each phase is
identified with a particular "level set" function. The zero
level set of this function is that particular phase boundary.
The level set functions all evolve in time through a
constrained gradient descent procedure so as to minimize an
energy functional. The functions are coupled through physical
constraints and through the requirements that different phases
do not overlap and vacuum regions do not develop. Both boundary
conditions and inequality constraints are cast in terms of
(either local or global) equality constraints. The gradient
projection method leads to a system of perturbed (by curvature,
if surface tension is involved) HamiltonJacobi equations
coupled through a constraint. The coupling is enforced using
the Lagrange multiplier associated with this constraint. The
numerical implementation requires much of the modem level set
technology; in particular, we achieve a significant speed up by
using the fast localization algorithm of H.K. Zhao, M. Kang,
B. Merriman, D. Peng, and S. Osher. (C) 1998 Academic Press.
 Litman, A, Lesselier, D, and Santosa, F, "Reconstruction of a twodimensional binary obstacle by controlled evolution of a levelset," INVERSE PROBLEMS, vol. 14, pp. 685706, 1998.
Abstract:
We are concerned with the retrieval of the unknown cross
section of a homogeneous cylindrical obstacle embedded in a
homogeneous medium and illuminated by timeharmonic
electromagnetic line sources. The dielectric parameters of the
obstacle and embedding materials are known and piecewise
constant. That is, the shape (here, the contour) of the
obstacle is sufficient for its full characterization. The
inverse scattering problem is then to determine the contour
from the knowledge of the scattered field measured for several
locations of the sources and/or frequencies. An iterative
process is implemented: given an initial contour, this contour
is progressively evolved such as to minimize the residual in
the data fit. This algorithm presents two main important
points. The first concerns the choice of the transformation
enforced on the contour. We will show that this involves the
design of a velocity field whose expression only requires the
resolution of an adjoint problem at each step. The second
concerns the use of a levelset function in order to represent
the obstacle. This levelset function will be of great use to
handle in a natural way splitting or merging of obstacles along
the iterative process. The evolution of this levelset is
controlled by a HamiltonJacobitype equation which will be
solved by using an appropriate finitedifference scheme.
Numerical results of inversion obtained from both noiseless and
noisy synthetic data illustrate the behaviour of the algorithm
for a variety of obstacles.
 Angenent, S, Sapiro, G, and Tannenbaum, A, "On the affine heat equation for nonconvex curves," JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 11, pp. 601634, 1998.
Abstract:
We are concerned with the retrieval of the unknown cross
section of a homogeneous cylindrical obstacle embedded in a
homogeneous medium and illuminated by timeharmonic
electromagnetic line sources. The dielectric parameters of the
obstacle and embedding materials are known and piecewise
constant. That is, the shape (here, the contour) of the
obstacle is sufficient for its full characterization. The
inverse scattering problem is then to determine the contour
from the knowledge of the scattered field measured for several
locations of the sources and/or frequencies. An iterative
process is implemented: given an initial contour, this contour
is progressively evolved such as to minimize the residual in
the data fit. This algorithm presents two main important
points. The first concerns the choice of the transformation
enforced on the contour. We will show that this involves the
design of a velocity field whose expression only requires the
resolution of an adjoint problem at each step. The second
concerns the use of a levelset function in order to represent
the obstacle. This levelset function will be of great use to
handle in a natural way splitting or merging of obstacles along
the iterative process. The evolution of this levelset is
controlled by a HamiltonJacobitype equation which will be
solved by using an appropriate finitedifference scheme.
Numerical results of inversion obtained from both noiseless and
noisy synthetic data illustrate the behaviour of the algorithm
for a variety of obstacles.
 Sussman, M, Fatemi, E, Smereka, P, and Osher, S, "An improved level set method for incompressible twophase flows," COMPUTERS & FLUIDS, vol. 27, pp. 663680, 1998.
Abstract:
A level set method for capturing the interface between two
fluids is combined with a variable density projection method to
allow for computation of a twophase flow where the interface
can merge/break and the flow can have a high Reynolds number. A
distance function formulation of the level set method enables
us to compute flows with large density ratios (1000/1) and
flows that are surface tension driven, with no emotional
involvement. Recent work has improved the accuracy of the
distance function formulation and the accuracy of the advection
scheme. We compute flows involving air bubbles and water drops,
among others. We validate our code against experiments and
theory. (C) 1998 Elsevier Science Ltd. All rights reserved.
 Kimmel, R, Kiryati, N, and Bruckstein, AM, "Multivalued distance maps for motion planning on surfaces with moving obstacles," IEEE TRANSACTIONS ON ROBOTICS AND AUTOMATION, vol. 14, pp. 427436, 1998.
Abstract:
This paper presents a new algorithm for planning the time
optimal motion of a robot traveling with limited velocity from
a given location to a given destination on a surface in the
presence of moving obstacles. Additional constraints such as
space variant terrain traversability and fuel economy can be
accommodated. A multivalued distance map is defined and applied
in computing optimal trajectories. The multivalued distance map
incorporates constraints imposed by the moving obstacles,
surface topography, and terrain traversability, It is generated
by an efficient numerical curve propagation technique.
 Shyue, KM, "An efficient shockcapturing algorithm for compressible multicomponent problems," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 142, pp. 208242, 1998.
Abstract:
A simple shockcapturing approach to multicomponent flow
problems is developed for the compressible Euler equations with
a stiffened gas equation of state in multiple space dimensions.
The algorithm uses a quasiconservative formulation of the
equations that is derived to ensure the correct fluid mixing
when approximating the equations numerically with interfaces. A
gammabased model and a volumefraction model have been
described, and both of them are solved using the standard high
resolution wave propagation method for general hyperbolic
systems of partial differential equations. Several calculations
are presented with a Roe approximate Riemann solver that show
accurate results obtained using the method without any spurious
oscillations in the pressure near the interfaces, Convergence
of the computed solutions to the correct weak ones has been
verified for a twodimensional RichtmyerMeshkov unstable
interface problem where we have performed a meshrefinement
study and also shown fronttracking results for comparison. (C)
1998 Academic Press.
 Steiner, A, Kimmel, R, and Bruckstein, AM, "Planar shape enhancement and exaggeration," GRAPHICAL MODELS AND IMAGE PROCESSING, vol. 60, pp. 112124, 1998.
Abstract:
A local smoothing operator applied in the reverse direction is
used to obtain planar shape enhancement and exaggeration.
Inversion of a smoothing operator is an inherently unstable
operation. Therefore, a stable numerical scheme simulating the
inverse smoothing effect is introduced. Enhancement is obtained
for short time spans of evolution. Carrying the evolution
further yields shape exaggeration or caricaturization effect.
Introducing attraction forces between the evolving shape and
the initial one yields an enhancement process that converges to
a steady state. These forces depend on the distance of the
evolving curve from the original one and on local properties.
Results of applying the unrestrained and restrained evolution
on planar shapes, based on a stabilized inverse geometric heat
equation, are presented showing enhancement and
caricaturization effects. (C) 1998 Academic Press.
 King, MJ, and DattaGupta, A, "Streamline simulation: A current perspective," IN SITU, vol. 22, pp. 91140, 1998.
Abstract:
Recent developments in reservoir characterization and in the
management of uncertainty have lead to the ability of the
petroleum industry to routinely generate large, multimillion
cell detailed geologic models. This has resulted in a steadily
increasing gap between flow simulation and the static model,
not only because of the size of these models, but also because
of our desire to obtain reservoir performance predictions for
multiple realizations of such models. Threedimensional
streamlinebased computation offers significant potential to
meet some of these challenges, leading to a rapidly developing
technology. The purpose of this paper is to review current
streamline technology: its foundations (the 'time of flight'
formulation), historical precedents (streamtubes and front
trackers), current applications, open questions, and potential
limitations. A wide range of applications will be used to
demonstrate the utility of both streamline simulation and the
underlying formulation. Where required, new material will be
presented (analysis of field tracer response, streamline
modeling in corner point cells, evaluation of grid orientation
effects, discussion of open questions). Otherwise, existing
results will be drawn from the literature.
 Carlson, NN, and Miller, K, "Design and application of a gradientweighted moving finite element code II: In two dimensions," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 19, pp. 766798, 1998.
Abstract:
In part I the authors reported on the design of a robust and
versatile gradientweighted moving finite element (GWMFE) code
in one dimension and on its application to a variety of PDEs
and PDE systems. This companion paper does the same for the
twodimensional (2D) case. These moving node methods are
especially suited to problems which develop sharp moving
fronts, especially problems where one needs to resolve the
finescale structure of the fronts. The many potential pitfalls
in the design of GWMFE codes and the special features of the
implicit onedimensional (1D) and 2D codes which contribute to
their robustness and efficiency are discussed at length in part
I; this paper concentrates on issues unique to the 2D case.
Brief explanations are given of the variational interpretation
of GWMFE, the geometricalmechanical interpretation, simplified
regularization terms, and the treatment of systems. A catalog
of inner products which occur in GWMFE is given, with
particular attention paid to those involving secondorder
operators. After presenting an example of the 2D phenomenon of
grid collapse and discussing the need for longtime
regularization, the paper reports on the application of the 2D
code to several nontrivial problems  nonlinear arsenic
diffusion in the manufacture of semiconductors, the drift
diffusion equations for semiconductor device simulation, the
BuckleyLeverett black oil equations for reservoir simulation,
and the motion of surfaces by mean curvature.
 Welch, SWJ, "Direct simulation of vapor bubble growth," INTERNATIONAL JOURNAL OF HEAT AND MASS TRANSFER, vol. 41, pp. 16551666, 1998.
Abstract:
This paper presents a numerical method directed towards the
local simulation of axisymmetric vapor bubble growth. We use an
interface tracking method in conjunction with a finite volume
method on a moving unstructured mesh. We allow metastable bulk
slates and assume the interface exists in thermal and chemical
equilibrium. The bulk fluids are viscous, conducting, and
compressible. The control volume continuity, momentum and
energy equations are modified in the presence of a phase
interface to include surface tension and discontinuous pressure
and velocity. A solid wall model is included to allow for
conjugate heat transfer modes. (C) 1998 Elsevier Science Ltd.
All rights reserved.
 Visintin, A, "Introduction to the models of phase transitions.," BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, vol. 1B, pp. 147, 1998.
Abstract:
This paper presents a numerical method directed towards the
local simulation of axisymmetric vapor bubble growth. We use an
interface tracking method in conjunction with a finite volume
method on a moving unstructured mesh. We allow metastable bulk
slates and assume the interface exists in thermal and chemical
equilibrium. The bulk fluids are viscous, conducting, and
compressible. The control volume continuity, momentum and
energy equations are modified in the presence of a phase
interface to include surface tension and discontinuous pressure
and velocity. A solid wall model is included to allow for
conjugate heat transfer modes. (C) 1998 Elsevier Science Ltd.
All rights reserved.
 Verdi, C, "Numerical methods for phase transition problems.," BOLLETTINO DELLA UNIONE MATEMATICA ITALIANA, vol. 1B, pp. 83108, 1998.
Abstract:
This paper presents a numerical method directed towards the
local simulation of axisymmetric vapor bubble growth. We use an
interface tracking method in conjunction with a finite volume
method on a moving unstructured mesh. We allow metastable bulk
slates and assume the interface exists in thermal and chemical
equilibrium. The bulk fluids are viscous, conducting, and
compressible. The control volume continuity, momentum and
energy equations are modified in the presence of a phase
interface to include surface tension and discontinuous pressure
and velocity. A solid wall model is included to allow for
conjugate heat transfer modes. (C) 1998 Elsevier Science Ltd.
All rights reserved.
 Barles, G, and Souganidis, PE, "A new approach to front propagation problems: Theory and applications," ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, vol. 141, pp. 237296, 1998.
Abstract:
In this paper we present a new definition for the globalin
time propagation (motion) of fronts (hypersurfaces, boundaries)
with a prescribed normal velocity, past the first time they
develop singularities. We show that if this propagation
satisfies a geometric maximum principle (inclusionavoidance
type property), then the normal velocity must depend only on
the position of the front, its normal direction and principal
curvatures. This new approach, which is more geometric and, as
it turns out, equivalent to the levelset method, is then used
to develop a very general and simple method to rigorously
validate the appearance of moving interne faces at the
asymptotic limit of general evolving systems like interacting
particles and reactiondiffusion equations. We finally present
a number of new asymptotic results. Among them are the
asymptotics of (i) reactiondiffusion equations with rapidly
oscillating coefficients, (ii) fully nonlinear nonlocal
(integral differential) equations, and (iii) stochastic Ising
models with longrange anisotropic interactions and general
spinflip dynamics.
 Zhu, JY, "A numerical study of chemical front propagation in a HeleShaw flow under buoyancy effects," PHYSICS OF FLUIDS, vol. 10, pp. 775788, 1998.
Abstract:
We consider the propagation of chemical fronts in a HeleShaw
flow where the front is assumed to propagate with a curvature
dependent velocity. The motivation is to model some recent
experiments that employ aqueous autocatalytic chemical
reactions in such a device. The density change across the front
in such experiments is quite small so the Boussinesq
approximation can be used, and the flow field generated is
exclusively due to buoyancy effects. We derive a free boundary
formulation based on Darcy's law and potential theory, and
describe the evolution in terms of the thetaL formulation, in
which the tangent angle and the perimeter of the closed front
are followed in time. Numerical solutions are obtained for this
formulation with a rising and expanding bubble. As observed in
the experiments, a fingering phenomenon which is different from
the surface tension associated phenomenon appears in our
calculations. The mechanisms that control the wavelength
selection of the fingers, and a comparison with the result of a
linear stability analysis for flat fronts are discussed. (C)
1998 American Institute of Physics.
 Chertkov, M, and Yakhot, V, "Propagation of a huygens front through turbulent medium," PHYSICAL REVIEW LETTERS, vol. 80, pp. 28372840, 1998.
Abstract:
The dynamics of a thin Huygens front propagating through a
turbulent medium is considered. A rigorous asymptotic
expression for the effective velocity v(F) proportional to the
front area is derived. The smallscale fluctuations of the
front position are shown to be strongly intermittent. This
intermittency prays a crucial role in establishing a steady
state magnitude of the front velocity. The results are compared
with experimental data.
 Li, ZL, "A fast iterative algorithm for elliptic interface problems," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 35, pp. 230254, 1998.
Abstract:
A fast, secondorder accurate iterative method is proposed for
the elliptic equation del .(beta(x,y)del u) = f(x,y) in a
rectangular region Omega in twospace dimensions. We assume
that there is an irregular interface across which the
coefficient beta, the solution u and its derivatives, and/or
the source term f may have jumps. We are especially interested
in the cases where the coefficients beta are piecewise constant
and the jump in beta is large. The interface may or may not
align with an underlying Cartesian grid. The idea in our
approach is to precondition the differential equation before
applying the immersed interface method proposed by LeVeque and
Li [SIAM J. Numer. Anal., 4 (1994), pp. 10191044]. In order to
take advantage of fast Poisson solvers on a rectangular region,
an intermediate unknown function, the jump in the normal
derivative across the interface, is introduced. Our
discretization is equivalent to using a secondorder difference
scheme for a corresponding Poisson equation in the region, and
a secondorder discretization for a Neumannlike interface
condition. Thus secondorder accuracy is guaranteed. A GMRES
iteration is employed to solve the Schur complement system
derived from the discretization. A new weighted least squares
method is also proposed to approximate interface quantities
from a grid function. Numerical experiments are provided and
analyzed. The number of iterations in solving the Schur
complement system appears to be independent of both the jump in
the coefficient and the mesh size.
 Calabi, E, Olver, PJ, Shakiban, C, Tannenbaum, A, and Haker, S, "Differential and numerically invariant signature curves applied to object recognition," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 26, pp. 107135, 1998.
Abstract:
We introduce a new paradigm, the differential invariant
signature curve or manifold, for the invariant recognition of
visual objects. A general theorem of E. Cartan implies that two
curves are related by a group transformation if and only if
their signature curves are identical. The important examples of
the Euclidean and equiaffine groups are discussed in detail.
Secondly, we show how a new approach to the numerical
approximation of differential invariants, based on suitable
combination of joint invariants of the underlying group action,
allows one to numerically compute differential invariant
signatures in a fully groupinvariant manner. Applications to a
variety of fundamental issues in vision, including detection of
symmetries, visual tracking, and reconstruction of occlusions,
are discussed.
 Shaked, D, and Bruckstein, AM, "Pruning medial axes," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 69, pp. 156169, 1998.
Abstract:
The medial axis is an attractive shape feature; however, its
high sensitivity to boundary noise hinders its use in many
applications. In order to overcome the sensitivity problem some
regularization has to be performed. Pruning is a family of
medial axis regularization processes, incorporated in most
skeletonization and thinning algorithms. Pruning algorithms
usually appear in a variety of applicationdependent
formulations. Inconsistent terminology used until now prevented
analysis and comparison of the various pruning methods. Indeed
many seemingly different algorithms are in fact equivalent. In
this paper we suggest the rate pruning paradigm as a standard
for pruning methods. The proposed paradigm is a framework in
which it is easy to analyze, compare, and tailor new pruning
methods. We analyze existing pruning methods, propose two new
methods, and compare the methods via a modelbased analysis.
The theoretical analysis is supported by simulation results of
the various pruning methods. (C) 1998 Academic Press.
 Visintin, A, "Nucleation and mean curvature flow," COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, vol. 23, pp. 1753, 1998.
Abstract:
In this paper the equation of mean curvature flow (with forcing
term) is modified, to account not only for surface motion but
also for nucleation and other discontinuities in set evolution.
Existence of a solution is proved for a weak formulation, which
is written in terms of the characteristic function of the
evolving set. The argument is based on implicit time
discretization, derivation of uniform estimates, and passage to
the limit.
 Ceniceros, HD, and Hou, TY, "Convergence of a nonstiff boundary integral method for interfacial flows with surface tension," MATHEMATICS OF COMPUTATION, vol. 67, pp. 137182, 1998.
Abstract:
Boundary integral methods to simulate interfacial flows are
very sensitive to numerical instabilities. In addition, surface
tension introduces nonlinear terms with high order spatial
derivatives into the interface dynamics. This makes the spatial
discretization even more difficult and, at the same time,
imposes a severe time step constraint for stable explicit time
integration methods. A proof of the convergence of a
reformulated boundary integral method for twodensity fluid
interfaces with surface tension is presented. The method is
based on a scheme introduced by Hou, Lowengrub and Shelley [J.
Comp. Phys. 114 (1994), pp. 312338] to remove the high order
stability constraint or stiffness. Some numerical filtering is
applied carefully at certain places in the discretization to
guarantee stability. The key of the proof is to identify the
most singular terms of the method and to show, through energy
estimates, that these terms balance one another. The analysis
is at a time continuousspace discrete level but a fully
discrete case for a simple HeleShaw interface is also studied.
The time discrete analysis shows that the high order stiffness
is removed and also provides an estimate of how the CFL
constraint depends on the curvature and regularity of the
solution. The robustness of the method is illustrated with
several numerical examples. A numerical simulation of an
unstably stratified twodensity interfacial flow shows the
rollup of the interface; the computations proceed up to a time
where the interface is about to pinch off and trapped bubbles
of fluid are formed. The method remains stable even in the full
nonlinear regime of motion. Another application of the method
shows the process of drop formation in a falling single fluid.
 Caselles, V, Morel, JM, Sapiro, G, and Tannenbaum, A, "Introduction to the special issue on partial differential equations and geometrydriven diffusion in image processing and analysis," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 7, pp. 269273, 1998.
Abstract:
Boundary integral methods to simulate interfacial flows are
very sensitive to numerical instabilities. In addition, surface
tension introduces nonlinear terms with high order spatial
derivatives into the interface dynamics. This makes the spatial
discretization even more difficult and, at the same time,
imposes a severe time step constraint for stable explicit time
integration methods. A proof of the convergence of a
reformulated boundary integral method for twodensity fluid
interfaces with surface tension is presented. The method is
based on a scheme introduced by Hou, Lowengrub and Shelley [J.
Comp. Phys. 114 (1994), pp. 312338] to remove the high order
stability constraint or stiffness. Some numerical filtering is
applied carefully at certain places in the discretization to
guarantee stability. The key of the proof is to identify the
most singular terms of the method and to show, through energy
estimates, that these terms balance one another. The analysis
is at a time continuousspace discrete level but a fully
discrete case for a simple HeleShaw interface is also studied.
The time discrete analysis shows that the high order stiffness
is removed and also provides an estimate of how the CFL
constraint depends on the curvature and regularity of the
solution. The robustness of the method is illustrated with
several numerical examples. A numerical simulation of an
unstably stratified twodensity interfacial flow shows the
rollup of the interface; the computations proceed up to a time
where the interface is about to pinch off and trapped bubbles
of fluid are formed. The method remains stable even in the full
nonlinear regime of motion. Another application of the method
shows the process of drop formation in a falling single fluid.
 Blomgren, P, and Chan, TF, "Color TV: Total variation methods for restoration of vector valued images," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 7, pp. 304309, 1998.
Abstract:
We propose a new definition of the total variation norm for
vectorvalued functions that can be applied to restore color
and other vectorvalued images, The new TV norm has the
desirable properties of 1) not penalizing discontinuities
(edges) in the image, 2) being rotationally invariant in the
image space, and 3) reducing to the usual TV norm in the scalar
case. Some numerical experiments on deionising simple color
images in redgreenblue (RGB) color space are presented.
 Faugeras, O, and Keriven, R, "Variational principles, surface evolution, PDE's, level set methods, and the stereo problem," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 7, pp. 336344, 1998.
Abstract:
We present a novel geometric approach for solving the stereo
problem for an arbitrary number of images (greater than or
equal to 2). It is based upon the definition of a variational
principle that must be satisfied by the surfaces of the objects
in the scene and their images, The EulerLagrange equations
that are deduced from the variational principle provide a set
of partial differential equations (PDE's) that are used to
deform an initial set of surfaces which then move toward the
objects to he detected, The level set implementation of these
PDE's potentially provides an efficient and robust way of
achieving the surface evolution and to deal automatically with
changes in the surface topology during the deformations, i.e.,
to deal with multiple objects, Results of an implementation of
our theory also dealing with occlusion and vilility are
presented on sydnthetic and real images.+
 Moisan, L, "Affine plane curve evolution: A fully consistent scheme," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 7, pp. 411420, 1998.
Abstract:
We present an accurate numerical scheme for the affine plane
curve evolution and its morphological extension to greylevel
images, This scheme is based on the iteration of a nonlocal,
fully affine invariant and numerically stable operator, which
can be exactly computed on polygons, The properties of this
operator ensure that a fea iterations are sufficient to achieve
a very good accuracy, unlike classical finite difference
schemes that generally require a tot of iterations, Convergence
results are provided, as well as theoretical examples and
experiments.
 Rey, JC, Li, JL, Boksha, V, Adalsteinsson, D, and Sethian, JA, "Topography simulation for interconnect deposition," SOLID STATE TECHNOLOGY, vol. 41, pp. 77+, 1998.
Abstract:
New process simulation programs can closely model the complex
structures of ULSI interconnects. Levelset methods simulate
and predict the structure of evolving surfaces in three
dimensions, such as that seen in thinfilm deposition. Models
rely upon iterative calibration using empirical results. The
ramifications of process or design changes can be predicted,
and reliabilityrelated problems such as void formation during
thin film deposition can be prevented.
 Anderson, DM, McFadden, GB, and Wheeler, AA, "Diffuseinterface methods in fluid mechanics," ANNUAL REVIEW OF FLUID MECHANICS, vol. 30, pp. 139165, 1998.
Abstract:
We review the development of diffuseinterface models of
hydrodynamics and their application to a wide variety of
interfacial phenomena. These models have been applied
successfully to situations in which the physical phenomena of
interest have a length scale commensurate with the thickness of
the interfacial region (e.g. nearcritical interfacial
phenomena or smallscale flows such as those occurring near
contact lines) and fluid flows involving large interface
deformations and/or topological changes (e.g. breakup and
coalescence events associated with fluid jets, droplets, and
largedeformation waves). We discuss the issues involved in
formulating diffuseinterface models for singlecomponent and
binary fluids. Recent applications and computations using these
models are discussed in each case. Further, we address issues
including sharpinterface analyses that relate these models to
the classical freeboundary problem, computational approaches
to describe interfacial phenomena, and models of fully miscible
fluids.
 Bordemann, M, and Hoppe, J, "Diffeomorphism invariant integrable field theories and hypersurface motions in Riemannian manifolds," JOURNAL OF MATHEMATICAL PHYSICS, vol. 39, pp. 683694, 1998.
Abstract:
We discuss hypersurface motions in Riemannian manifolds whose
normal velocity is a function of the induced hypersurface
volume element and derive a secondorder partial differential
equation for the corresponding time function tau(x) at which
the hypersurface passes the point x. Equivalently, these
motions may be described in a Hamiltonian formulation as the
singlet sector of certain diffeomorphisminvariant held
theories. At least in some (infinite class of) cases, which
could be viewed as a largevolume limit of Euclidean Mbranes
moving in an arbitrary M+1dimensional Riemannian manifold, the
models are integrable: In the timefunction formulation the
equation becomes linear [with tau(x) a harmonic function on the
embedding Riemannian manifold]. We explicitly compute solutions
to the large volume limit of Euclidean membrane dynamics in R3
by methods used in electrostatics and point out an additional
gradient how structure in Rn. In the Hamiltonian formulation
we discover infinitely many hierarchies of integrable,
multidimensional, Ncomponent theories possessing infinitely
many diffeomorphism invariant, Poisson commuting, conserved
charges. (C) 1998 American Institute of Physics. [S0022
2488(97)009122].


1999 
 Chung, DH, and Sapiro, G, "A windowsbased user friendly system for image analysis with partial differential equations," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 453458, 1999.
Abstract:
In this paper we present and briefly describe a Windows user
friendly system designed to assist with the analysis of images
in general, and biomedical images in particular. The system,
which is being made publicly available to the research
community, implements basic 2D image analysis operations based
on partial differential equations (PDE's). The system is under
continuous development, and already includes a large number of
image enhancement and segmentation routines that have been
tested for several applications.
 Guo, YL, and Vemuri, BC, "Hybrid geometric active models for shape recovery in medical images," INFORMATION PROCESSING IN MEDICAL IMAGING, PROCEEDINGS, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1613, pp. 112125, 1999.
Abstract:
In this paper, we propose extensions to a powerful geometric
shape modeling scheme introduced in [14]. The extension allows
the model to automatically cope with topological changes and
for the first time, introduces the concept of a global shape
into geometric/geodesic snake models. The ability to
characterize global shape of an object using very few
parameters facilitates shape learning and recognition. In this
new modeling scheme, object shapes are represented using a
parameterized function  called the generator  which accounts
for the global shape of an object and the pedal curve/surface
of this global shape with respect to a geometric snake to
represent any local detail. Traditionally, pedal
curves/surfaces are defined as the loci of the feet of
perpendiculars to the tangents of the generator from a fixed
point called the pedal point. We introduce physicsbased
control for shaping these geometric models by using distinct
pedal points  lying on a snake  for each point on the
generator. The model dubbed as a "snake pedal" allows for
interactive manipulation via forces applied to the snake.
Automatic topological changes of the model may be achieved by
implementing the geometric active contour in a levelset
framework. We demonstrate the applicability of this modeling
scheme via examples of shape estimation from a variety of
medical image data.
 Sussman, M, Almgren, AS, Bell, JB, Colella, P, Howell, LH, and Welcome, ML, "An adaptive level set approach for incompressible twophase flows," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 148, pp. 81124, 1999.
Abstract:
We present a numerical method using the level set approach for
serving incompressible twophase flow with surface tension. In
the level set approach, the free surface is represented as the
zero level set of a smooth function; this has the effect of
replacing the advection of density, which has steep gradients
at the free surface, with the advection of the level set
function, which is smooth. In addition, the free surface can
merge or break up with no special treatment. We maintain the
level set function as the signed distance from the free surface
in order to accurately compute flows with high density ratios
and stiff surface tension effects. In this work, we couple the
level set scheme to an adaptive projection method for the
incompressible NavierStokes equations, in order to achieve
higher resolution of the free surface with a minimum of
additional expense. We present twodimensional axisymmetric and
fully threedimensional results of air bubble and water drop
computations. (C) 1999 Academic Press.
 Koren, B, and Venis, ACJ, "A fed back levelset method for moving materialvoid interfaces," JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS, vol. 101, pp. 131152, 1999.
Abstract:
This report is a feasibility study of a levelset method for
the computation of moving materialvoid interfaces in an
Eulerian formulation. The report briefly introduces levelset
methods and focuses on the development of such a method, that
does not just accurately resolve the geometry of the interface,
but also the physical quantities at and near the interface.
Results are presented for illustrative model problems. As
concerns its ability to improve the geometrical resolution of
free boundaries, as expected, the levelset method performs
excellently. Concerning the improvement of physical (all other
than merely geometrical) freeboundary properties, the method
performs very well for downstreamfacing fronts and is
promising for upstreamfacing ones. (C) 1999 Elsevier Science
B.V. All rights reserved. AMS classification: 65M20; 65M99;
76M99; 76T05.
 Helenbrook, BT, Martinelli, L, and Law, CK, "A numerical method for solving incompressible flow problems with a surface of discontinuity," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 148, pp. 366396, 1999.
Abstract:
A numerical method for solving problems in which a moving
surface of discontinuity separates regions of incompressible
how is presented. The method developed is notable in that it
does not introduce any artificial smoothing of the change in
fluid properties across the surface of discontinuity. This
results in an increase in accuracy relative to methods which
introduce smoothing effects. The method was also shown to be
fairly versatile; problems describing a free surface, an
immiscible fluid interface, and a premixed flame discontinuity
were solved. There is a limitation, however, in that the method
appears to be most suitable for application to inviscid
problems. The reason for this limitation and possible
approaches toward resolving it are discussed. (C) 1999 Academic
Press.
 Adalsteinsson, D, and Sethian, JA, "The fast construction of extension velocities in level set methods," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 148, pp. 222, 1999.
Abstract:
Level set techniques are numerical techniques for tracking the
evolution of interfaces. They rely on two central embeddings;
first, the embedding of the interface as the zero level set of
a higher dimensional function, and second, the embedding (or
extension) of the interface's velocity to this higher
dimensional level set function. This paper applies Sethian's
Fast Marching Method, which is a very fast technique for
solving the eikonal and related equations, to the problem of
building fast and appropriate extension velocities for the
neighboring level sets. Our choice and construction of
extension velocities serves several purposes. First, it
provides a way of building velocities for neighboring level
sets in the cases where the velocity is defined only on the
front itself. Second, it provides a subgrid resolution not
present in the standard level set approach. Third, it provides
a way to update an interface according to a given velocity
field prescribed on the front in such a way that the signed
distance function is maintained, and the front is never re
initialized; this is valuable in many complex simulations. In
this paper, we describe the details of such implementations,
together with speed and convergence tests and applications to
problems in visibility relevant to semiconductor manufacturing
and thin film physics. (C) 1999 Academic Press.
 Caselles, V, Lisani, JL, Morel, JM, and Sapiro, G, "Shape preserving local histogram modification," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 8, pp. 220230, 1999.
Abstract:
A novel approach for shape preserving contrast enhancement is
presented in this paper. Contrast enhancement is achieved by
means of a local histogram equalization algorithm which
preserves the levelsets of the image. This basic property is
violated by common local schemes, thereby introducing spurious
objects and modifying the image information, The scheme is
based on equalizing the histogram in all the connected
components of the image, which are defined based both on the
greyvalues and spatial relations between pixels in the image,
and following mathematical morphology, constitute the basic
objects in the scene. We give examples for both greyvalue and
color images.
 Fischl, B, and Schwartz, EL, "Adaptive nonlocal filtering: A fast alternative to anisotropic diffusion for image enhancement," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 21, pp. 4248, 1999.
Abstract:
Nonlinear anisotropic diffusion algorithms provide significant
improvement in image enhancement as compared to linear filters.
However, the excessive computational cost of solving nonlinear
PDEs precludes their use in realtime vision applications. In
the present paper, we show that two orders of magnitude speed
improvement is provided by a new image filtering paradigm in
which an adaptively determined vector field specifies nonlocal
application points for an image filter.
 Chakraborty, A, and Duncan, JS, "Gametheoretic integration for image segmentation," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 21, pp. 1230, 1999.
Abstract:
Robust segmentation of structures from an image is essential
for a variety of image analysis problems. However, the
conventional methods of regionbased segmentation and gradient
based boundary finding are often frustrated by poor image
quality. Here we propose a method to integrate the two
approaches using game theory in an effort to form a unified
approach that is robust to noise and poor initialization. This
combines the perceptual notions of complete boundary
information using edge data and shape priors with graylevel
homogeneity using two computational modules. The novelty of the
method is that this is a bidirectional framework, whereby both
computational modules improve their results through mutual
information sharing. A number of experiments were performed
both on synthetic datasets and datasets of real images to
evaluate the new approach and it is shown that the integrated
method typically performs better than conventional gradient
based boundary finding.
 Hamaguchi, S, "Modeling and simulation methods for plasma processing," IBM JOURNAL OF RESEARCH AND DEVELOPMENT, vol. 43, pp. 199215, 1999.
Abstract:
Methods used for the modeling and numerical simulation of the
plasma processes used in semiconductor integratedcircuit
fabrication are reviewed. In the first part of the paper, we
review continuum and kinetic methods, A model based on the
driftdiffusion equations is presented as an example of a
continuum model; the model and associated numerical solutions
are discussed. The most widely used simulation method for
kinetic modeling is the ParticleInCell/MonteCarloCollision
(PIC/MCC) method, in which the plasma is modeled by a system of
charged superparticles (each of which represents a collection
of a large number of ions or electrons) that move in self
consistent electromagnetic fields and collide via given
collision cross sections. In the second part of the paper, we
review the modeling and simulation of the evolution of surface
topography in plasma etching and deposition.
 Hu, CQ, and Shu, CW, "Weighted essentially nonoscillatory schemes on triangular meshes," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 150, pp. 97127, 1999.
Abstract:
In this paper we construct highorder weighted essentially non
oscillatory schemes on twodimensional unstructured meshes
(triangles) in the finite volume formulation. We present third
order schemes using a combination of linear polynomials and
fourthorder schemes using a combination of quadratic
polynomials. Numerical examples are shown to demonstrate the
accuracies and robustness of the methods for shock
calculations. (C) 1999 Academic Press.
 Siddiqi, K, Kimia, BB, Tannenbaum, A, and Zucker, SW, "Shapes, shocks and wiggles," IMAGE AND VISION COMPUTING, vol. 17, pp. 365373, 1999.
Abstract:
We earlier introduced an approach to categorical shape
description based on the singularities (shocks) of curve
evolution equations. The approach relates to many techniques in
computer vision, such as Blum's grassfire transform, but since
the motivation was abstract it is not clear that it should also
relate to human perception. We now report that this shockbased
computational model can account for recent psychophysical data
collected by Burbeck and Pizer. In these experiments subjects
were asked to estimate the local centers of stimuli consisting
of rectangles with 'wiggles' (sides modulated by sinusoids).
Since the experiments were motivated by their 'core' model, in
which the scale of boundary detail is proportional to object
width, we conclude that such properties are also implicit in
shockbased shape descriptions. More generally, the results
suggest that significance is a structural notion, not an image
based one, and that scale should be defined primarily in terms
of relationships between abstract entities, not concrete
pixels. (C) 1999 Elsevier Science B.V. All rights reserved.
 Oka, H, and Ishii, K, "Numerical analysis on the motion of gas bubbles using level set method," JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN, vol. 68, pp. 823832, 1999.
Abstract:
In this paper, we study the behavior of gas bubbles rising
through a viscous liquid in a vertical square duct numerically.
The level set formulation developed by Sussman et al. is
successfully generalized for threedimensional incompressible
twophase flows including large density and viscosity ratios as
well as surface tension effect. Numerical simulations are
carried out for gasliquid flows with different ratios of
density. It is shown that the effect of variation of the
density ratio on the bubble shape and the flow field is
extremely weak when the ratio is larger than 1:50. The
simulations of flows with a single bubble in a duct are also
carried out to investigate the influence of the duct walls on
the flow field. It is clarified that the bubble shape and the
rising velocity strongly depend upon the ratio of a duct width
to an initial bubble radius, but that there is hardly the
effect of the ratio on the rising velocity when it exceeds ten.
Finally, we present a numerical result on the interaction of
two bubbles. The result is in qualitatively agreement with
previous experimental data.
 Aslam, TD, and Stewart, DS, "Detonation shock dynamics and comparisons with direct numerical simulation," COMBUSTION THEORY AND MODELLING, vol. 3, pp. 77101, 1999.
Abstract:
Comparisons between direct numerical simulation (DNS) of
detonation and detonation shock dynamics (DSD) is made. The
theory of DSD defines the motion of the detonation shock in
terms of the intrinsic geometry of the shock surface, in
particular for condensed phase explosives the shock normal
velocity, Dn, the normal acceleration, (D) over dot(n), and
the total curvature, kappa. In particular, the properties of
three intrinsic front evolution laws are studied and compared.
These are (i) constant speed detonation (Huygens construction),
(ii) curvaturedependent speed propagation (Dnkappa relation)
and (iii) curvature and speeddependent acceleration ((D) over
dot(n)Dnkappa relation). We show that it is possible to
measure shock dynamics directly from simulation of the reactive
Euler equations and that subsequent numerical solution of the
intrinsic partial differential equation for the shock motion
(e.g. a (D) over dot(n)Dnkappa relation) reproduces the
computed shock motion with high precision.
 Sethian, JA, and Popovici, AM, "3D traveltime computation using the fast marching method," GEOPHYSICS, vol. 64, pp. 516523, 1999.
Abstract:
We present a fast algorithm for solving the eikonal equation in
three dimensions: based on the fast marching method. The
algorithm is of the order O(N log N), where N is the total
number of grid points in the computational domain. The
algorithm can be used in any orthogonal coordinate system and
globally constructs the solution to the eikonal equation for
each point in the coordinate domain. The method is
unconditionally stable and constructs solutions consistent with
the exact solution for arbitrarily large gradient jumps in
velocity. In addition, the method resolves any overturning
propagation wavefronts. We begin with the mathematical
foundation for solving the eikonal equation using the fast
marching method and follow with the numerical details. We then
show examples of traveltime propagation through the SEG/EAGE
salt model using pointsource and planewave initial conditions
and analyze the error in constant velocity media. The algorithm
allows for any shape of the initial wavefront. While a point
source is the most commonly used initial condition, initial
plane waves can be used for controlled illumination or for
downward continuation of the traveltime field from one depth to
another or from a topographic depthsurface to another. The
algorithm presented here is designed for computing first
arrival traveltimes. Nonetheless, since it exploits the fast
marching method for solving the eikonal equation, we believe it
is the fastest of all possible consistent schemes to compute
first arrivals.
 Hou, TY, Rosakis, P, and LeFloch, P, "A levelset approach to the computation of twinning and phase transition dynamics," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 150, pp. 302331, 1999.
Abstract:
A computational method is proposed for the dynamics of solids
capable of twinning and phase transitions. In a two
dimensional, sharpinterface model of twinning, the stored
energy function is a nonconvex potential with multiple wells.
The evolution of twin interfaces is governed by held equations
and jump conditions of momentum balance, and by a kinetic
relation expressing the interface velocity as a function of the
local driving traction and interfacial orientation. A
regularized version of the model is constructed based on the
levelset method. A levelset function which changes signs
across the interface is introduced, The evolution of this
function is described by a HamiltonJacobi equation, whose
velocity coefficient is determined by the kinetic relation.
Jump conditions are thereby eliminated, allowing finite
difference discretization. Numerical simulations exhibit
complex evolution of the interface, including cusp formation,
needle growth, spontaneous tip splitting, and topological
changes that result in microstructure refinement. The results
capture experimentally observed phenomena in martensitic
crystals. (C) 1999 Academic Press.
 Barles, G, "Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 154, pp. 191224, 1999.
Abstract:
We prove comparison results between viscosity sub and
supersolutions of degenerate elliptic and parabolic equations
associated to, possibly nonlinear, Neumann boundary conditions.
These results are obtained under more general assumptions on
the equation tin particular the dependence in the gradient of
the solution and they allow applications to quasilinear,
possibly singular, elliptic or parabolic equations. One of the
main applications is the extension of the socalled level set
approach for equations set in bounded domains with nonlinear
Neumann boundary conditions, In such a framework, the level set
approach provides a weak notion for the motion of hypersurfaces
with curvature dependent velocities and a prescribed contact
angle at the boundary. (C) 1999 Academic Press.
 Sussman, M, and Fatemi, E, "An efficient, interfacepreserving level set redistancing algorithm and its application to interfacial incompressible fluid flow," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 20, pp. 11651191, 1999.
Abstract:
In Sussman, Smereka, and Osher [J. Comp. Phys., 94 (1994), pp.
146159], a numerical scheme was presented for computing
incompressible airwater flows using the level set method.
Crucial to the above method was a new iteration method for
maintaining the level set function as the signed distance from
the zero level set. In this paper we implement a "constraint"
along with higher order difference schemes in order to make the
iteration method more accurate and efficient. Accuracy is
measured in terms of the new computed signed distance function
and the original level set function having the same zero level
set. We apply our redistancing scheme to incompressible flows
with noticeably better resolved results at reduced cost. We
validate our results with experiment and theory. We show that
our "distance level set scheme" with the added constraint
competes well with available interface tracking schemes for
basic advection of an interface. We perform basic accuracy
checks and more stringent tests involving complicated
interfacial structures. As with all level set schemes, our
method is easy to implement.
 Koo, Y, "A fattening principle for fronts propagating by mean curvature plus a driving force," COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, vol. 24, pp. 10351053, 1999.
Abstract:
We demonstrate a general situation in which a hypersurface in
Rn propagating by mean curvature, plus a nonzero driving
force, develops an interior after finite time in the level set
formulation of the problem.
 Alhanaty, M, and Bercovier, M, "Shapes with offsets of nearly constant surface area," COMPUTERAIDED DESIGN, vol. 31, pp. 287296, 1999.
Abstract:
This article addresses several issues of offset sizes. A new
type of isoperimetric problems is introduced: find the shapes,
which have offsets of minimal surface area change. This problem
arises in some physical and chemical processes with constant
energy release. A computational solution is presented for two
cases: convex sets, and a subclass of nonconvex sets (star
like sets). Both cases are discussed in the plane and in the
space. The solution uses methods from convex theory including
the theory of mixed volumes and also some optimization
techniques. The formulas developed in the optimization problem
can be applied to get analytical formulas of the length of the
curve offsets, as well as of the surface area and the volume of
the surface offsets. Evaluating properties of offsets without
constructing them proves useful for preliminary design in solid
modeling, for approximating offsets curves and for planning the
velocity of offset paths. (C) 1999 Published by Elsevier
Science Ltd. All rights reserved.
 Strain, J, "Tree methods for moving interfaces," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 151, pp. 616648, 1999.
Abstract:
Fast adaptive numerical methods for solving moving interface
problems are presented. The methods combine a level set
approach with frequent redistancing and semiLagrangian time
stepping schemes which are explicit yet unconditionally stable.
A quadtree mesh is used to concentrate computational effort on
the interface, so the methods move an interface with N degrees
of freedom in O(N log N) work per time step. Efficiency is
increased by taking large time steps even for parabolic
curvature flows. The methods compute accurate viscosity
solutions to a wide variety of difficult moving interface
problems involving merging, anisotropy, faceting, and
curvature. (C) 1999 Academic Press.
 Strain, J, "SemiLagrangian methods for level set equations," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 151, pp. 498533, 1999.
Abstract:
A new numerical method for solving geometric moving interface
problems is presented. The method combines a level set approach
and a semiLagrangian time stepping scheme which is explicit
yet unconditionally stable. The combination decouples each mesh
point from the others and the time step from the CFL stability
condition, permitting the construction of methods which are
efficient, adaptive, and modular. Analysis of a linear one
dimensional model problem suggests a surprising convergence
criterion which is supported by heuristic arguments and
confirmed by an extensive collection of twodimensional
numerical results. The new method computes comet viscosity
solutions to problems involving geometry, anisotropy,
curvature, and complex topological events. (C) 1999 Academic
Press.
 Li, ZL, Zhao, HK, and Gao, HJ, "A numerical study of electromigration voiding by evolving level set functions on a fixed Cartesian grid," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 152, pp. 281304, 1999.
Abstract:
A numerical method for studying migration of voids driven by
surface diffusion and electric current in a metal conducting
line is developed. The mathematical model involves moving
boundaries governed by a fourth order nonlinear partial
differential equation which contains a nonlocal term
corresponding to the electrical field and a nonlinear term
corresponding to the curvature. Numerical challenges include
efficient computation of the electrical field with sufficient
accuracy to afford fourth order differentiation along the void
boundary and to capture singularities arising in topological
changes. We use the modified immersed interface method with a
fixed Cartesian grid to solve for the electrical field, and the
fast local level set method to update the position of moving
voids, Numerical examples are performed to demonstrate the
physical mechanisms by which voids interact under
electromigration. (C) 1999 Academic Press.
 Niessen, WJ, Vincken, KL, Weickert, J, Romeny, BMT, and Viergever, MA, "Multiscale segmentation of threedimensional MR brain images," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 31, pp. 185202, 1999.
Abstract:
Segmentation of MR brain images using intensity values is
severely limited owing to field inhomogeneities, susceptibility
artifacts and partial volume effects. Edge based segmentation
methods suffer from spurious edges and gaps in boundaries. A
multiscale method to MRI brain segmentation is presented which
uses both edge and intensity information. First a multiscale
representation of an image is created, which can be made edge
dependent to favor intratissue diffusion over intertissue
diffusion. Subsequently a multiscale linking model (the
hyperstack) is used to group voxels into a number of objects
based on intensity. It is shown that both an improvement in
accuracy and a reduction in image postprocessing can be
achieved if edge dependent diffusion is used instead of linear
diffusion. The combination of edge dependent diffusion and
intensity based linking facilitates segmentation of grey
matter, white matter and cerebrospinal fluid with minimal user
interaction. To segment the total brain (white matter plus grey
matter) morphological operations are applied to remove small
bridges between the brain and cranium. If the total brain is
segmented, grey matter, white matter and cerebrospinal fluid
can be segmented by joining a small number of segments. Using a
supervised segmentation technique and MRI simulations of a
brain phantom for validation it is shown that the errors are in
the order of or smaller than reported in literature.
 Li, ZL, and Soni, B, "Fast and accurate numerical approaches for Stefan problems and crystal growth," NUMERICAL HEAT TRANSFER PART BFUNDAMENTALS, vol. 35, pp. 461484, 1999.
Abstract:
New numerical approaches for moving boundary/interface
applications tailored for Stefan problems and crystal growth
simulation are proposed in this article. The focus is on the
issues of accuracy and speedup. A modified CrankNicolson
method that is secondorder accurate and stable is developed.
The alternating directional implicit (ADI) method is also
developed to speed up the simulation for a certain class of
problems. The ADI method is shown to be asymptotically stable
and at least firstorder accurate. Numerical results, however,
show that the ADI method actually provides secondorder
accuracy if the velocity can be calculated accurately. The
level set method is used to update the moving interface so that
the topological changes can be handled easily. Numerical
experiments are compared to exact solutions and results in the
literature.
 Sethian, JA, "Fast marching methods," SIAM REVIEW, vol. 41, pp. 199235, 1999.
Abstract:
Fast Marching Methods are numerical schemes for computing
solutions to the nonlinear Eikonal equation and related static
HamiltonJacobi equations. Based on entropysatisfying upwind
schemes and fast sorting techniques, they yield consistent,
accurate, and highly efficient algorithms. They are optimal in
the sense that the computational complexity of the algorithms
is O(N log N), where N is the total number of points in the
domain. The schemes are of use in a variety of applications,
including problems in shape offsetting, computing distances
from complex curves and surfaces, shapefromshading,
photolithographic development, computing first arrivals in
seismic travel times, construction of shortest geodesics on
surfaces, optimal path planning around obstacles, and
visibility and reflection calculations. In this paper, we
review the development of these techniques, including the
theoretical and numerical underpinnings; provide details of the
computational schemes, including higher order versions; and
demonstrate the techniques in a collection of different areas.
 Dinh, TN, Bui, VA, Nourgaliev, RR, Green, JA, and Sehgal, BR, "Experimental and analytical studies of melt jetcoolant interactions: a synthesis," NUCLEAR ENGINEERING AND DESIGN, vol. 189, pp. 299327, 1999.
Abstract:
Instability and fragmentation of a core melt jet in water have
been actively studied during the past 10 years. Several models,
and a few computer codes, have been developed. However, there
are, still, large uncertainties, both, in interpreting
experimental results and in predicting reactorscale processes.
Steam explosion and debris coolability, as reactor safety
issues, are related to the jet fragmentation process. A better
understanding of the physics of jet instability and
fragmentation is crucial for assessments of fuelcoolant
interactions (FCIs). This paper presents research, conducted at
the Division of Nuclear Power Safety, Royal Institute of
Technology (RIT/NPS), Stockholm, concerning molten jetcoolant
interactions, as a precursor for premixing. First, observations
were obtained from scoping experiments with simulant fluids.
Second, the linear perturbation method was extended and applied
to analyze the interfacialinstability characteristics. Third,
two innovative approaches to computational fluid dynamics (CFD)
modeling of jet fragmentation were developed and employed for
analysis. The focus of the studies was placed on (a)
identifying potential factors, which may affect the jet
instability, (b) determining the scaling laws, and (c)
predicting the jet behavior for severe accident conditions. In
particular, the effects of melt physical properties, and the
thermal hydraulics of the mixing zone, on jet fragmentation
were investigated. Finally, with the insights gained from a
synthesis of the experimental results and analysis results, a
new phenomenological concept, named 'macrointeractions concept
of jet fragmentation' is proposed. (C) 1999 Elsevier Science
S.A. All rights reserved.
 Bertalmio, M, Sapiro, G, and Randall, G, "Region tracking on levelsets methods," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 18, pp. 448451, 1999.
Abstract:
Since the work by Osher and Sethian on levelsets algorithms
for numerical shape evolutions, this technique has been used
for a large number of applications in numerous fields. In
medical imaging, this numerical technique has been successfully
used for example, in segmentation and cortex unfolding
algorithms. The migration from a Lagrangian implementation to a
Eulerian one via implicit representations or levelsets brought
some of the main advantages of the technique, i.e., topology
independence and stability. This migration means also: that the
evolution is parametrization free. Therefore, we do not know
exactly how each part of the shape is deforming and the point
wise correspondence is lost. In this note we present a
technique to numerically track regions on surfaces that are
being deformed. using the levelsets method. The basic idea is
to represent the region of interest as the intersection of two
implicit surfaces and then track its deformation from the
deformation of these surfaces. This technique then solves one
of the main shortcomings of the very useful levelsets
approach. Applications include lesion localization in medical
images, region tracking in functional MRI (fMRI) visualization,
and geometric surface mapping.
 Strain, J, "Fast treebased redistancing for level set computations," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 152, pp. 664686, 1999.
Abstract:
Level set methods for moving interface problems require
efficient techniques for transforming an interface to a
globally defined function whose zero set is the interface, such
as the signed distance to the interface. This paper presents
efficient algorithms for this "redistancing" problem. The
algorithms use quadtrees and triangulation to compute global
approximate signed distance functions. A quadtree mesh is built
to resolve the interface and the vertex distances are evaluated
exactly with a robust search strategy to provide both
continuous and discontinuous interpolants. Given a polygonal
interface with N elements, our algorithms run in O (N) space
and O(N log N) time. Twodimensional numerical results show
they are highly efficient in practice. (C) 1999 Academic Press.
 He, XY, Chen, SY, and Zhang, RY, "A lattice Boltzmann scheme for incompressible multiphase flow and its application in simulation of RayleighTaylor instability," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 152, pp. 642663, 1999.
Abstract:
In this pager, we propose a new lattice Boltzmann scheme for
simulation of multiphase flow in the nearly incompressible
limit. The new scheme simulates fluid flows based on
distribution functions. The interfacial dynamics, such as phase
segregation and surface tension, are modeled by incorporating
molecular interactions. The lattice Boltzmann equations are
derived from the continuous Boltzmann equation with appropriate
approximations suitable for incompressible flow. The numerical
stability is improved by reducing the effect of numerical
errors in calculation of molecular interactions. An index
function is used to track interfaces between different phases.
Simulations of the twodimensional RayleighTaylor instability
yield satisfactory results. The interface thickness is
maintained at 34 grid spacings throughout simulations without
artificial reconstruction steps. (C) 1999 Academic Press.
 Reinecke, M, Hillebrandt, W, and Niemeyer, JC, "Thermonuclear explosions of Chandrasekharmass C+O white dwarfs," ASTRONOMY AND ASTROPHYSICS, vol. 347, pp. 739747, 1999.
Abstract:
First results of simulations are presented which compute the
dynamical evolution of a Chandrasekharmass white dwarf,
consisting of equal amounts of carbon and oxygen, from the
onset of violent thermonuclear burning, by means of a new two
dimensional numerical code. Since in the interior of such a
massive white dwarf nuclear burning progresses on microscopic
scales as a sharp discontinuity, a socalled flamelet, which
cannot be resolved by any numerical scheme, and since on
macroscopic scales the burning front propagates due to
turbulence, we make an attempt to model both effects explicitly
in the framework of a finitevolume hydrodynamics code.
Turbulence is included by a subgrid model, following the
spirit of large eddy simulations, and the welllocalized
burning front is treated by means of a level set, which allows
us to compute the geometrical structure of the front more
accurately than with previous methods. The only free parameters
of our simulations are the location and the amount of nuclear
fuel that is ignited as an initial perturbation. We find that
models in which explosive carbon burning is ignited at the
center remain bound by the time the front reaches low
densities, where we stopped the computations because our
description of combustion is no longer applicable. In contrast,
offcenter ignition models give rise to explosions which,
however, are still too weak for typical Type Ia supernovae.
Possible reasons for this rather disappointing result are
discussed.
 Reinecke, M, Hillebrandt, W, Niemeyer, JC, Klein, R, and Grobl, A, "A new model for deflagration fronts in reactive fluids," ASTRONOMY AND ASTROPHYSICS, vol. 347, pp. 724733, 1999.
Abstract:
We present a new way of modeling deflagration fronts in
reactive fluids, the main emphasis being on turbulent
thermonuclear deflagration fronts in white dwarfs undergoing a
Type Ia supernova explosion. Our approach is based on a level
set method which treats the front as a mathematical
discontinuity and allows full coupling between the front
geometry and the flow field (Smiljanovski et al., 1997). With
only minor modifications, this method can also be applied to
describe contact discontinuities. Two different implementations
are described and their physically correct behaviour for simple
testcases is shown. First results of the method applied to the
concrete problems of Type Ia supernovae and chemical hydrogen
combustion are briefly discussed; a more extensive analysis of
our astrophysical simulations is given in Reinecke et al.
(1998).
 Ubbink, O, and Issa, RI, "A method for capturing sharp fluid interfaces on arbitrary meshes," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 153, pp. 2650, 1999.
Abstract:
The paper describes a high resolution method (CICSAM) for the
accurate capturing of fluid interfaces on meshes of arbitrary
topology. It is based on the finitevolume technique and is
fully conservative. The motion of the interface is tracked by
the solution of a scalar transport equation for a phase
indicator held that is discontinuous at the interface and
uniform elsewhere; no explicit interface reconstruction, which
is perceived to be difficult to implement on unstructured
meshes, is needed. The novelty of the method lies in the
adaptive combination of high resolution discretisation schemes
which ensure the preservation of the sharpness and shape of the
interface while retaining boundedness of the field. The special
implicit implementation presented herein makes it applicable to
unstructured meshes and an extension to such grids is
presented. The method is capable of handling interface rupture
and coalescence. The paper outlines the methodology of CICSAM
and its validation against academic test cases used to verify
its accuracy. (C) 1999 Academic Press.
 Ishii, H, Pires, GE, and Souganidis, PE, "Threshold dynamics type approximation schemes for propagating fronts," JOURNAL OF THE MATHEMATICAL SOCIETY OF JAPAN, vol. 51, pp. 267308, 1999.
Abstract:
We study the convergence of general threshold dynamics type
approximation schemes to hypersurfaces moving with normal
velocity depending on the normal direction and the curvature
tensor. We also present results about the asymptotic shape of
fronts propagating by threshold dynamics. Our results
generalize and extend models introduced in the theories of
cellular automaton and motion by mean curvature.
 Schulze, TP, and Kohn, RV, "A geometric model for coarsening during spiralmode growth of thin films," PHYSICA D, vol. 132, pp. 520542, 1999.
Abstract:
We study the coarsening observed in spiralmode growth of thin
films. The hightemperature superconductor YBa2Cu3O7delta
provides a suitable model system. The density of spirals at the
surface decreases as the him gets thicker. In other words, the
grain size coarsens with distance from the substrate. We
propose a simple mechanism for this coarsening, based on
geometrical competition of spirals with different vertical
growth rates. The consequences of this mechanism are developed
both analytically and numerically in the limit where adatom
attachment is controlled by surface diffusion. In particular,
we show how the timeevolution of spiral density, film
thickness, and surface roughness depend on the spiral growth
rate statistics. (C)1999 Elsevier Science B.V. All rights
reserved.
 Udaykumar, HS, Mittal, R, and Shyy, W, "Computation of solidliquid phase fronts in the sharp interface limit on fixed grids," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 153, pp. 535574, 1999.
Abstract:
A finitedifference formulation is applied to track solid
liquid boundaries on a fixed underlying grid. The interface is
not of finite thickness but is treated as a discontinuity and
is explicitly tracked. The imposition of boundary conditions
exactly on a sharp interface that passes through the Cartesian
grid is performed using simple stencil readjustments in the
vicinity of the interface. Attention is paid to formulating
difference schemes that are globally secondorder accurate in x
and t. Error analysis and grid refinement studies are performed
for test problems involving the diffusion and convection
diffusion equations, and for stable solidification problems.
Issues concerned with stability and change of phase of grid
points in the evolution of solidliquid phase fronts are also
addressed. It is demonstrated that the field calculation is
secondorder accurate while the position of the phase front is
calculated to firstorder accuracy. Furthermore, the accuracy
estimates hold for the cases where there is a property jump
across the interface. Unstable solidification phenomena are
simulated and an attempt is made to compare results with
previously published work. The results indicate the need to
begin an effort to benchmark computations of instability
phenomena. (C) 1999 Academic Press.
 Vese, L, "A method to convexify functions via curve evolution," COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, vol. 24, pp. 15731591, 1999.
Abstract:
This paper is devoted to a new method which allows to compute
the convex envelope of a given function, by an evolution
equation and techniques of curve evolution. We study the
problem in the context of viscosity solutions and we propose
numerical algorithms, to convexify a function, in one and two
dimensions. In the end, we validate the model by presenting
various numerical results.
 Gyure, MF, "Bridging time and length scales in semiconductor process model development," COMPUTING IN SCIENCE & ENGINEERING, vol. 1, pp. 100103, 1999.
Abstract:
We call "natural" image any photograph of an outdoor or indoor
scene taken by a standard camera. We discuss the physical
generation process of natural images as a combination of
occlusions, transparencies and contrast changes. This
description fits to the phenomenological description of Gaetano
Kanizsa according to which visual perception tends to remain
stable with respect to these basic operations. We define a
contrast invariant presentation of the digital image, the
topographic map, where the subjacent occlusiontransparency
structure is put into evidence by the interplay of level lines.
We prove that each topographic map represents a class of images
invariant with respect to local contrast changes. Several
visualization strategies of the topographic map are proposed
and implemented and mathematical arguments are developed to
establish stability properties of the topographic map under
digitization.
 Caselles, V, Coll, B, and Morel, JM, "Topographic maps and local contrast changes in natural images," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 33, pp. 527, 1999.
Abstract:
We call "natural" image any photograph of an outdoor or indoor
scene taken by a standard camera. We discuss the physical
generation process of natural images as a combination of
occlusions, transparencies and contrast changes. This
description fits to the phenomenological description of Gaetano
Kanizsa according to which visual perception tends to remain
stable with respect to these basic operations. We define a
contrast invariant presentation of the digital image, the
topographic map, where the subjacent occlusiontransparency
structure is put into evidence by the interplay of level lines.
We prove that each topographic map represents a class of images
invariant with respect to local contrast changes. Several
visualization strategies of the topographic map are proposed
and implemented and mathematical arguments are developed to
establish stability properties of the topographic map under
digitization.
 Mikula, K, and Sevcovic, D, "Solution of nonlinearly curvature driven evolution of plane curves," APPLIED NUMERICAL MATHEMATICS, vol. 31, pp. 191207, 1999.
Abstract:
The evolution of plane curves obeying the equation v = beta
(k), where v is normal velocity and k curvature of the curve is
studied. Morphological image and shape multiscale analysis of
Alvarez, Guichard, Lions and Morel and affine invariant scale
space of curves introduced by Sapiro and Tannenbaum as well as
isotropic motions of plane phase interfaces studied by Angenent
and Gurtin are included in the model. We introduce and analyze
a numerical scheme for solving the governing equation and
present numerical experiments, (C) 1999 Elsevier Science B.V.
and IMACS. All rights reserved.
 Fedkiw, RP, Aslam, T, and Xu, SJ, "The ghost fluid method for deflagration and detonation discontinuities," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 154, pp. 393427, 1999.
Abstract:
The level set method for multiphase compressible flows is
simple to implement, especially in the presence of topological
changes. However, this method was shown to suffer from large
spurious oscillations. A new Ghost Fluid Method (GFM) was
developed to remove these spurious oscillations by minimizing
the numerical smearing in the entropy field with the help of an
Isobaric Fix technique. The GFM was designed for traditional
contact discontinuities where the interface moves with the
fluid velocity only. In this paper, the GFM is extended to
treat multimaterial interfaces where the interface velocity
includes a regression rate due to the presence of chemical
reactions converting one material into another. Specifically,
interface models for deflagration and detonation
discontinuities are considered. The resulting numerical method
is robust and easy to implement. (C) 1999 Academic Press.
 Oparin, A, and Abarzhi, S, "Threedimensional bubbles in RayleighTaylor instability," PHYSICS OF FLUIDS, vol. 11, pp. 33063311, 1999.
Abstract:
We study the highly nonlinear stages of the RayleighTaylor
instability (RTI) for threedimensional flow. The proposed
numerical and analytical methods are original approaches to the
problem. They validate each other and the obtained results
agree well. (C) 1999 American Institute of Physics. [S1070
6631(99)003116].
 Salden, AH, Romeny, BMT, and Viergever, MA, "Linearised euclidean shortening flow of curve geometry," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 34, pp. 2967, 1999.
Abstract:
The geometry of a space curve is described in terms of a
Euclidean invariant frame field, metric, connection, torsion
and curvature. Here the torsion and curvature of the connection
quantify the curve geometry. In order to retain a stable and
reproducible description of that geometry, such that it is
slightly affected by nonuniform protrusions of the curve, a
linearised Euclidean shortening flow is proposed. (Semi)
discretised versions of the flow subsequently physically
realise a concise and exact (semi)discrete curve geometry.
Imposing special ordering relations the torsion and curvature
in the curve geometry can be retrieved on a multiscale basis
not only for simply closed planar curves but also for open,
branching, intersecting and space curves of nontrivial knot
type. In the context of the shortening flows we revisit the
maximum principle, the semigroup property and the comparison
principle normally required in scalespace theories. We show
that our linearised flow satisfies an adapted maximum
principle, and that its Green's functions possess a semigroup
property. We argue that the comparison principle in the case of
knots can obstruct topological changes being in contradiction
with the required curve simplification principle. Our
linearised flow paradigm is not hampered by this drawback; all
nonsymmetric knots tend to trivial ones being infinitely small
circles in a plane. Finally, the differential and integral
geometry of the multiscale representation of the curve
geometry under the flow is quantified by endowing the scale
space of curves with an appropriate connection, and calculating
related torsion and curvature aspects. This multiscale modern
geometric analysis forms therewith an alternative for curve
description methods based on entropy scalespace theories.
 Aubert, G, and BlancFeraud, L, "Some remarks on the equivalence between 2D and 3D classical snakes and geodesic active contours," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 34, pp. 1928, 1999.
Abstract:
Recently, Caselles et al. have shown the equivalence between a
classical snake problem of Kass et al. and a geodesic active
contour model. The PDE derived from the geodesic problem gives
an evolution equation for active contours which is very
powerfull for image segmentation since changes of topology are
allowed using the level set implementation. However in
Caselles' paper the equivalence with classical snake is only
shown for 2D images and 1D curves, by using concepts of
Hamiltonian theory which have no meanings for active surfaces.
This paper propose to examine the notion of equivalence and to
revisite Caselles et al. arguments. Then a notion equivalence
is introduced and shown for classical snakes and geodesic
active contours in the 2D (active contour) and 3D (active
surface) case.
 Angenent, S, Haker, S, Tannenbaum, A, and Kikinis, R, "On the LaplaceBeltrami operator and brain surface flattening," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 18, pp. 700711, 1999.
Abstract:
In this paper, using certain conformal mappings from
uniformization theory, we give an explicit method far
flattening the brain surface in a way which preserves angles.
From a triangulated surface representation of the cortex, we
indicate how the procedure may be implemented using finite
elements. Further, we show how the geometry of the brain
surface may be studied using this approach.
 Jin, S, Katsoulakis, MA, and Xin, ZP, "Relaxation schemes for curvaturedependent front propagation," COMMUNICATIONS ON PURE AND APPLIED MATHEMATICS, vol. 52, pp. 15871615, 1999.
Abstract:
In this paper we study analytically and numerically a novel
relaxation approximation for front evolution according to a
curvaturedependent local law. In the ChapmanEnskog expansion,
this relaxation approximation leads to the levelset equation
for transportdominated front propagation, which includes the
mean curvature as the nextorder term. This approach yields a
new and possibly attractive way of calculating numerically the
propagation of curvaturedependent fronts. Since the relaxation
system is a symmetrizable, semilinear, and linearly convective
hyperbolic system without singularities, the relaxation scheme
captures the curvaturedependent front propagation without
discretizing directly the complicated yet singular mean
curvature term. (C) 1999 John Wiley & Sons, Inc.
 Tomlin, C, Lygeros, J, and Sastry, S, "Computing controllers for nonlinear hybrid systems," HYBRID SYSTEMS: COMPUTATION AND CONTROL, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1569, pp. 238255, 1999.
Abstract:
We discuss a procedure for synthesizing controllers for safety
specifications for hybrid systems. The procedure depends on the
construction of the set of states of a continuous dynamical
system that can be driven to a subset of the state space,
avoiding another subset of the state space (the ReachAvoid
set). We present a new characterization of the ReachAvoid set
in terms of the solution of a pair of coupled HamiltonJacobi
partial differential equations. We also discuss a computational
algorithm for solving such partial differential equations and
demonstrate its effectiveness on numerical examples.
 Wang, KC, Dutton, RW, and Taylor, CA, "Improving geometric model construction for blood flow modeling  Geometric image segmentation and imagebased model construction for computational hemodynamics," IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE, vol. 18, pp. 3339, 1999.
Abstract:
We develop a fast method to localize the level set method of
Osher and Sethian (1988, J. Comput. Phys. 79, 12) and address
two important issues that are intrinsic to the level set
method: (a) how to extend a quantity that is given only on the
interface to a neighborhood of the interface; (b) how to reset
the level set function to be a signed distance function to the
interface efficiently without appreciably moving the interface.
This fast local level set method reduces the computational
effort by one order of magnitude, works in as much generality
as the original one, and is conceptually simple and easy to
implement. Our approach differs from previous related works in
that we extract all the information needed from the level set
function (or functions in multiphase flow) and do not need to
find explicitly the location of the interface in the space
domain. The complexity of our method to do tasks such as
extension and distance reinitialization is O (N), where N is
the number of points in space, not O(N log N) as in works by
Sethian (1996, Proc. Not. Acad. Sci. 93, 1591) and Helmsen and
coworkers (1996, SPIE Microlithography IX, p. 253). This
complexity estimation is also valid for quite general
geometrically based front motion for our localized method. (C)
1999 Academic Press.
 Peng, DP, Merriman, B, Osher, S, Zhao, HK, and Kang, MJ, "A PDEbased fast local level set method," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 155, pp. 410438, 1999.
Abstract:
We develop a fast method to localize the level set method of
Osher and Sethian (1988, J. Comput. Phys. 79, 12) and address
two important issues that are intrinsic to the level set
method: (a) how to extend a quantity that is given only on the
interface to a neighborhood of the interface; (b) how to reset
the level set function to be a signed distance function to the
interface efficiently without appreciably moving the interface.
This fast local level set method reduces the computational
effort by one order of magnitude, works in as much generality
as the original one, and is conceptually simple and easy to
implement. Our approach differs from previous related works in
that we extract all the information needed from the level set
function (or functions in multiphase flow) and do not need to
find explicitly the location of the interface in the space
domain. The complexity of our method to do tasks such as
extension and distance reinitialization is O (N), where N is
the number of points in space, not O(N log N) as in works by
Sethian (1996, Proc. Not. Acad. Sci. 93, 1591) and Helmsen and
coworkers (1996, SPIE Microlithography IX, p. 253). This
complexity estimation is also valid for quite general
geometrically based front motion for our localized method. (C)
1999 Academic Press.
 Xiao, F, "A computational model for suspended large rigid bodies in 3D unsteady viscous flows," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 155, pp. 348379, 1999.
Abstract:
A 3D numerical model for computing large rigid objects
suspended in fluid flow has been developed. Rather than
calculating the surface pressure upon the solid body, we
evaluate the net force and torque based on a volume force
formulation. The total effective force is obtained by summing
up the forces at the Eulerian grids occupied by the rigid body.
The effects of the moving bodies are coupled to the fluid flow
by imposing the velocity field of the bodies to the fluid. A
Poisson equation is used to compute the pressure over the whole
domain. The objects are identified by color functions and
calculated by the PPM scheme and a tangent function
transformation which scales the transition region of the
computed interface to a compact thickness. The model is then
implemented on a parallel computer of distributed memory and
validated with Stokes and low Reynolds number hows. (C) 1999
Academic Press.
 Hu, CQ, and Shu, CW, "A discontinuous Galerkin finite element method for Hamilton Jacobi equations," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 21, pp. 666690, 1999.
Abstract:
In this paper, we present a discontinuous Galerkin finite
element method for solving the nonlinear HamiltonJacobi
equations. This method is based on the RungeKutta
discontinuous Galerkin finite element method for solving
conservation laws. The method has the flexibility of treating
complicated geometry by using arbitrary triangulation, can
achieve highorder accuracy with a local, compact stencil, and
is suited for efficient parallel implementation. One and two
dimensional numerical examples are given to illustrate the
capability of the method. At least kth order of accuracy is
observed for smooth problems when kth degree polynomials are
used, and derivative singularities are resolved well without
oscillations, even without limiters.
 Gremaud, PA, and Ide, NR, "Computation of nonclassical solutions to HamiltonJacobi problems," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 21, pp. 502521, 1999.
Abstract:
This paper is devoted to the construction of numerical methods
for the approximation of nonclassical solutions to
multidimensional Hamilton{Jacobi equations for both scalar and
vectorial problems. Recent theoretical results have yielded
existence of solutions in many cases for which the usual
viscosity approach was illsuited or not applicable. The
selection criterion used here is based on a
viscoelasticity/capillarity approach, common in solid
mechanics. Numerical methods adapted to this framework are
built. Consistency of the model equation with the given
selection criterion is essential. It is achieved here through
the use of highorder finite difference schemes. By considering
applications to potential well problems, the convergence of the
methods are investigated.
 August, J, Siddiqi, K, and Zucker, SW, "Contour fragment grouping and shared, simple occluders," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 76, pp. 146162, 1999.
Abstract:
Bounding contours of physical objects are often fragmented by
other occluding objects. Longdistance perceptual grouping
seeks to join fragments belonging to the same object.
Approaches to grouping based on invariants assume objects are
in restricted classes, while those based on minimal energy
continuations assume a shape for the missing contours and
require this shape to drive the grouping process. While these
assumptions may be appropriate for certain specific tasks or
when contour gaps are small, in general occlusion can give rise
to large gaps, and thus longdistance contour fragment grouping
is a different type of perceptual organization problem. We
propose the longdistance principle that those fragments should
be grouped whose fragmentation could have arisen from a shared,
simple occluder. The gap skeleton is introduced as a
representation of this virtual occluder, and an algorithm for
computing it is given. Finally, we show that a view of the
virtual occluder as a disk can be interpreted as an equivalence
class of curves interpolating the fragment endpoints. (C) 1999
Academic Press.
 Chopp, D, Evans, LC, and Ishii, H, "Waiting time effects for Gauss curvature flows," INDIANA UNIVERSITY MATHEMATICS JOURNAL, vol. 48, pp. 311334, 1999.
Abstract:
R. Hamilton in [Ham1] proved that a planar region on a convex
hypersurface does not "instantly bend", and so instantly
vanish, under Gauss curvature flow. We demonstrate that if the
surface is smooth, the planar region in fact does not move at
all for some positive time. This is a sort of geometric
analogue of "waiting time" phenomena for the porous medium
equation.
 Chung, EH, and Kwon, S, "The effect of volume expansion on the propagation of wrinkled laminar premixed flame," COMBUSTION SCIENCE AND TECHNOLOGY, vol. 146, pp. 85103, 1999.
Abstract:
Past studies using Gequation successfully described the effect
of flame stretch on the laminar flame propagation. In those
studies, flames were regarded as a passive interface that did
not influence the flow field. The experimental evidences,
however, suggested that flow field was significantly modified
by the flames as the burned gas expanded at the flame. A method
using Gequation and BiotSavart law to approximate induced
velocity field is described to estimate the effect of volume
expansion. Present method was applied to initially wrinkled and
planar flames propagating in an imposed velocity field and the
average flame speed was evaluated from the ratio of simulated
flame surface area and projected area of unburned stream
channel. It was found that the initial wrinkling of flame could
not sustain itself without Velocity disturbance but decayed
into planar flame. The rate of decay of the wrinkles increased
as the volume expansion ratio increased, The asymptotic change
in the average burning speed occurred only in a disturbed
velocity field. The average burning speed was always affected
by the volume expansion that directly influenced the velocity
field. With relatively small expansion ratio of 3, the average
flame speed increased 10%. The comparison of the relative
significance of volume expansion and flame stretch suggested
that the effect of volume expansion was no less important than
that of flame stretch and warranted that both of the effects
should be taken into account in the simulation of flame
propagation.
 Siddiqi, K, Shokoufandeh, A, Dickinson, SJ, and Zucker, SW, "Shock graphs and shape matching," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 35, pp. 1332, 1999.
Abstract:
We have been developing a theory for the generic representation
of 2D shape, where structural descriptions are derived from
the shocks (singularities) of a curve evolution process, acting
on bounding contours. We now apply the theory to the problem of
shape matching. The shocks are organized into a directed,
acyclic shock graph, and complexity is managed by attending to
the most significant (central) shape components first. The
space of all such graphs is highly structured and can be
characterized by the rules of a shock graph grammar. The
grammar permits a reduction of a shock graph to a unique rooted
shock tree. We introduce a novel tree matching algorithm which
finds the best set of corresponding nodes between two shock
trees in polynomial time. Using a diverse database of shapes,
we demonstrate our system's performance under articulation,
occlusion, and moderate changes in viewpoint.
 McInerney, T, and Terzopoulos, D, "Topology adaptive deformable surfaces for medical image volume segmentation," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 18, pp. 840850, 1999.
Abstract:
Deformable models, which include deformable contours (the
popular snakes) and deformable Surfaces, are a powerful model
based medical image analysis technique. We develop a new class
of deformable models by formulating deformable surfaces in
terms of an affine cell image decomposition (ACID). Our
approach significantly extends standard deformable surfaces,
while retaining their interactivity and other desirable
properties. In particular, the ACID induces an efficient
reparameterization mechanism that enables parametric deformable
surfaces to evolve into complex geometries, even modifying
their topology as necessary. We demonstrate that our new ACID
based deformable surfaces, dubbed Tsurfaces, can effectively
segment complex anatomic structures from medical volume images.
 Olver, PJ, Sapiro, G, and Tannenbaum, A, "Affine invariant detection: Edge maps, anisotropic diffusion, and active contours," ACTA APPLICANDAE MATHEMATICAE, vol. 59, pp. 4577, 1999.
Abstract:
In this paper we undertake a systematic investigation of affine
invariant object detection and image denoising. Edge detection
is first presented from the point of view of the affine
invariant scalespace obtained by curvature based motion of the
image levelsets. In this case, affine invariant maps are
derived as a weighted difference of images at different scales.
We then introduce the affine gradient as an affine invariant
differential function of lowest possible order with qualitative
behavior similar to the Euclidean gradient magnitude. These
edge detectors are the basis for the extension of the affine
invariant scalespace to a complete affine flow for image
denoising and simplification, and to define affine invariant
active contours for object detection and edge integration. The
active contours are obtained as a gradient flow in a
conformally Euclidean space defined by the image on which the
object is to be detected. That is, we show that objects can be
segmented in an affine invariant manner by computing a path of
minimal weighted affine distance, the weight being given by
functions of the affine edge detectors. The gradient path is
computed via an algorithm which allows to simultaneously detect
any number of objects independently of the initial curve
topology. Based on the same theory of affine invariant gradient
flows we show that the affine geometric heat flow is
minimizing, in an affine invariant form, the area enclosed by
the curve.
 Kim, S, and Cook, R, "3D traveltime computation using secondorder ENO scheme," GEOPHYSICS, vol. 64, pp. 18671876, 1999.
Abstract:
We consider a secondorder finite difference scheme to solve
the eikonal equation. Upwind differences are requisite to
sharply resolve discontinuities in the traveltime derivatives,
whereas centered differences improve the accuracy of the
computed traveltime, A secondorder upwind essentially non
oscillatory (ENO) scheme satisfies these requirements. It is
implemented with a dynamic down 'n' out (DNO) marching, an
expanding box approach. To overcome the instability of such an
expanding box scheme, the algorithm incorporates an efficient
post sweeping (PS), a correctionbyiteration method. Near the
source, an efficient and accurate meshrefinement
initialization scheme is suggested for the DNO marching, The
resulting algorithm, ENODNOPS, turns out to be
unconditionally stable, of secondorder accuracy, and
efficient; for various synthetic and real velocity models
having large contrasts, two PS iterations produce traveltimes
accurate enough to complete the computation.
 Rekeczky, C, and Chua, LO, "Computing with front propagation: Active contour and skeleton models in continuoustime CNN," JOURNAL OF VLSI SIGNAL PROCESSING SYSTEMS FOR SIGNAL IMAGE AND VIDEO TECHNOLOGY, vol. 23, pp. 373402, 1999.
Abstract:
In this paper, a linear CNN template class is studied with a
symmetric feedback matrix capable of generating triggerwaves,
a special type of binary travelingwave. The qualitative
properties of these waves are examined and some simple control
strategies are derived based on modifying the bias and feedback
terms in a CNN template. It is shown that a properly controlled
wavefront can be efficiently used in segmentation, shape and
structure detection/recovery tasks. Shape is represented by the
contour of an evolving front. An algorithmic framework is
discussed that incorporates bias controlled triggerwaves in
tracking the active contour of the objects during rigid and
nonrigid motion. The object skeleton (structure) is obtained
as a composition of stable annihilation lines formed during the
collision of trigger wavefronts. The shortest path problem in
a binary labyrinth is also formulated as a special type of
skeletonization task and solved by combined triggerwave based
techniques.
 Marquina, A, and Osher, S, "A new time dependent model based on level set motion for nonlinear deblurring and noise removal," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 429434, 1999.
Abstract:
In this paper we summarize the main features of a new time
dependent model to approximate the solution to the nonlinear
total variation optimization problem for deblurring and noise
removal introduced by Rudin, Osher and Fatemi. Our model is
based on level set motion whose steady state is quickly reached
by means of an explicit procedure based on an ENO Hamilton
Jacobi version of Roe's scheme. We show numerical evidence of
the speed, resolution and stability of this simple explicit
procedure in two representative 1D and 2D numerical examples.
 Maragos, P, and Meyer, F, "Nonlinear PDEs and numerical algorithms for modeling levelings and reconstruction filters," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 363374, 1999.
Abstract:
In this paper we develop partial differential equations (PDEs)
that model the generation of a large class of morphological
filters, the levelings and the openings/closings by
reconstruction. These types of filters are very useful in
numerous image analysis and vision tasks ranging from
enhancement, to geometric feature detection, to segmentation.
The developed PDEs are nonlinear functions of the first spatial
derivatives and model these nonlinear filters as the limit of a
controlled growth starting from an initial seed signal. This
growth is of the multiscale dilation or erosion type and the
controlling mechanism is a switch that reverses the growth when
the difference between the current evolution and a reference
signal switches signs. We discuss theoretical aspects of these
PDEs, propose discrete algorithms for their numerical solution
and corresponding filter implementation, and provide insights
via several experiments. Finally, we outline the use of these
PDEs for improving the Gaussian scalespace by using the latter
as initial seed to generate: multiscale levelings that have a
superior preservation of image edges and boundaries.
 Meyer, F, and Maragos, P, "Multiscale morphological segmentations based on watershed, flooding, and eikonal PDE," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 351362, 1999.
Abstract:
The classical morphological segmentation paradigm is based on
the watershed transform, constructed by flooding the gradient
image seen as a topographic surface. For flooding a topographic
surface, a topographic distance is defined from which a minimum
distance algorithm is derived for the watershed. In a
continuous formulation, this is modeled via the eikonal PDE,
which can be solved using curve evolution algorithms. Various
ultrametric distances between the catchment basins may then be
associated to the flooding itself. To each ultrametric distance
is associated a multiscale segmentation; each scale being the
closed balls of the ultrametric distance.
 Bertalmio, M, Sapiro, G, and Randall, G, "Region tracking on surfaces deforming via levelsets methods," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 330338, 1999.
Abstract:
Since the work by Osher and Sethian on levelsets algorithms
for numerical shape evolutions, this technique has been used
for a large number of applications in numerous fields. In
medical imaging, this numerical technique has been successfully
used for example in segmentation and cortex unfolding
algorithms. The migration from a Lagrangian implementation to
an Eulerian one via implicit representations or levelsets
brought some of the main advantages of the technique, mainly,
topology independence and stability. This migration means also
that the evolution is parametrization free, and therefore we do
not know exactly how each part of the shape is deforming, and
the pointwise correspondence is lost. In this note we present
a technique to numerically track regions on surfaces that are
being deformed using the levelsets method. The basic idea is
to represent the region of interest as the intersection of two
implicit surfaces, and then track its deformation from the
deformation of these surfaces. This technique then solves one
of the main shortcomings of the very useful levelsets
approach. Applications include lesion localization in medical
images, region tracking in functional MRI visualization, and
geometric surface mapping.
 Samson, C, BlancFeraud, L, Aubert, G, and Zerubia, J, "A level set model for image classification," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 306317, 1999.
Abstract:
We present a supervised classification model based on a
variational approach. This model is devoted to find an optimal
partition compound of homogeneous classes with regular
interfaces. We represent the regions of the image defined by
the classes and their interfaces by level set functions, and we
define a functional whose minimum is an optimal partition. The
coupled Partial Differential Equations (PDE) related to the
minimization of the functional axe considered through a
dynamical scheme. Given an initial interface set (zero level
set), the different terms of the PDE's are governing the motion
of interfaces such that, at convergence, we get an optimal
partition as defined above. Each interface is guided by
internal forces (regularity of the interface), and external
ones (data term, no vacuum, no regions overlapping). Several
experiments were conducted on both synthetic an real images.
 van den Boomgaard, R, "Numerical solution schemes for continuousscale morphology," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 199210, 1999.
Abstract:
The partial differential equations describing the propagation
of (wave) fronts in space are closely connected with the
morphological erosion and dilation. Strangely enough this
connection has not been explored in the derivation of numerical
schemes to solve the differential equations. In this paper the
morphological facet model is introduced in which an analytical
function is locally fitted to the data. This function is then
dilated analytically with an infinitesimal small structuring
element. These subpixel dilations form the core of the
numerical solution schemes presented in this paper. One of the
simpler morphological facet models leads to a numerical scheme
that is identical with a well known classical upwind finite
difference scheme. Experiments show that the morphological
facet model provides stable numerical solution schemes for
these partial differential equations.
 Chan, T, and Vese, L, "An active contour model without edges," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 141151, 1999.
Abstract:
In this paper, we propose a new model for active contours to
detect objects in a given image, based on techniques of curve
evolution, MumfordShah functional for segmentation and level
sets. Our model can detect objects whose boundaries are not
necessarily defined by gradient. The model is a combination
between more classical active contour models using mean
curvature motion techniques, and the MumfordShah model for
segmentation. We minimize an energy which can be seen as a
particular case of the socalled minimal partition problem. In
the level set formulation, the problem becomes a "mean
curvature flow" like evolving the active contour, which will
stop on the desired boundary. However, the stopping term does
not depend on the gradient of the image, as in the classical
active contour models, but is instead related to a particular
segmentation of the image. Finally, we will present various
experimental results and in particular some examples for which
the classical snakes methods based on the gradient are not
applicable.
 Gomes, J, and Faugeras, O, "Reconciling distance functions and level sets," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 7081, 1999.
Abstract:
This paper is concerned with the simulation of the Partial
Differential Equation (PDE) driven evolution of a closed
surface by means of an implicit representation. In most
applications, the natural choice for the implicit
representation is the signed distance function to the closed
surface. Osher and Sethian propose to evolve the distance
function with a HamiltonJacobi equation. Unfortunately the
solution to this equation is not a distance function. As a
consequence, the practical application of the level set method
is plagued with such questions as when do we have to
"reinitialize" the distance function? How do we "reinitialize"
the distance function? Etc... which reveal a disagreement
between the theory and its implementation. This paper proposes
an alternative to the use of HamiltonJacobi equations which
eliminates this contradiction: in our method the implicit
representation always remains a distance function by
construction, and the implementation does not differ from the
theory anymore. This is achieved through the introduction of a
new equation. Besides its theoretical advantages, the proposed
method also has several practical advantages which we
demonstrate in two applications: (i) the segmentation of the
human cortex surfaces from MRI images using two coupled
surfaces [26], (ii) the construction of a hierarchy of
Euclidean skeletons of a 3D surface.
 Hermosillo, G, Faugeras, O, and Gomes, J, "Unfolding the cerebral cortex using level set methods," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 5869, 1999.
Abstract:
Level set methods provide a robust way to implement geometric
flows, but they suffer from two problems which are relevant
when using smoothing flows to unfold the cortex: the lack of
pointcorrespondence between scales and the inability to
implement tangential velocities. In this paper, we suggest to
solve these problems by driving the nodes of a mesh with an
ordinary differential equation. We state that this approach
does not suffer from the known problems of Lagrangian methods
since all geometrical properties axe computed on the fixed
(Eulerian) grid. Additionally, tangential velocities can be
given to the nodes, allowing the mesh to follow general
evolution equations, which could be crucial to achieving the
final goal of minimizing local metric distortions. To
experiment with this approach, we derive area and volume
preserving mean curvature flows and use them to unfold surfaces
extracted from MRI data of the human brain.
 Bertalmio, M, Sapiro, G, and Randall, G, "Morphing active contours," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 4657, 1999.
Abstract:
A method for deforming curves in a given image to a desired
position in a second image is introduced in this paper. The
algorithm is based on deforming the first image toward the
second one via a partial differential equation, while tracking
the deformation of the curves of interest in the first image
with an additional, coupled, partial differential equation. The
tracking is performed by projecting the velocities of the first
equation into the second one. In contrast with previous PDE
based approaches, both the images and the curves on the
frames/slices of interest axe used for tracking. The technique
can be applied to object tracking and sequential segmentation.
The topology of the deforming curve can change, without any
special topology handling procedures added to the scheme. This
permits for example the automatic tracking of scenes where, due
to occlusions, the topology of the objects of interest changes
from frame to frame. In addition, this work introduces the
concept of projecting velocities to obtain systems of coupled
partial differential equations for image analysis applications.
We show examples for object tracking and segmentation of
electronic microscopy. We also briefly discuss possible uses of
this framework for three dimensional morphing.
 Goldenberg, R, Kimmel, R, Rivlin, E, and Rudzsky, M, "Fast geodesic active contours," SCALESPACE THEORIES IN COMPUTER VISION, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1682, pp. 3445, 1999.
Abstract:
We use an unconditionally stable numerical scheme to implement
a fast version of the geodesic active contour model. The
proposed scheme is useful for object segmentation in images,
like tracking moving objects in a sequence of images. The
method is based on the WeickertRomeneyViergever [33] AOS
scheme. It is applied at small regions, motivated by
AdalsteinssonSethian [1] level set narrow band approach, and
uses Sethian's fast marching method [26] for reinitialization.
Experimental results demonstrate the power of the new method
for tracking in color movies.


2000 
 Betelu, SI, Aronson, DG, and Angenent, SB, "Renormalization study of twodimensional convergent solutions of the porous medium equation," PHYSICA D, vol. 138, pp. 344359, 2000.
Abstract:
Tn the Focusing problem, we study a solution of the porous
medium equation u(t) = Delta(u(m)) whose initial distribution
is positive in the exterior of a closed noncircular two
dimensional region, and zero inside. We implement a numerical
scheme that renormalizes the solution each time that the
average size of the empty region reduces by a half. The initial
condition is a function with circular level sets distorted with
a small sinusoidal perturbation of wave number k > 3. We find
that for nonlinearity exponents m smaller than a critical value
which depends on k, the solution tends to a selfsimilar
regime, characterized by rounded polygonal interfaces and
similarity exponents that depend on m and on the discrete
rotational symmetry number k. For m greater than the critical
value, the final form of the interface is circular. (C) 2000
Elsevier Science B.V. All rights reserved.
 Smereka, P, "Spiral crystal growth," PHYSICA D, vol. 138, pp. 282301, 2000.
Abstract:
We numerically study the spiral mode of crystal growth using a
theory developed by Burton, Cabrera and Frank using a level set
method. This method is novel in that it can handle not only
closed curves but open curves as well. We use our method to
compute interacting spirals and make estimates of growth rates.
We also propose a possible coarsening mechanism for a large
number of interacting spirals. (C) 2000 Elsevier Science B.V.
All rights reserved.
 Mayer, UF, "A numerical scheme for moving boundary problems that are gradient flows for the area functional," EUROPEAN JOURNAL OF APPLIED MATHEMATICS, vol. 11, pp. 6180, 2000.
Abstract:
Many moving boundary problems that are driven in some way by
the curvature of the free boundary are gradient flows for the
area of the moving interface. Examples are the MullinsSekerka
flow, the HeleShaw flow, flow by mean curvature, and flow by
averaged mean curvature. The gradient flow structure suggests
an implicit finite differences approach to compute numerical
solutions. The proposed numerical scheme will allow us to treat
such free boundary problems in both R2 and R3. The advantage
of such an approach is the reusability of much of the setup for
all of the different problems. As an example of the method, we
compute solutions to the averaged mean curvature flow that
exhibit the formation of a singularity.
 Ruuth, SJ, and Merriman, B, "Convolutiongenerated motion and generalized Huygens' principles for interface motion," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 60, pp. 868890, 2000.
Abstract:
A physical interface can often be modeled as a surface that
moves with a velocity determined by the local geometry.
Accordingly, there is great interest in algorithms that
generate such geometric interface motion. In this paper we
unify and generalize two simple algorithms for constant and
mean curvature based interface motion: the classical Huygens'
principle and diffusiongenerated motion. We show that the
resulting generalization can be viewed both geometrically as a
type of Huygens' principle and algebraically as a convolution
generated motion. Using the geometricalgebraic duality from
the unification, we construct specific convolutiongenerated
motion algorithms for a common class of anisotropic, curvature
dependent motion laws. We validate these algorithms with
numerical experiments and show that they can be implemented
accurately and efficiently with adaptive resolution and fast
Fourier transform techniques.
 Liao, GJ, Liu, F, de la Pena, GC, Peng, DP, and Osher, S, "Levelsetbased deformation methods for adaptive grids," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 159, pp. 103122, 2000.
Abstract:
A new method for generating adaptive moving grids is formulated
based on physical quantities. Level set functions are used to
construct the adaptive grids, which are solutions of the
standard level set evolution equation with the Cartesian
coordinates as initial values. The intersection points of the
level sets of the evolving functions form a new grid at each
time. The velocity vector in the evolution equation is chosen
according to a monitor function and is equal to the node
velocity. A uniform grid is then deformed to a moving grid with
desired cell volume distribution at each time. The method
achieves precise control over the Jacobian determinant of the
grid mapping as the traditional deformation method does. The
new method is consistent with the level set approach to dynamic
moving interface problems. (C) 2000 Academic Press.
 Lepsky, O, Hu, CQ, and Shu, CW, "Analysis of the discontinuous Galerkin method for Hamilton Jacobi equations," APPLIED NUMERICAL MATHEMATICS, vol. 33, pp. 423434, 2000.
Abstract:
HamiltonJacobi equations are frequently encountered in
applications, e.g., calculus of variations, control theory and
differential games. in this paper a discontinuous Galerkin
finite element method for nonlinear HamiltonJacobi equations
(first proposed by Hu and Shu (to appear)) is investigated.
This method handles the complicated geometry by using arbitrary
triangulation, achieves the high order accuracy in smooth
regions and the high resolution of the derivatives
discontinuities. Theoretical results on accuracy and stability
properties of the method are proved for certain cases and
related numerical examples are presented. (C) 2000 IMACS.
Published by Elsevier Science B.V. Ail rights reserved.
 Paragios, N, and Deriche, R, "Geodesic active contours and level sets for the detection and tracking of moving objects," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 22, pp. 266280, 2000.
Abstract:
This paper presents a new variational framework for detecting
and tracking multiple moving objects in image sequences. Motion
detection is performed using a statistical framework for which
the observed interframe difference density function is
approximated using a mixture model. This model is composed of
two components, namely, the static (background) and the mobile
(moving objects) one. Both components are zeromean and obey
Laplacian or Gaussian law. This statistical framework is used
to provide the motion detection boundaries. Additionally, the
original frame is used to provide the moving object boundaries.
Then, the detection and the tracking problem are addressed in a
common framework that employs a geodesic active contour
objective function. This function is minimized using a gradient
descent method, where a flow deforms the initial curve towards
the minimum of the objective function, under the influence of
internal and external image dependent forces. Using the level
set formulation scheme, complex curves can be detected and
tracked white topological changes for the evolving curves are
naturally managed. To reduce the computational cost required by
a direct implementation of the level set formulation scheme, a
new approach named Hermes is proposed. Hermes exploits aspects
from the wellknown front propagation algorithms (Narrow Band.
Fast Marching) and compares favorably to them. Very promising
experimental results are provided using real video sequences.
 Liu, XD, Fedkiw, RP, and Kang, MJ, "A boundary condition capturing method for Poisson's equation on irregular domains," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 160, pp. 151178, 2000.
Abstract:
Interfaces have a variety of boundary conditions (or jump
conditions) that need to be enforced. The Ghost Fluid Method
(GFM) was developed to capture the boundary conditions at a
contact discontinuity in the inviscid Euler equations and has
been extended to treat more general discontinuities such as
shocks, detonations, and deflagrations and compressible viscous
flows. In this paper, a similar boundary condition capturing
approach is used to develop a new numerical method for the
variable coefficient Poisson equation in the presence of
interfaces where both the variable coefficients and the
solution itself may be discontinuous. This new method is robust
and easy to implement even in three spatial dimensions.
Furthermore, the coefficient matrix of the associated linear
system is the standard symmetric matrix for the variable
coefficient Poisson equation in the absence of interfaces
allowing for straightforward application of standard "black
box" solvers. (C) 2000 Academic Press.
 Jia, W, "Transport coordinate (TC) method for the dynamics of multiple materials," JOURNAL OF FLUIDS ENGINEERINGTRANSACTIONS OF THE ASME, vol. 122, pp. 125133, 2000.
Abstract:
In this paper we propose a new idea of tracking material
interface. Since the regions filled with different materials at
the initial time are merely transported by the velocity field,
the material type at present is determined by its original
location. me introduce the advection equation of the base
coordinates to specify the material type, and solve this
equation in the Euler framework. Thanks to the initial linear
distribution, this method is free of numerical diffusion for
the problems with a constant or a rigid body rotation velocity
field and can produce accurate results for the general case.
Moreover, it is applicable to the advection function of
arbitrary distribution, for example, problems with move than
two types of fluids. The new method is incorporated into a
newly developed flow solver employing the semiLagrangian model
to successfully solve the flow problems with multiple types of
fluids. [S00982202(00)020010].
 Ravi, D, "A new active contour model for shape extraction," MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol. 23, pp. 709722, 2000.
Abstract:
We propose a new active contour model for shape extraction of
objects in greyvalued twodimensional images based on an
energyminimization formulation. The energy functional that we
consider takes into account the two requirements of object
isolation and smoothness of the contour. After deriving the
EulerLagrange equations corresponding to the energy
functional, we bring out some important geometric properties of
a solution to these equations. The discussion on our solution
methodwith the help of which we try to minimize the energy
functional by evolving an initial curvealso focuses on how to
prescribe the initial curve fully automatically. The
effectiveness of our algorithms is demonstrated with the help
of experimental results. Copyright (C) 2000 John Wiley & Sons,
Ltd.
 Ratsch, C, Gyure, MF, Chen, S, Kang, M, and Vvedensky, DD, "Fluctuations and scaling in aggregation phenomena," PHYSICAL REVIEW B, vol. 61, pp. 1059810601, 2000.
Abstract:
We introduce a method which enables us to isolate different
sources of fluctuations during a typical aggregation process.
As an example, we focus on the evolution of islands during
irreversible submonolayer epitaxy. We show that only spatial
fluctuations in island nucleation are required to produce the
scaling of their size distribution as determined by Monte Carlo
simulations. In particular, once the islands are seeded, their
growth can be described in a purely deterministic manner.
 Hillebrandt, W, Reinecke, M, and Niemeyer, JC, "Thermonuclear supernovae," COMPUTER PHYSICS COMMUNICATIONS, vol. 127, pp. 5358, 2000.
Abstract:
We present a new way of modeling turbulent thermonuclear
deflagration fronts in white dwarfs undergoing a type Ia
supernova explosion. Our approach is based on a level set
method which treats the front as a mathematical discontinuity
and allows for full coupling between the front geometry and the
flow field. First results of the method applied to the problem
of type Ia supernovae are discussed. It will be shown that even
in 2D and even with a physically motivated subgrid model
numerically "converged" results are difficult to obtain. (C)
2000 Published by Elsevier Science B.V. All rights reserved.
 Chan, CK, Lau, KS, and Zhang, BL, "Simulation of a premixed turbulent flame with the discrete vortex method," INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, vol. 48, pp. 613627, 2000.
Abstract:
The behaviour of a premixed turbulent flame is numerically
studied in this paper. The numerical model is based on solving
turbulent flow field by the discrete vortex method. The flame
is considered to be of zero thickness boundary which separates
burnt and unburnt regions with different constant density and
propagates into the fresh mixture at a local curvature
dependent flame speed. The flame front is located by means of
levelset algorithm. The flow turbulence is simulated through
the unsteady vortexshedding mechanism. The computed velocity
fields, turbulence scalar statistics as well as flame brush
thickness for the turbulent Vflame are well comparable to
experimental results. The computed Reynolds stresses in the
flame brush region based on unconditioned velocities are
substantial, but the two conditioned Reynolds stresses are
negligible. These results show that the intermittency effect is
a major influence on turbulent statistics in premixed flame and
should require careful consideration in numerical models.
Copyright (C) 2000 John Wiley & Sons, Ltd.
 Meyer, F, and Maragos, P, "Nonlinear scalespace representation with morphological levelings," JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION, vol. 11, pp. 245265, 2000.
Abstract:
In this paper we present a nonlinear scalespace representation
based on a general class of morphological strong filters, the
levelings, which include the openings and closings by
reconstruction. These filters are very useful for image
simplification and segmentation. From one scale to the next,
details vanish, but the contours of the remaining objects are
preserved sharp and perfectly localized. Both the lattice
algebraic and the scalespace properties of levelings are
analyzed and illustrated. We also develop a nonlinear partial
differential equation that models the generation of levelings
as the limit of a controlled growth starting from an initial
seed signal. Finally, we outline the use of levelings in
improving the Gaussian scalespace by using the latter as an
initial seed to generate multiscale levelings that have a
superior preservation of image edges. (C) 2000 Academic Press.
 Gomes, J, and Faugeras, O, "Reconciling distance functions and level sets," JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION, vol. 11, pp. 209223, 2000.
Abstract:
This paper is concerned with the simulation of the partial
differential equation driven evolution of a closed surface by
means of an implicit representation. In most applications, the
natural choice for the implicit representation is the signed
distance function to the closed surface. Osher and Sethian have
proposed to evolve the distance function with a HamiltonJacobi
equation. Unfortunately the solution to this equation is not a
distance function. As a consequence, the practical application
of the level set method is plagued with such questions as When
do we have to reinitialize the distance function? How do we
reinitialize the distance function?, which reveal a
disagreement between the theory and its implementation. This
paper proposes an alternative to the use of HamiltonJacobi
equations which eliminates this contradiction: in our method
the implicit representation always remains a distance function
by construction, and the implementation does not differ from
the theory anymore. This is achieved through the introduction
of a new equation. Besides its theoretical advantages, the
proposed method also has several practical advantages which we
demonstrate in three applications: (i) the segmentation of the
human cortex surfaces from MRI images using two coupled
surfaces (X. Zeng, et al., in Proceedings of the International
Conference on Computer Vision and Pattern Recognition, June
1998), (ii) the construction of a hierarchy of Euclidean
skeletons of a 3D surface, (iii) the reconstruction of the
surface of 3D objects through stereo (O. Faugeras and R.
Keriven, Lecture Notes in Computer Science, Vol. 1252, pp. 272
283). (C) 2000 Academic Press.
 Shah, J, "Riemannian drums, anisotropic curve evolution, and segmentation," JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION, vol. 11, pp. 142153, 2000.
Abstract:
The method of curve evolution is a popular method for
recovering shape boundaries. However, isotropic metrics have
always been used to induce the how of the curve and potential
steady states tend to be difficult to determine numerically,
especially in noisy or towcontrast situations. Initial curves
shrink past the steady slate and soon vanish. In this paper,
anisotropic metrics are considered to remedy the situation by
taking the orientation of the feature gradient into account.
The problem of shape recovery or segmentation is formulated as
the problem of finding minimum cuts of a Riemannian manifold.
Approximate methods, namely anisotropic geodesic flows and the
solution of an eigenvalue problem, are discussed. (C) 2000
Academic Press.
 Chan, TE, Sandberg, BY, and Vese, LA, "Active contours without edges for vectorvalued images," JOURNAL OF VISUAL COMMUNICATION AND IMAGE REPRESENTATION, vol. 11, pp. 130141, 2000.
Abstract:
In this paper, we propose an active contour algorithm for
object detection in vectorvalued images (such as RGB or
multispectral). The model is an extension of the scalar Chan
Vese algorithm to the vectorvalued case [1]. The model
minimizes a MumfordShah functional over the length of the
contour, plus the sum of the fitting error over each component
of the vectorvalued image. Like the ChanVese model, our
vectorvalued model can detect edges both with or without
gradient. We show examples where our model detects vector
valued objects which are undetectable in any scalar
representation. For instance, objects with different missing
parts in different channels are completely detected (such as
occlusion). Also, in color images, objects which are invisible
in each channel or in intensity can be detected by our
algorithm. Finally, the model is robust with respect to noise,
requiring no a priori denoising step. (C) 2000 Academic Press.
 White, B, "The size of the singular set in mean curvature flow of mean convex sets," JOURNAL OF THE AMERICAN MATHEMATICAL SOCIETY, vol. 13, pp. 665695, 2000.
Abstract:
The Fast Marching Method is a numerical algorithm for solving
the Eikonal equation on a rectangular orthogonal mesh in O(M
log M) steps, where M is the total number of grid points. The
scheme relies on an upwind finite difference approximation to
the gradient and a resulting causality relationship that lends
itself to a Dijkstralike programming approach. In this paper,
we discuss several extensions to this technique, including
higher order versions on unstructured meshes in Rn and on
manifolds and connections to more general static Hamilton
Jacobi equations.
 Sethian, JA, and Vladimirsky, A, "Fast methods for the Eikonal and related HamiltonJacobi equations on unstructured meshes," PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, vol. 97, pp. 56995703, 2000.
Abstract:
The Fast Marching Method is a numerical algorithm for solving
the Eikonal equation on a rectangular orthogonal mesh in O(M
log M) steps, where M is the total number of grid points. The
scheme relies on an upwind finite difference approximation to
the gradient and a resulting causality relationship that lends
itself to a Dijkstralike programming approach. In this paper,
we discuss several extensions to this technique, including
higher order versions on unstructured meshes in Rn and on
manifolds and connections to more general static Hamilton
Jacobi equations.
 Sarti, A, Malladi, R, and Sethian, JA, "Subjective surfaces: A method for completing missing boundaries," PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, vol. 97, pp. 62586263, 2000.
Abstract:
We present a model and algorithm for segmentation of images
with missing boundaries. In many situations. the human visual
system fills in missing gaps in edges and boundaries, building
and completing information that is not present This presents a
considerable challenge in computer vision, since most
algorithms attempt to exploit existing data. Completion models,
which postulate how to construct missing data, are popular but
are often trained and specific to particular images. In this
paper, we take the following perspective: We consider a
reference point within an image as given and then develop an
algorithm that tries to build missing information on the basis
of the given point of view and the available information as
boundary data to the algorithm. We test the algorithm on some
standard images, including the classical triangle of Kanizsa
and low signal:noise ratio medical images.
 Lin, CT, and Tadmor, E, "Highresolution nonoscillatory central schemes for Hamilton Jacobi equations," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 21, pp. 21632186, 2000.
Abstract:
In this paper, we construct secondorder central schemes for
multidimensional Hamilton Jacobi equations and we show that
they are nonoscillatory in the sense of satisfying the maximum
principle. Thus, these schemes provide the rst examples of
nonoscillatory secondorder Godunovtype schemes based on
global projection operators. Numerical experiments are
performed; L1/Linfinityerrors and convergence rates are
calculated. For convex Hamiltonians, numerical evidence con rms
that our central schemes converge with secondorder rates, when
measured in the L1norm advocated in our recent paper [Numer.
Math, to appear]. The standard Linfinitynorm, however, fails
to detect this secondorder rate.
 Jiang, GS, and Peng, DP, "Weighted ENO schemes for HamiltonJacobi equations," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 21, pp. 21262143, 2000.
Abstract:
In this paper, we present a weighted ENO (essentially
nonoscillatory) scheme to approximate the viscosity solution of
the Hamilton Jacobi equation: phi(t) + H (x(1),...,x(d), t,
phi, phi(x1),...,phi(xd)) = 0. This weighted ENO scheme is
constructed upon and has the same stencil nodes as the third
order ENO scheme but can be as high as fifth order accurate in
the smooth part of the solution. In addition to the accuracy
improvement, numerical comparisons between the two schemes also
demonstrate that the weighted ENO scheme is more robust than
the ENO scheme.
 Russo, G, and Smereka, P, "A levelset method for the evolution of faceted crystals," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 21, pp. 20732095, 2000.
Abstract:
A levelset formulation for the motion of faceted interfaces is
presented. The evolving surface of a crystal is represented as
the zerolevel of a phase function. The crystal is identified
by its orientation and facet speeds. Accuracy is tested on a
single crystal by comparison with the exact evolution. The
method is extended to study the evolution of a polycrystal.
Numerical examples in two and three dimensions are presented.
 Zheng, LL, and Zhang, H, "An adaptive level set method for movingboundary problems: Application to droplet spreading and solidification," NUMERICAL HEAT TRANSFER PART BFUNDAMENTALS, vol. 37, pp. 437454, 2000.
Abstract:
A threedimensional adaptive level set method has Been
developed for deformable free surface problems with or without
solidification. In the new scheme, a threedimensional
multizone adaptive grid generation (MAGG) scheme is employed to
track the moving boundaries and a level set method is used to
capture the free surface deformation. The effectiveness and
robustness of the algorithm are demonstrated by solving the
droplet spreading and solidification problem, in which both
free surface and solidification interface movements are
important.
 Benjamin, MA, "Fuel atomization for nextgeneration gas turbine combustors," ATOMIZATION AND SPRAYS, vol. 10, pp. 427U3, 2000.
Abstract:
The push toward higher specific fuel consumption and smaller,
lighter packaging for reducedcost aerospace gas turbine
engines has resulted in large increases in engine operating
pressures and temperatures, as well as major efforts to reduce
gas path losses and increase component Efficiencies. This is a
tread that is expected to continue, and as a result, thermal
management of the hot engine section, including the fuel
nozzle, combustor, and turbine, has emerged as a critical
technology area requiring further research and development. For
the fuel injection system, nozzle thermal management, turndown
ratio, and atomization performance while maintaining correct
combustor aerodynamics and low pollutant emissions are the most
important performance features that necessitate optimization.
Complex and expensive heatshielded designs are often required
to reduce nettle wettedwall temperatures and prevent the
formation of carbonaceous deposits within the fuel delivery
passages. Optimization of designs using current computational
methods is limited in capability and expensive. Significant
advances infuel injection concepts, physical understanding, and
computational methods are required to merl these increasingly
demanding combustor requirements, with configurations at or
below current cost levels. Five injector designs are presented,
which include an advanced hybrid air blast (HAB) atomizer, a
ball directinjection (LDI) concept and three lean prevaporized
premixer (LPP) designs that exemplify advanced fuel injection
technology and ideas to address the challenges of next
generation gas turbine combustors.
 Chopp, DL, "A levelset method for simulating island coarsening," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 162, pp. 104122, 2000.
Abstract:
Modeling of microstructural evolution during thinfilm
deposition requires a knowledge of several key activation
energies (surface diffusion, island edge atom diffusion, adatom
migration over descending step edges, etc.). These and other
parameters must be known as a function of crystal orientation.
In order to generate values for these parameters, we have
developed a numerical simulation in tandem with physical
experiments. By tuning the simulation to the results from
experiments Lye can extract and verify approximate values for
these parameters. The numerical method we use is based upon the
level set method. Our model is a continuum model in directions
parallel to the crystal facet, and resolves each discrete
atomic layer in the normal direction. The model includes
surface diffusion, step edge dynamics, and
attachment/detachment rates all of which may depend upon the
local geometry of the step edge. The velocity field for
advancing the island edges in the level set framework is
generated by computing the equilibrium adatom density on the
flat terraces resulting in Laplace's equation with mixed
boundary conditions at the step edges. We have turned to the
finite element method for solving this equation, which results
in very good agreement with analytically known solutions and
with experiment, (C) 2000 Academic Press.
 Kim, S, "Wavefronts of linear elastic waves: local convexity and modeling," WAVE MOTION, vol. 32, pp. 203216, 2000.
Abstract:
Seismic techniques incorporating high frequency asymptotic
representation of the 3D elastic Green's function require
efficient solution methods for the computation of traveltimes.
For finite difference eikonal solvers, upwind differences are
requisite to sharply resolve discontinuities in the traveltime
derivatives. In anisotropic media, the direction of energy
propagation is not in general tangent to the wavefront normal,
while finite difference eikonal solvers compute the solution
based on the traveltime gradients and wavefront normal. Local
convexity of the wavefronts in transverse isotropic (TI) media
is proved to shaw that wavefront normal determines the upwind
direction of the energy propagation. The eikonal equations for
the traveltimes in TI media of a generally inclined symmetry
axis (ITI) are derived in a way that the eikonal solvers fit
conveniently. A stable, secondorder, shockcapturing, upwind
finite difference scheme is suggested for solving ITI eikonal
equations in regular grids in 3D. Numerical experiments are
presented to demonstrate the efficiency of the algorithm. (C)
2000 Elsevier Science B.V. All rights reserved.
 Mitchell, I, and Tomlin, CJ, "Level set methods for computation in hybrid systems," HYBRID SYSTEMS: COMPUTATION AND CONTROL, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1790, pp. 310323, 2000.
Abstract:
Reachability analysis is frequently used to study the safety of
control systems. We present an implementation of an exact
reachability operator for nonlinear hybrid systems. After a
brief review of a previously presented algorithm for
determining reachable sets and synthesizing control laws  upon
whose theory the new implementation rests  an equivalent
formulation is developed of the key equations governing the
continuous state reachability. The new formulation is
implemented using level set methods, and its effectiveness is
shown by the numerical solution of three examples.
 Sussman, M, and Puckett, EG, "A coupled level set and volumeoffluid method for computing 3D and axisymmetric incompressible twophase flows," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 162, pp. 301337, 2000.
Abstract:
We present a coupled level set/volumeoffluid (CLSVOF) method
for computing 3D and axisymmetric incompressible twophase
flows. This method combines some of the advantages of the
volumeoffluid method with the level set method to obtain a
method which is generally superior to either method alone. We
present direct comparisons between computations made with the
CLSVOF method and computations made with the level set method,
the volumeoffluid method, and the boundary integral method.
We also compare our computations to the exact solution for an
oscillating ellipse due to Lamb and experimental results
obtained for a rising gas bubble in liquid obtained by Hnat and
Buckmaster. Our computational examples focus on flows in which
surface tension forces and changes in topology are dominant
features Of the flow. (C) 2000 Academic Press.
 Hansen, U, Rodgers, S, and Jensen, KF, "Modeling of metal thin film growth: Linking angstromscale molecular dynamics results to micronscale film topographies," PHYSICAL REVIEW B, vol. 62, pp. 28692878, 2000.
Abstract:
A general method for modeling ionized physical vapor deposition
is presented. As an example, the method is applied to growth of
an aluminum film in the presence of an ionized argon flux.
Molecular dynamics techniques are used to examine the surface
adsorption, reflection, and sputter reactions taking place
during ionized physical vapor deposition. We predict their
relative probabilities and discuss their dependence on energy
and incident angle. Subsequently, we combine the information
obtained from molecular dynamics with a line of sight transport
model in a twodimensional feature, incorporating all effects
of reemission and resputtering. This provides a complete growth
rate model that allows inclusion of energy and angular
dependent reaction rates. Finally, a levelset approach is used
to describe the morphology of the growing film. We thus arrive
at a computationally highly efficient and accurate scheme to
model the growth of thin films. We demonstrate the capabilities
of the model predicting the major differences on Al film
topographies between conventional and ionized sputter
deposition techniques studying thin film growth under ionized
physical vapor deposition conditions with different Ar fluxes.
 Kim, YT, Goldenfeld, N, and Dantzig, J, "Computation of dendritic microstructures using a level set method," PHYSICAL REVIEW E, vol. 62, pp. 24712474, 2000.
Abstract:
We compute timedependent solutions of the sharpinterface
model of dendritic solidification in two dimensions by using a
level set method. The steadystate results are in agreement
with solvability theory. Solutions obtained from the level set
algorithm are compared with dendritic growth simulations
performed using a phasefield model and the two methods are
found to give equivalent results. Furthermore, we perform
simulations with unequal diffusivities in the solid and liquid
phases and find reasonable agreement with the available theory.
 Bertalmio, M, Sapiro, G, and Randall, G, "Morphing active contours," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 22, pp. 733737, 2000.
Abstract:
A method for deforming curves in a given image to a desired
position in the second image is introduced in this paper. The
algorithm is based on deforming the first image toward the
second one via a Partial Differential Equation (PDE), while
tracking the deformation of the curves of interest in the first
image with an additional, coupled, PDE. The tracking is
performed by projecting the velocities of the first equation
into the second one. In contrast with previous PDEbased
approaches, both the images and the curves on the frames/slices
of interest are used for tracking. The technique can be applied
to object tracking and sequential segmentation. The topology of
the deforming curve can change without any special topology
handling procedures added to the scheme. This permits, for
example, the automatic tracking of scenes where, due to
occlusions, the topology of the objects of interest changes
from frame to frame. In addition, this work introduces the
concept of projecting velocities to obtain systems of coupled
PDEs for image analysis applications We show examples for
object tracking and segmentation of electronic microscopy.
 Marquina, A, and Osher, S, "Explicit algorithms for a new time dependent model based on level set motion for nonlinear deblurring and noise removal," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 22, pp. 387405, 2000.
Abstract:
In this paper we formulate a time dependent model to
approximate the solution to the nonlinear total variation
optimization problem for deblurring and noise removal
introduced by Rudin and Osher [ Total variation base image
restoration with free local constraints, in Proceedings IEEE
Internat. Conf. Imag. Proc., IEEE Press, Piscataway, NJ, (
1994), pp. 3135] and Rudin, Osher, and Fatemi [ Phys. D, 60 (
1992), pp. 259268], respectively. Our model is based on level
set motion whose steady state is quickly reached by means of an
explicit procedure based on Roe's scheme [ J. Comput. Phys., 43
( 1981), pp. 357372], used in fluid dynamics. We show
numerical evidence of the speed of resolution and stability of
this simple explicit procedure in some representative 1D and 2D
numerical examples.
 Ruuth, SJ, Merriman, B, and Osher, S, "A fixed grid method for capturing the motion of self intersecting wavefronts and related PDEs," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 163, pp. 121, 2000.
Abstract:
Moving surfaces that selfintersect arise naturally in the
geometric optics model of wavefront motion. Standard ray
tracing techniques can be used to compute these motions, but
they lose resolution as rays diverge. In this paper we develop
numerical methods that maintain uniform spatial resolution of
the front at all times. Our approach is a fixed grid, wavefront
capturing formulation based on the Dynamic Surface Extension
method of Steinhoff and Fan (Technical report, University of
Tennessee Space Institute). The new methods can treat
arbitrarily complicated self intersecting fronts, as well as
refraction, reflection, and focusing. We also develop methods
fur curvaturedependent front motions and the motion of
filaments. We validate our methods with numerical experiments.
(C) 2000 Academic Press.
 Ida, M, "An improved unified solver for compressible and incompressible fluids involving free surfaces. Part I. Convection," COMPUTER PHYSICS COMMUNICATIONS, vol. 132, pp. 4465, 2000.
Abstract:
An improved numerical solver for the unified solution of hows
involving an interface between either compressible or
incompressible fluids is proposed. This method is based on the
CIPCUP (Cubic Interpolated Propagation/Combined, Unified
Procedure) which is a semiimplicit solver for the Euler
equations of fluid flows. In the CIPCUP method, each of
convection and acoustic parts of the Euler equations are
treated individually by a splitting manner. Namely, the
convection part is solved by CTP method and the acoustic part
is solved by CUP method. As Part I of this series of articles,
we propose an improved scheme for the convection part. The
ability of the CIP method to capture interfaces is highly
improved by replacing the cubic interpolation function used in
the CIP scheme with a quadratictype extrapolation function
only around the interface. With this improvement, oscillation
and diffusion in the solution at interfaces (or phase
boundaries) are removed, which are the most significant
problems on freesurface flow simulations especially in the
case where we treat some materials which have quite different
properties. By giving some constraints on the extrapolation
function, its stability was guaranteed. Furthermore, we propose
a simple scheme for recognizing interface location and motion
with a color function or a level set function. This scheme is
very useful for the extrapolation process. Effectiveness,
accuracy and stability of the improved method were demonstrated
with some examples. (C) 2000 Published by Elsevier Science B.V.
 Tannenbaum, A, "On the eye tracking problem: a challenge for robust control," INTERNATIONAL JOURNAL OF ROBUST AND NONLINEAR CONTROL, vol. 10, pp. 875888, 2000.
Abstract:
Eye tracking is one of the key problems in controlled active
vision. Because of modelling uncertainty and noise in the
signals, it becomes a challenging problem for robust control.
In this paper, we outline some of the key issues involved as
well as some possible solutions. We will need to make contact
with techniques from machine vision and multiscale image
processing in carrying out this task. In particular, we will
sketch some of the necessary methods from computer vision and
image processing including optical flow, active contours
('snakes'), and geometric driven flows. The paper will thus
have a tutorial flavor as well. Copyright (C) 2000 John Wiley &
Sons, Ltd.
 O'Sullivan, PL, Baumann, FH, and Gilmer, GH, "Simulation of physical vapor deposition into trenches and vias: Validation and comparison with experiment," JOURNAL OF APPLIED PHYSICS, vol. 88, pp. 40614068, 2000.
Abstract:
We have performed twodimensional (2D) and threedimensional
(3D) (axisymmetric) numerical simulations of physical vapor
deposition into high aspect ratio trenches and vias used for
modern very largescale integration interconnects. The
topographic evolution is modeled using (continuum) level set
methods. The level set approach is a powerful computational
technique for accurately tracking moving interfaces or
boundaries, where the advancing front is embedded as the zero
level set (isosurface) of a higher dimensional mathematical
function. We have validated both codes against analytic
formulas for step coverage. First, we study the 2D case of long
rectangular trenches including 3D outofplane target flux. The
3D flux can be obtained from molecular dynamics computations,
and hence our approach represents a hybrid atomistic/continuum
model. Second, we report results of axisymmetric 3D simulations
of high aspect ratio vias, which we compare with experimental
data for Ti/TiN barrier layers. We find that the simulations
(using a cosine angular distribution for the flux from the
target) overpredict bottom coverage in some cases by
approximately 20%30% for both collimated and uncollimated
deposition, but in other cases provide a reasonably accurate
comparison with experiment. (C) 2000 American Institute of
Physics. [S00218979(00)091209].
 Karlsen, KH, Lie, KA, and Risebro, NH, "A fast marching method for reservoir simulation," COMPUTATIONAL GEOSCIENCES, vol. 4, pp. 185206, 2000.
Abstract:
We present a fast marching level set method for reservoir
simulation based on a fractional flow formulation of twophase,
incompressible, immiscible flow in two or three space
dimensions. The method uses a fast marching approach and is
therefore considerably faster than conventional finite
difference methods. The fast marching approach compares
favorably with a front tracking method as regards both
efficiency and accuracy. In addition, it maintains the
advantage of being able to handle changing topologies of the
front structure.
 You, YL, and Kaveh, M, "Fourthorder partial differential equations for noise removal," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 9, pp. 17231730, 2000.
Abstract:
A class of fourthorder partial differential equations (PDEs)
are proposed to optimize the tradeoff between noise removal
and edge preservation. The time evolution of these PDEs seeks
to minimize a cost functional which is an increasing function
of the absolute value of the Laplacian of the image intensity
function. Since the Laplacian of an image at a pixel is zero if
the image is planar in its neighborhood, these PDEs attempt to
remove noise and preserve edges by approximating an observed
image with a piecewise planar image. Piecewise planar images
look more natural than step images which anisotropic diffusion
(second order PDEs) uses to approximate an observed image. So
the proposed PDEs are able to avoid the blocky effects widely
seen in images processed by anisotropic diffusion, while
achieving the degree of noise removal and edge preservation
comparable to anisotropic diffusion. Although both approaches
seem to be comparable in removing speckles in the observed
images, speckles are more visible in images processed by the
proposed PDEs, because piecewise planar images are less likely
to mask speckles than step images and anisotropic diffusion
tends to generate multiple false edges. Speckles can be easily
removed by simple algorithms such as the one presented in this
paper.
 Sethian, JA, and Wiegmann, A, "Structural boundary design via level set and immersed interface methods," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 163, pp. 489528, 2000.
Abstract:
We develop and test an algorithmic approach to the boundary
design of elastic structures. The goal of our approach is two
fold: first, to develop a method which allows one to rapidly
solve the twodimensional Lame equations in arbitrary domains
and compute, for example, the stresses, and second, to develop
a systematic way of modifying the design to optimize chosen
properties. At the core, our approach relies on two distinct
steps. Given a design, we first apply an explicit jump immersed
interface method to compute the stresses for a given design
shape. We then use a narrow band level set method to perturb
this shape and progress towards an improved design. The
equations of 2D linear elastostatics in the displacement
formulation on arbitrary domains are solved quickly by domain
embedding and the use of fast elastostatic solvers. This
effectively reduces the dimensionality of the problem by one.
Once the stresses are found, the level set method, which
represents the design structure through an embedded implicit
function, is used in the second step to alter the shape, with
velocities depending on the stresses in the current design,
Criteria are provided for advancing the shape in an appropriate
direction and fur correcting the evolving shape when given
constraints are violated. (C) 2000 Academic Press.
 Groetsch, CW, and Scherzer, O, "Nonstationary iterated TikhonovMorozov method and thirdorder differential equations for the evaluation of unbounded operators," MATHEMATICAL METHODS IN THE APPLIED SCIENCES, vol. 23, pp. 12871300, 2000.
Abstract:
In this paper we analyse the nonstationary iterative Tikhonov
Morozov method analytically and numerically for the stable
evaluation of differential operators and for denoizing images.
A relationship between nonstationary iterative Tikhonov
Morozov regularization and a filtering technique based on a
differential equation of third order is established and both
methods are shown to be effective for denoizing images and for
the stable evaluation of differential operators. The
theoretical results are verified numerically on model problems
in ultrasound imaging and numerical differentiation. Copyright
(C) 2000 John Wiley & Sons, Ltd.
 Tomlin, CJ, Lygeros, J, and Sastry, SS, "A game theoretic approach to controller design for hybrid systems," PROCEEDINGS OF THE IEEE, vol. 88, pp. 949970, 2000.
Abstract:
We present a method to design controllers for safety
specifications in hybrid systems. The hybrid system combines
discrete event dynamics with nonlinear continuous dynamics: the
discrete event dynamics model linguistic and qualitative
information and naturally accommodate mode switching logic, and
the continuous dynamics model the physical processes
themselves, such as the continous response of an aircraft to
the forces of aileron and throttle. Input variables model both
continuous and discrete control and disturbance parameters. We
translate safety specifications into restrictions on the
system's reachable sets of states. Then, using analysis based
on optimal control and game theory for automata and continuous
dynamical systems, we derive HamiltonJacobi equations whose
solutions describe the boundaries of reachable sets. These
equations are the heart of our general controller synthesis
technique for hybrid systems, in which we calculate feedback
control laws for the continuous and discrete variables, which
guarantee that the hybrid system remains in the "safe subset"
of the reachable set. We discuss issues related to computing
solutions to HamiltonJacobi equations. Throughout, we
demonstrate our techniques on examples of hybrid automata
modeling aircraft conflict resolution, autopilot flight mode
switching, and vehicle collision avoidance.
 Xie, WS, and Tao, JH, "Interaction of a solitary wave and a front step simulated by level set method," APPLIED MATHEMATICS AND MECHANICSENGLISH EDITION, vol. 21, pp. 761766, 2000.
Abstract:
As a new method, the Level Set method had been developed to
compute the interface of twophase flow. The basic mathematical
theory and the detailed method to solve the free surface
hydrodynamic problem had been investigated. By using the Level
Set method, the transformation of a solitary wave over a front
step was simulated. The results were in good agreement with
laboratory experiments.
 Ulitsky, M, Ghenai, C, Gokalp, I, Wang, LP, and Collins, LR, "Comparison of a spectral model for premixed turbulent flame propagation to DNS and experiments," COMBUSTION THEORY AND MODELLING, vol. 4, pp. 241264, 2000.
Abstract:
A recently developed spectral model for premixed turbulent
combustion in the flamelet regime (based on the EDQNM
turbulence theory) has been compared with both direct numerical
simulations (DNS) and experimental data. The 128(3) DNS is
performed at a Reynolds number of 223 based on the integral
length scale. Good agreement is observed for both single and
twopoint quantities (i.e. ratio of the turbulent to laminar
burning velocities, scalar autocorrelation. dissipation and
scalarvelocity cross correlation spectral for the two
different values of u'/s(LO) considered. The model also
predicts the rapid transient behaviour of the flame at early
times. An experimental setup is then described for generating
a lean methaneah flame and measuring two point spatial
correlations along the midpoint of the flame brush (i.e. along
the (C) over bar = 0.5 contour). The experimental measurements
in the flamelet regime take the form of a discontinuous or
'telegraph' signal. The EDQNM model, in contrast, describes an
'ensemble' of flames, and thus is based solely on continuous
variables. A theoretical relationship between the correlation
obtained from the EDQNM model and the equivalent correlation
for a discontinuous (experimental) flame is derived. The
relationship is used to enable a meaningful comparison between
experimentally observed and model correlations. In general, the
agreement is good for the three different cases considered in
this study, with most of the error occurring at the lowest
Reynolds number (ReL = 22). Furthermore, it is shown that
considerably more error would result if no attempt is made to
convert the ensemble representation in the model to an
equivalent singleflame or 'telegraph' signal.
 Chung, DH, and Sapiro, G, "Segmenting skin lesions with partialdifferentialequations based image processing algorithms," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 19, pp. 763767, 2000.
Abstract:
In this paper, a partialdifferential equations (PDE)based
system for detecting the boundary of skin lesions in digital
clinical skin images is presented. The image is first
preprocessed via contrastenhancement and anisotropic
diffusion. If the lesion is covered by hairs, a PDEbased
continuous morphological filter that removes them is used as an
additional preprocessing step. Following these steps, the skin
lesion is segmented either by the geodesic active contours
model or the geodesic edge tracing approach. These techniques
are based on computing, again via PDEs, a geodesic curve in a
space defined by the image content. Examples showing the
performance of the algorithm are given.
 Whitaker, RT, "A levelset approach to image blending," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 9, pp. 18491861, 2000.
Abstract:
This paper presents a novel method for blending images, Image
blending refers to the process of creating a set of discrete
samples of a continuous, oneparameter family of images that
connects a pair of input images. Image blending has uses in a
variety of computer graphics and image processing applications.
In particular, it can be used for image morphing, which is a
method for creating video streams that depict transformations
of objects in scenes based solely on pairs of images and sets
of userdefined fiducial points. Image blending also has
applications for video compression and imagebased rendering.
The proposed method for image blending relies on the
progressive minimization of a difference metric which compares
the level sets between two images. This strategy results in an
image blend which is the solution of a pair of coupled,
nonlinear, firstorder, partial differential equations that
model multidimensional levelset propagations. When compared to
interpolation this method produces more natural appearances of
motion because it manipulates the shapes of image contours
rather than simply interpolating intensity values. This
strategy results in a process that has the qualitative property
of deforming greyscale objects in images rather than producing
a simple fade from one object to another. This paper presents
the mathematics that underlie this new method, a numerical
implementation, and results on real images that demonstrate its
effectiveness.
 Kaminski, CF, Bai, XS, Hult, J, Dreizler, A, Lindenmaier, S, and Fuchs, L, "Flame growth and wrinkling in a turbulent flow," APPLIED PHYSICS BLASERS AND OPTICS, vol. 71, pp. 711716, 2000.
Abstract:
Highspeed planar laserinduced fluorescence (PLIF) and 3D
large eddy simulations (LES) are used to study turbulent flame
kernel growth, wrinkling and the formation of separated flame
pockets in methane/air mixtures. Turbulence was effected by a
set of rotary fans situated in a cylindrical enclosure. Flame
wrinkling was followed on sequential 2D OH images captured at
kHz repetition rates. Under stoichiometric conditions and low
turbulence levels the flame kernel remains singly connected and
close to spherical in shape. By increasing turbulence or
reducing the stoichiometry of the mixture the formation of
separated pockets could be observed and studied. The mechanisms
behind these phenomena are investigated qualitatively by LES of
a levelset Gequation describing the flame surface propagation
in turbulent flows.
 Sarti, A, de Solorzano, CO, Lockett, S, and Malladi, R, "A geometric model for 3D confocal image analysis," IEEE TRANSACTIONS ON BIOMEDICAL ENGINEERING, vol. 47, pp. 16001609, 2000.
Abstract:
In this paper, we use partialdifferentialequationbased
filtering as a preprocessing add post processing strategy for
computeraided cytology, We wish to accurately extract and
classify. the shapes of nuclei from confocal microscopy images,
which is a prerequisite to an accurate quantitative
intranuclear (genotypic and phenotypic) and internuclear
(tissue structure) analysis of tissue and cultured specimens.
First, we study the use of a geometrydriven edgepreserving
image smoothing mechanism before nuclear segmentation. We show
how this biter outperforms other widelyused filters in that it
provides higher edge fidelity. Then we apply the same
filter,,vith a different initial condition, to smooth nuclear
surfaces and obtain subpixel accuracy. Finally we use another
instance of the geometrical filter to correct for
misinterpretations of the nuclear surface by the segmentation
algorithm. Our prefiltering and post filtering nicely
complements our initial segmentation strategy, in that it
provides substantial and measurable improvement in the
definition of the nuclear surfaces.
 Bourlioux, A, "Semianalytical validation of a dynamic largeeddy simulation procedure for turbulent premixed flames via the Gequation," COMBUSTION THEORY AND MODELLING, vol. 4, pp. 363389, 2000.
Abstract:
The performance of a dynamic subgrid model for the turbulent
burning speed of a premixed flame is investigated for a series
of idealized test cases where the flame front is wrinkled by a
multiplescale shear flow; a rigorous asymptotic subgrid model
is also implemented for comparison. Explicit formulae for the
flame wrinkled shape and turbulent speed are available to
generate a reference database. The role of the subgrid
wrinkling models is to achieve the same overall flame shape and
propagation speed in a simulation where only the largest scales
of the flow are explicitly accounted for. Very good results are
obtained when the subgrid burning speed enhancement is
estimated using the asymptotic subgrid model. On the other
hand, the dynamic model attempts to exploit the scaling
observable in the simulation to extrapolate the turbulent
burning speed enhancement in the original system The
performance of this strategy is adequate for some regimes but
poor for others; the source of the problem is traced back to
the existence of a scaling transition that occurs as the flame
propagating speed is adjusted during the largeeddy simulation.
A modification to the scaling of the enhanced burning is
implemented to account for the existence of the two distinct
scaling ranges; it improves significantly the predictions of
the dynamic model away from the transition, but results in the
nearcritical range remain predictably very poor compared with
the rigorous asymptotic model results. These conclusions based
on apriori performance for the reference steady data are
confirmed by comparing unsteady largeeddy and direct
simulations. Results based on rigorous mathematical tools are
possible here because of the separation of length scales in the
special class of idealized flow fields used in this study:
their relevance to more realistic flows is also discussed.
 Shoemaker, DM, Huq, MF, and Matzner, RA, "Generic tracking of multiple apparent horizons with level flow  art. no. 124005," PHYSICAL REVIEW D, vol. 6212, pp. 4005+, 2000.
Abstract:
We report the development of the first apparent horizon locator
capable of finding multiple apparent horizons in a "generic"
numerical black hole spacetime. We use a levelhow method
which, starting from a single arbitrary initial guess surface,
can undergo topology changes as it flows towards disjoint
apparent horizons. The level flow method has two advantages:
(1) The solution is independent of changes in the initial guess
and (2) the solution can have multiple components. We
illustrate our method of locating apparent horizons in a short
KerrSchild binary black hole grazing collision.
 Pereyra, V, "Ray tracing methods for inverse problems," INVERSE PROBLEMS, vol. 16, pp. R1R35, 2000.
Abstract:
We discuss the origin, use and implementation of ray tracing
methods for nonlinear inverse modelling problems associated
with wave propagation phenomena. These methods have a long
tradition in acoustic and elastodynamic wave propagation
problems for Various important applications, and they can
surely be helpful in other realms, including electromagnetic
wave propagation and diffusion dominated phenomena. The
subjacent models used for forward simulation have increased in
complexity and dimension as computer power has enabled us to
solve such problems in an acceptable amount of time. Thus, we
will devote part of this paper to discuss modelling issues and
the parallel implementation of algorithms for applications in
earth sciences.
 Zhao, HK, Osher, S, Merriman, B, and Kang, M, "Implicit and nonparametric shape reconstruction from unorganized data using a variational level set method," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 80, pp. 295314, 2000.
Abstract:
In this paper we consider a fundamental visualization problem:
shape reconstruction from an unorganized data set. A new
minimalsurfacelike model and its variational and partial
differential equation (PDE) formulation are introduced. In our
formulation only distance to the data set is used as our input.
Moreover, the distance is computed with optimal speed using a
new numerical PDE algorithm. The data set can include points,
curves, and surface patches. Our model has a natural scaling in
the nonlinear regularization that allows flexibility close to
the data set while it also minimizes oscillations between data
points. To find the final shape, we continuously deform an
initial surface following the gradient flow of our energy
functional. An offset (an exterior contour) of the distance
function to the data set is used as our initial surface. We
have developed a new and efficient algorithm to find this
initial surface. We use the level set method in our numerical
computation in order to capture the deformation of the initial
surface and to find an implicit representation (using the
signed distance function) of the final shape on a fixed
rectangular grid. Our variational/PDE approach using the level
set method allows us to handle complicated topologies and noisy
or highly nonuniform data sets quite easily. The constructed
shape is smoother than any piecewise linear reconstruction.
Moreover, our approach is easily scalable for different
resolutions and works in any number of space dimensions. (C)
2000 Academic Press.
 Allaire, G, Clerc, S, and Kokh, S, "A fiveequation model for the numerical simulation of interfaces in twophase flows," COMPTES RENDUS DE L ACADEMIE DES SCIENCES SERIE IMATHEMATIQUE, vol. 331, pp. 10171022, 2000.
Abstract:
In the Eulerian approach for simulating interfaces in twophase
Rows, the main difficulties arise from the tired character of
the mesh which does not follow the interface. Therefore. near
the interface there are computational cells containing both
fluids which require a suitable modelling of the mixture.
Furthermore. most numerical algorithms, such as the volume of
fluid or the level set method, involve the transport of a
function indicating the localization of each phase. Due to
unavoidable numerical diffusion. they have the tendency to
thicken this mixture layer around the interface. It is thus
necessary to model correctly the twophase mixture. In the
context of compressible gas dynamics we propose such a model.
valid for any type of state laws, which satisfies an important
property of pressure stability through the interface. (C) 2000
Academie des sciences/Editions scientifiques et medicales
Elsevier SAS.
 Samson, C, BlancFeraud, L, Aubert, G, and Zerubia, J, "A level set model for image classification," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 40, pp. 187197, 2000.
Abstract:
We present a supervised classification model based on a
variational approach. This model is devoted to find an optimal
partition composed of homogeneous classes with regular
interfaces. The originality of the proposed approach concerns
the definition of a partition by the use of level sets. Each
set of regions and boundaries associated to a class is defined
by a unique level set function. We use as many level sets as
different classes and all these level sets are moving together
thanks to forces which interact in order to get an optimal
partition. We show how these forces can be defined through the
minimization of a unique fonctional. The coupled Partial
Differential Equations (PDE) related to the minimization of the
functional are considered through a dynamical scheme. Given an
initial interface set (zero level set), the different terms of
the PDE's are governing the motion of interfaces such that, at
convergence, we get an optimal partition as defined above. Each
interface is guided by internal forces (regularity of the
interface), and external ones (data term, no vacuum, no regions
overlapping). Several experiments were conducted on both
synthetic and real images.
 Ninokata, H, Muramatsu, T, Nishimura, M, Tomiyama, A, Minato, A, Kunugi, T, Takagi, S, Aoki, T, Fujii, S, Morii, T, Morita, K, Koshizuka, S, Tanaka, N, Shirakawa, N, Chen, Y, Matsukuma, Y, and Watanabe, T, "Microscopic simulation of nuclear reactor thermalhydraulics," JOURNAL OF THE ATOMIC ENERGY SOCIETY OF JAPAN, vol. 42, pp. 12421259, 2000.
Abstract:
In this paper, we propose a new model for active contours to
detect objects in a given image, based on techniques of curve
evolution, MumfordShah functional for segmentation and level
sets. Our model can detect objects whose boundaries are not
necessarily defined by gradient. We minimize an energy which
can he seen as a particular case of the minimal partition
problem, In the level set formulation, the problem becomes a
"meancurvature flow"like evolving the active contour, which
will stop on the desired boundary. However, the stopping term
does not depend on the gradient of the. image, as in the
classical active contour models, hut is instead related to a
particular segmentation of the image. We will give a numerical
algorithm using finite differences. Finally, we will present
various experimental results and in particular some examples
for which the classical snakes methods based on the gradient
are not applicable. Also, the initial curve can be anywhere in
the image, and interior contours are automatically detected.
 Rifai, H, Bloch, I, Hutchinson, S, Wiart, J, and Garnero, L, "Segmentation of the skull in MRI volumes using and taking the partial volume effect into account deformable model," MEDICAL IMAGE ANALYSIS, vol. 4, pp. 219233, 2000.
Abstract:
Segmentation of the skull in medical imagery is an important
stage in applications that require the construction of
realistic models of the head. Such models are used, for
example, to simulate the behavior of electromagnetic fields in
the head and to model the electrical activity of the cortex in
EEG and MEG data. in this paper, we present a new approach for
segmenting regions of bone in MRI volumes using deformable
models. Our method takes into account the partial volume
effects that occur with MRI data, thus permitting a precise
segmentation of these bone regions. At each iteration of the
propagation of the model, partial volume is estimated in a
narrow band around the deformable model, Our segmentation
method begins with a presegmentation stage, in which a
preliminary segmentation of the skull is constructed using a
regiongrowing method. The surface that bounds the pre
segmented skull region offers an automatic 3D initialization of
the deformable model. This surface is then propagated (in 3D)
in the direction of its normal. This propagation is achieved
using level set method, thus permitting changes to occur in the
topology of the surface as it evolves, an essential capability
for our problem. The speed at which the surface evolves is a
function of the estimated partial volume. This provides a sub
voxel accuracy in the resulting segmentation. (C) 2000 Elsevier
Science B.V. All rights reserved.
 Audette, MA, Ferrie, FP, and Peters, TM, "An algorithmic overview of surface registration techniques for medical imaging," MEDICAL IMAGE ANALYSIS, vol. 4, pp. 201217, 2000.
Abstract:
This paper presents a literature survey of automatic 3D surface
registration techniques emphasizing the mathematical and
algorithmic underpinnings of the subject. The relevance of
surface registration to medical imaging is that there is much
useful anatomical information in the form of collected surface
points which originate from complimentary modalities and which
must be reconciled. Surface registration can be roughly
partitioned into three issues: choice of transformation,
elaboration of surface representation and similarity criterion,
and matching and global optimization. The first issue concerns
the assumptions made about the nature of relationships between
the two modalities, e.g. whether a rigidbody assumption
applies, and if nor, what type and how general a relation
optimally maps one modality onto the other. The second issue
determines what type of information we extract from the 3D
surfaces, which typically characterizes their local or global
shape, and how we organize this information into a
representation of the surface which will lead to improved
efficiency and robustness in the last stage. The last issue
pertains to how we exploit this information to estimate the
transformation which best aligns local primitives in a globally
consistent manner or which maximizes a measure of the
similarity in global shape of two surfaces. Within this
framework, this paper discusses in detail each surface
registration issue and reviews the stateoftheart among
existing techniques. (C) 2000 Elsevier Science BN. All rights
reserved.
 McInerney, T, and Terzopoulos, D, "Tsnakes: Topology adaptive snakes," MEDICAL IMAGE ANALYSIS, vol. 4, pp. 7391, 2000.
Abstract:
We present a new class of deformable contours (snakes) and
apply them to the segmentation of medical images. Our snakes
are defined in terms of an affine cell image decomposition
(ACID). The 'snakes in ACID' framework significantly extends
conventional snakes, enabling topological flexibility among
other features. The resulting topology adaptive snakes, or 'T
snakes', can be used to segment some of the most complexshaped
biological structures from medical images in an efficient and
highly automated manner. (C) 2000 Elsevier Science BN. All
rights reserved.
 Volkov, VI, Gordeychuk, VA, Es'kov, NS, and Kozyrev, OM, "Numerical simulation by the MAH3 code of the interfaces using an unstructured mesh of markers," LASER AND PARTICLE BEAMS, vol. 18, pp. 197205, 2000.
Abstract:
The paper addresses RayleighTaylor instability (RTI) problems
and presents a method of describing an interface with markers
on a Eulerian mesh, which is implemented in the MAH3 code. The
proposed method allows a more accurate description of the
evolution of the interface caused by RayleighTaylor
perturbations and presences symmetry of the interface under
appropriate symmetry of the problem (planar, cylindrical, and
spherical). The method employs an unstructured triangular mesh
of makers. Method capabilities are demonstrated on 2D and 3D
RayleighTaylor instability problems.
 Steinhoff, J, Fan, M, and Vang, LS, "A new Eulerian method for the computation of propagating short acoustic and electromagnetic pulses," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 157, pp. 683706, 2000.
Abstract:
A new method is described to compute short acoustic or
electromagnetic pulses that propagate according to geometrical
optics. The pulses are treated as zero thickness sheets that
can propagate over long distances through inhomogeneous media
with multiple reflections. The method has many of the
advantages of Lagrangian ray tracing, but is completely
Eulerian, typically using a uniform Cartesian grid,
Accordingly, it can treat arbitrary configurations of pulses
that can reflect from surfaces and pass through each other
without requiring special computational marker arrays for each
pulse, Also, information describing the pulses, which are
treated as continuous surfaces, can be available throughout the
computational grid, rather than only at isolated individual
markers. The method uses a new type of representation, which we
call "Dynamic Surface Extension." The basic idea is to
propagate or "broadcast" defining fields from each pulse
surface through a computational grid. These fields carry
information about a nearby pulse surface that is used at each
node to compute the location of the pulse surfaces and other
attributes, such as amplitude. Thus the emphasis is on the
dynamics of these propagating defining fields, which obey only
local Eulerian equations at each node. The Dynamic Surface
Extension representation can be thought of as dual to level set
representation: The defining fields involve single valued
variables which are constant at each time along lines that are
normal to the evolving surface, whereas level set techniques
involve a function which has constant values on the evolving
surface and neighboring surfaces. In this way the new method
overcomes the inability of level set or Eikonal methods to
treat intersecting pulses that obey a wave equation and can
pass through each other, while still using only singlevalued
variables. Propagating thin pulse surfaces in 1D, 2D, and 3D
that can reflect from boundaries and pass through each other
are computed using the new method. The method was first
presented as a new, general representation of surfaces,
filaments, and particles by J. Steinhoff and M. Fan (1998,
Eulerian computation of evolving surfaces, curves and
discontinuous fields, UTSI preprint). (C) 2000 Academic Press.
 Yokoi, K, and Xiao, F, "Relationships between a roller and a dynamic pressure distribution in circular hydraulic jumps," PHYSICAL REVIEW E, vol. 61, pp. R1016R1019, 2000.
Abstract:
We investigated numerically the relation between a roller and
the pressure distribution to clarify the dynamics of the roller
in circular hydraulic jumps. We found that a roller which
characterizes a type II jump is associated with two high
pressure regions after the jump, while a type I jump (without
the roller) is associated with only one high pressure region.
Our numerical results show that building up an appropriate
pressure field is essential for a roller.
 Chen, YM, Vemuri, BC, and Wang, L, "Image denoising and segmentation via nonlinear diffusion," COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 39, pp. 131149, 2000.
Abstract:
Image denoising and segmentation are fundamental problems in
the field of image processing and computer vision with numerous
applications. In this paper, we present a nonlinear PDEbased
model for image denoising and segmentation which unifies the
popular model of Alvarez, Lions and Morel (ALM) for image
denoising and the Caselles, Kimmel and Sapiro model of geodesic
"snakes". Our model includes nonlinear diffusive as well as
reactive terms and leads to quality denoising and segmentation
results as depicted in the experiments presented here. We
present a proof for the existence, uniqueness, and stability of
the viscosity solution of this PDEbased model. The proof is in
spirit similar to the proof of the ALM model; how ever, there
are several differences which arise due to the presence of the
reactive terms that require careful treatment/consideration. A
fast implementation of our model is realized by embedding the
model in a scale space and then achieving the solution via a
dynamic system governed by a coupled system of firstorder
differential equations. The dynamic system finds the solution
at a coarse scale and tracks it continuously to a desired fine
scale. We demonstrate the smoothing and segmentation results on
several real images. (C) 2000 Elsevier Science Ltd. All rights
reserved.
 Barcelos, CAZ, and Chen, Y, "Heat flows and related minimization problem in image restoration," COMPUTERS & MATHEMATICS WITH APPLICATIONS, vol. 39, pp. 8197, 2000.
Abstract:
A new anisotropic diffusion model is proposed for image
restoration and segmentation, which is closely related to the
minimization problems for the unconstrained total variation
E(u) = integral(Omega) alpha(x)\del u\ + (beta/2)\u  I\(2).
Existence, uniqueness, and stability of the viscosity solutions
of the equation are proved. The experimental results are given
and compared with the existing models in the framework of image
restoration. The improvement on preserving sharp edges by using
the new model is visible. (C) 2000 Elsevier Science Ltd. All
rights reserved.
 Wang, HY, and Ghosh, B, "Geometric active deformable models in shape modeling," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 9, pp. 302308, 2000.
Abstract:
This paper analyzes the problem of shape modeling using the
principle of active geometric deformable models. While the
basic modeling technique already exists in the literature, we
highlight many of its drawbacks and discuss their source and
steps to overcome them. We propose a new stopping criterion to
address the stopping problem. We also propose to apply level
set algorithm to implement the active geometric deformable
models, thereby handling topology changes automatically. To
alleviate the numerical problems associated with the
implementation of the level set algorithm, we propose a new
adaptive multigrid narrow band algorithm. All the proposed new
changes have been illustrated with experiments with synthetic
images and medical images.
 Duncan, JS, and Ayache, N, "Medical image analysis: Progress over two decades and the challenges ahead," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 22, pp. 85106, 2000.
Abstract:
The analysis of medical images has been woven into the fabric
of the Pattern Analysis and Machine Intelligence (PAMI)
community since the earliest days of these Transactions.
Initially, the efforts in this area were seen as applying
pattern analysis and computer vision techniques to another
interesting dataset. However, over the last two to three
decades, the unique nature of the problems presented within
this area of study have led to the development of a new
discipline in its own right. Examples of these include: the
types of image information that are acquired, the fully three
dimensional image data, the nonrigid nature of object motion
and deformation, and the statistical variation of both the
underlying normal and abnormal ground truth. In this paper, we
look at progress in the field over the last 20 years and
suggest some of the challenges that remain for the years to
come.
 Katsoulakis, MA, and Vlachos, DG, "From microscopic interactions to macroscopic laws of cluster evolution," PHYSICAL REVIEW LETTERS, vol. 84, pp. 15111514, 2000.
Abstract:
We derive macroscopic governing laws of growth velocity,
surface tension, mobility, critical nucleus size, and
morphological evolution of clusters, from microscopic scale
master equations for a prototype surface reaction system with
long range adsorbateadsorbate interactions.
 Aivazis, M, Goddard, WA, Meiron, D, Ortiz, M, Pool, J, and Shepherd, J, "A virtual, test facility for simulating the dynamic response of materials," COMPUTING IN SCIENCE & ENGINEERING, vol. 2, pp. 4253, 2000.
Abstract:
The goal of the Caltech Center is to construct a Virtual Test
Facilitya problemsolving environment for full 3D parallel
simulation of the dynamic response of materials undergoing
compression due to shock waves.
 Oka, H, and Ishii, K, "Numerical simulation on the interaction of buoyant drops with a fluidfluid interface," JOURNAL OF THE PHYSICAL SOCIETY OF JAPAN, vol. 69, pp. 392400, 2000.
Abstract:
The fully threedimensional interactions of one or two buoyant
drops with a fluidfluid interface are studied numerically at
the moderate Reynolds number and Bond number. This complex
interaction process is successfully simulated using the three
dimensional level set approach to deal with the deformable
inter face and drops. The characteristics of vortex ring
generated at the coalescence between the interface and the drop
depend upon the viscosity ratio between two fluids, and they
have much to do with a jet formation process. In the case of
two drops, the motion of the trailing drop is influenced hy the
deformation of interfaces through the coalescence of the
leading drop with its homophase. The behavior of the trailing
drop is also classified into two types with respect to the
viscosity ratio. These studies show that the investigation in
the viewpoint of vorticity is effective on the understanding of
the interaction between drops and an interface.
 Dorst, L, and van den Boomgaard, R, "The support cone: A representational tool for the analysis of boundaries and their interactions," IEEE TRANSACTIONS ON PATTERN ANALYSIS AND MACHINE INTELLIGENCE, vol. 22, pp. 174178, 2000.
Abstract:
We present a directional boundary representation which deals
locally and consistently with the boundary's "inside." We show
that collision and wave propagation are reduced to addition on
the spectrum of directions and we derive transformation laws
for differential geometrical properties such as directed
curvature.
 Ying, LA, and Zhang, PW, "Vanishing curvature viscosity for front propagation," JOURNAL OF DIFFERENTIAL EQUATIONS, vol. 161, pp. 289306, 2000.
Abstract:
In this paper we study the front propagation with constant
speed and small curvature viscosity. We first investigate two
related problems of conservation laws, one of which is on the
nonlinear viscosity methods for the conservation laws, and the
other one is on the structure of solutions to conservation laws
with L1 initial data. We show that the nonlinear viscosity
methods approaching the piecewise smooth solutions with
finitely many discontinuity for convex conservation laws have
the firstorder rate of L1convergence. The solutions of
conservation laws with L1 initial data are shown to be bounded
after t > 0 if all singular points of initial data are from
shocks. These results suggest that the front propagation with
constant speed and a small curvature viscosity will approach
the front movements with a constant speed, as the small
parameter goes to zero. After the front breaks down, the cusps
will disappear promptly and corners will be formed. (C) 2000
Academic Press.
 Gomes, J, and Faugeras, O, "Level sets and distance functions," COMPUTER VISION  ECCV 2000, PT I, PROCEEDINGS, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1842, pp. 588602, 2000.
Abstract:
This paper is concerned with the simulation of the Partial
Differential Equation (PDE) driven evolution of a closed
surface by means of an implicit representation. In most
applications, the natural choice for the implicit
representation is the signed distance function to the closed
surface. Osher and Sethian propose to evolve the distance
function with a HamiltonJacobi equation. Unfortunately the
solution to this equation is not a distance function. As a
consequence, the practical application of the level set method
is plagued with such questions as when do we have to
"reinitialize" the distance function? How do we "reinitialize"
the distance function? Etc... which reveal a disagreement
between the theory and its implementation. This paper proposes
an alternative to the use of HamiltonJacobi equations which
eliminates this contradiction: in our method the implicit
representation always remains a distance function by
construction, and the implementation does not differ from the
theory anymore. This is achieved through the introduction of a
new equation. Besides its theoretical advantages, the proposed
method also has several practical advantages which we
demonstrate in three applications: (i) the segmentation of the
human cortex surfaces from MRI images using two coupled
surfaces [27], (ii) the construction of a hierarchy of
Euclidean skeletons of a 3D surface, (iii) the reconstruction
of the surface of 3D objects through stereo [13].
 Westin, CF, Lorigo, LM, Faugeras, O, Grimson, WEL, Dawson, S, Norbash, A, and Kikinis, R, "Segmentation by adaptive geodesic active contours," MEDICAL IMAGE COMPUTING AND COMPUTERASSISTED INTERVENTION  MICCAI 2000, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1935, pp. 266275, 2000.
Abstract:
This paper introduces the use of spatially adaptive components
into the geodesic active contour segmentation method for
application to volumetric medical images. These components are
derived from local structure descriptors and are used both in
regularization of the segmentation and in stabilization of the
imagebased vector field which attracts the contours to
anatomical structures in the images. They are further used to
incorporate prior knowledge about spatial location of the
structures of interest. These components can potentially
decrease the sensitivity to parameter settings inside the
contour evolution system while increasing robustness to image
noise. We show segmentation results on blood vessels in
magnetic resonance angiography data and bone in computed
tomography data.
 Tsai, A, Yezzi, A, and Willsky, AS, "A curve evolution approach to medical image magnification via the MumfordShah functional," MEDICAL IMAGE COMPUTING AND COMPUTERASSISTED INTERVENTION  MICCAI 2000, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1935, pp. 246255, 2000.
Abstract:
In this paper, we introduce a curve evolution approach to image
magnification based on a generalization of the MumfordShah
functional. This work is a natural extension of the curve
evolution implementation of the MumfordShah functional
presented by the authors in previous work. In particular, by
considering the image magnification problem as a structured
case of the missing data problem, we generalize the data
fidelity term of the original MumfordShah energy functional by
incorporating a spatially varying penalty to accommodate those
pixels with missing measurements. This generalization leads us
to a PDEbased approach for simultaneous image magnification,
segmentation, and smoothing, thereby extending the traditional
applications of the MumfordShah functional which only
considers simultaneous segmentation and smoothing. This novel
approach for image magnification is more global and much less
susceptible to blurring or blockiness artifacts as compared to
other more traditional magnification techniques, and has the
additional attractive denoising capability.
 Baillard, C, and Barillot, C, "Robust 3D segmentation of anatomical structures with level sets," MEDICAL IMAGE COMPUTING AND COMPUTERASSISTED INTERVENTION  MICCAI 2000, LECTURE NOTES IN COMPUTER SCIENCE, vol. 1935, pp. 236245, 2000.
Abstract:
This paper is concerned with the use of the level set formalism
to segment anatomical structures in 3D medical images
(ultrasound or magnetic resonance images.). A closed 3D surface
propagates towards the desired boundaries through the iterative
evolution of a 4D implicit function. The major contribution of
this work is the design of a robust evolution model based on
adaptive parameters depending on the data. First the iteration
step and the external propagation force, both usually constant,
are automatically computed at each iteration. Additionally,
regionbased information rather than the gradient is used, via
an estimation of intensity probability density functions over
the image. As a result, the method can be applied to various
kinds of data. Quantitative and qualitative results on brain MR
images and 3D echographies of carotid arteries are discussed.


2001 
 Siddiqi, K, Kimia, BB, Tannenbaum, A, and Zucker, SW, "On the psychophysics of the shape triangle," VISION RESEARCH, vol. 41, pp. 11531178, 2001.
Abstract:
We earlier introduced an approach to categorical shape
description based on the singularities (shocks) of curve
evolution equations. We now consider the simplest compositions
of shocks, and show that they lead to three classes of
parametrically ordered shape sequences, organized along the
sides of a shape triangle. By conducting several psychophysical
experiments we demonstrate that shockbased descriptions are
predictive of performance in shape perception. Most
significantly, the experiments reveal a fundamental difference
between perceptual effects dominated by when shocks form with
respect to one another, versus those dominated by where they
form. The shockbased theory provides a foundation for unifying
tasks as diverse as shape bisection, recognition, and
categorization. (C) 2001 Elsevier Science Ltd. All rights
reserved.
 Chen, YM, and Bose, P, "On the incorporation of timedelay regularization into curvaturebased diffusion," JOURNAL OF MATHEMATICAL IMAGING AND VISION, vol. 14, pp. 149164, 2001.
Abstract:
A new anisotropic nonlinear diffusion model incorporating time
delay regularization into curvaturebased diffusion is proposed
for image restoration and edge detection. A detailed
mathematical analysis of the proposed model in the form of the
proof of existence, uniqueness and stability of the "viscosity"
solution of the model is presented. Furthermore, implementation
issues and computational methods for the proposed model are
also discussed in detail. The results obtained from testing our
denoising and edge detection algorithm on several synthetic and
real images showed the effectiveness of the proposed model in
prserving sharp edges and fine structures while removing noise.
 Rhee, CW, "Evolution of flame shape to a vortex pair," KSME INTERNATIONAL JOURNAL, vol. 15, pp. 623629, 2001.
Abstract:
The PSC (Propagation of Surfaces under Curvature) algorithm is
adapted to the simulation of a flame propagation in a premixed
medium including the effect of volume expansion across the
flame front due to exothermicity. The algorithm is further
developed to incorporate the flame anchoring scheme. This
methodology is successfully applied to numerically simulate the
response of an anchored Vflame to two strong free stream
vortices, in accord with experimental observations of a passage
of Karman vortex street through a flame. The simulation
predicts flame cusping when a strong vortex pair interacts with
flame front. In other words, this algorithm handles merging and
breaking of the flame front and provides an accurate
calculation of the flame curvature which is needed for flame
propagation computation and estimation of curvaturedependent
flame speeds.
 Chen, Y, Barcelos, CAS, and Mair, BA, "Smoothing and edge detection by timevarying coupled nonlinear diffusion equations," COMPUTER VISION AND IMAGE UNDERSTANDING, vol. 82, pp. 85100, 2001.
Abstract:
In this paper, we develop new methods for denoising and edge
detection in images by the solution of nonlinear diffusion
partial differential equations. Many previous methods in this
area obtain a denoising u of the noisy image I as the solution
of an equation of the form partial derivative (t)u = L(g(\del
upsilon\), delu, u  I), when g controls the speed of the
diffusion and defines the edge map. The usual choice for g(s)
is (1 + ks(2))(1) and the function upsilon is always some
smoothing of u. Previous choices include upsilon = u, upsilon =
G(sigma) * u, and upsilon = G sigma * I. Numerical results
indicate that the choice of upsilon plays a very important role
in the quality of the images obtained. Notice that all these
choices involve an isotropic smoothing of u, which sometimes
fails to presence important corners and junctions, and this may
also fail to resolve small features which are closely grouped
together. This paper obtains u as the solution of a nonlinear
diffusion equation which depends on u. The equation can be
obtained as the energy descent equation for the total variation
of upsilon penalized by the mean squared error between u and
upsilon. The parameters in this energy descent equation are
regarded as functions of time rather than constants, to allow
for a reduction in the amount of smoothing as time progresses.
Numerical tests indicate that our new method is faster and able
to resolve small details and junctions better than standard
methods. (C) 2001 Academic Press.
 Han, C, Hatsukami, TS, Hwang, JN, and Yuan, C, "A fast minimal path active contour model," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 10, pp. 865873, 2001.
Abstract:
A new minimal path active contour model for boundary extraction
is presented. Implementing the new approach requires four steps
1) users place some initial end points on or near the desired
boundary through an interactive interface; 2) potential
searching window is defined between two end points; 3) graph
search method based on conic curves is used to search the
boundary; 4) "wriggling" procedure is used to calibrate the
contour and reduce sensitivity of the search results on the
selected initial end points. The last three steps are performed
automatically. In the proposed approach, the potential window
systematically provides a new node connection for the later
graph search, which is different from the rowbyrow and
columnbycolumn methods used in the classical graph search.
Furthermore, this graph search also suggests ways to design a
"wriggling" procedure to evolve the contour in the direction
nearly perpendicular to itself by creating a list of
displacement vectors in the potential window. The proposed
minimal path active contour model speeds up the search and
reduces the "metrication error" frequently encountered in the
classical graph search methods e,g,, the dynamic programming
minimal path (DPMP) method.
 Iwasaki, T, Nishimura, K, Tanaka, M, and Hagiwara, Y, "Direct numerical simulation of turbulent Couette flow with immiscible droplets," INTERNATIONAL JOURNAL OF HEAT AND FLUID FLOW, vol. 22, pp. 332342, 2001.
Abstract:
A direct numerical simulation has been carried out in order to
clarify the effects of the high viscosity and the interfacial
tension of a droplet on the interaction between the droplet and
nearwall turbulence. A liquid turbulent plane Couette flow
with an immiscible droplet of the same fluid density as that of
the: continuousphase has been used. The diameter of the
droplet is fixed at onefourth of the wall distance. which is
nearly equal to 41 wall units. The droplet has been assigned in
the range of 2060 wall units from one moving wall initially.
The modified volume of fluid (VOF) algorithm and local grid
refinement are used for tracking the phase interface. The
velocities for the fine grid are decided so that the equation
of continuity is satisfied in the fine cell. It is found that
the deformation of the droplet due to the surrounding fluid Row
is suppressed by the effect of the interfacial tension of the
droplet. The streamwise vortex is attenuated by the existence
of the droplet with the interfacial tension. The small vortex
is generated in the wake region of the droplet. The Reynolds
shear stress product becomes higher in a wide region around the
droplet. (C) 2001 Elsevier Science Inc. All rights reserved.
 Kim, S, "An O(N) level set method for eikonal equations," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 22, pp. 21782193, 2001.
Abstract:
A propagating interface can develop corners and discontinuities
as it advances. Level set algorithms have been extensively
applied for the problems in which the solution has advancing
fronts. One of the most popular level set algorithms is the so
called fast marching method (FMM), which requires total O(N
log(2)N) operations, where N is the number of grid points. The
article is concerned with the development of an O(N) level set
algorithm called the group marching method (GMM). The new
method is based on the narrow band approach as in the FMM.
However, it is incorporating a correctionbyiteration strategy
to advance a group of grid points at a time, rather than
sorting the solution in the narrow band to march forward a
single grid point. After selecting a group of grid points
appropriately, the GMM advances the group in tw iterations for
the cost of slightly larger than one iteration. Numerical
results are presented to show the efficiency of the method,
applied to the eikonal equation in tw and three dimensions.
 Angenent, SB, Aronson, DG, Betelu, SI, and Lowengrub, JS, "Focusing of an elongated hole in porous medium flow," PHYSICA D, vol. 151, pp. 228252, 2001.
Abstract:
In the focusing problem, we study solutions to the porous
medium equation partial derivative (t)u = Delta (u(m)) whose
initial distributions are positive in the exterior of a compact
2D region and zero inside. We assume that the initial interface
is elongated (i.e., has an aspect ratio > I), and possesses
reflectional symmetry with respect to both the x and yaxes.
We implement a numerical scheme that adapts the numerical grid
around the interface so as to maintain a high resolution as the
interface shrinks to a point. We find that as t tends to the
focusing time T, the interface becomes ovallike with lengths
of the major and minor axes O(rootT  t) and O(T  t),
respectively. Thus the aspect ratio is O(1/rootT  t). By
scaling and formal asymptotic arguments we derive an
approximate solution which is valid for all m. This
approximation indicates that the numerically observed power law
behavior for the major and minor axes is universal for all m >
1. (C) 2001 Elsevier Science B.V. All rights reserved.
 Breen, DE, and Whitaker, RT, "A levelset approach for the metamorphosis of solid models," IEEE TRANSACTIONS ON VISUALIZATION AND COMPUTER GRAPHICS, vol. 7, pp. 173192, 2001.
Abstract:
This paper presents a new approach to 3D shape metamorphosis.
We express the interpolation of two shapes as a process where
one shape deforms to maximize its similarity with another
shape. The process incrementally optimizes an objective
function while deforming an implicit surface model. We
represent the deformable surface as a level set (isosurface)
of a densely sampled scalar function of three dimensions. Such
levelset models have been shown to mimic conventional
parametric deformable surface models by encoding surface
movements as changes in the grayscale values of a volume data
set. Thus, a wellfounded mathematical structure leads to a set
of procedures that describes how voxel values can be
manipulated to create deformations that are represented as a
sequence of volumes. The result is a 3D morphing method that
offers several advantages over previous methods, including
minimal need for user input, no model parameterization,
flexible topology, and subvoxel accuracy.
 Yabe, T, Xiao, F, and Utsumi, T, "The constrained interpolation profile method for multiphase analysis," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 169, pp. 556593, 2001.
Abstract:
We present a review of the constrained interpolation profile
(CIP) method that is known as a general numerical solver for
solid, liquid, gas, and plasmas. This method is a kind of semi
Lagrangian scheme and has been extended to treat incompressible
flow in the framework of compressible fluid. Since it uses
primitive Euler representation, it is suitable for multiphase
analysis. The recent version of this method guarantees the
exact mass conservation even in the framework of a semi
Lagrangian scheme. We provide a comprehensive review of the
strategy of the CIP method, which has a compact support and
subcell resolution, including a frontcapturing algorithm with
functional transformation, a pressurebased algorithm, and
other miscellaneous physics such as the elasticplastic effect
an;l surface tension. Some practical applications are also
reviewed, such as milk crown or coronet, laserinduced melting,
and turbulent mixing layer of liquidgas interface. (C) 2001
Academic Press.
 Sethian, JA, "Evolution, implementation, and application of level set and fast marching methods for advancing fronts," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 169, pp. 503555, 2001.
Abstract:
A variety of numerical techniques are available for tracking
moving interfaces. In this review, we concentrate on techniques
that result from the link between the partial differential
equations that describe moving interfaces and numerical schemes
designed for approximating the solutions: to hyperbolic
conservation laws. This link gives rise to computational
techniques for tracking moving interfaces in two and three
space dimensions under complex speed laws. We discuss the
evolution of these techniques, the fundamental numerical
approximations, involved. implementation details, and
applications. Tn particular, we review some work on three
aspects of materials sciences: semiconductor process
simulations. seismic processing, and optimal structural
topology design. (C) 2001 Academic Press.
 Osher, S, and Fedkiw, RP, "Level set methods: An overview and some recent results," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 169, pp. 463502, 2001.
Abstract:
The level set method was devised by S. Osher and J. A. Sethian
(1988, J. Comput, Phys. 79, 1249) as a simple and versatile
method for computing and analyzing the motion of an interface
Gamma in two or three dimensions, Gamma bounds a (possibly
multiply connected) region Omega. The goal is to compute and
analyze the subsequent motion of Gamma under a velocity field
v. This velocity can depend on position, time. the geometry of
the interface, and the external physics. The interface is
captured for later time as the zero level set of a smooth (at
least Lipschitz continuous) function phi (x. t); i.e., Gamma
(t) = {x \ phi (x, t) = 0}. phi is positive inside Omega,
negative outside Omega. and is zero on Gamma (t). Topological
merging and breaking are well defined and easily performed. In
this review article we discuss recent variants and extensions,
including the motion of curves in three dimensions, the dynamic
surface extension method. fast methods for steady state
problems, diffusion generated motion, and the variational level
set approach. We also give a user's guide to the level set
dictionary and technology and couple the method to a wide
variety of problems involving external physics, such as
compressible and incompressible (possibly reacting) flow,
Stefan problems. kinetic crystal growth, epitaxial growth of
thin films, vortexdominated flows, and extensions to
multiphase motion, We conclude with a discussion of
applications to computer vision and image processing. (C) 2001
Academic Press.
 Im, YH, Hahn, YB, and Pearton, SJ, "Level set approach to simulation of feature profile evolution in a highdensity plasmaetching system," JOURNAL OF VACUUM SCIENCE & TECHNOLOGY B, vol. 19, pp. 701710, 2001.
Abstract:
The simulation of feature profile evolution in highdensity
plasmaetching processes has been carried out using a levelset
technique. The main feature of this work is the inclusion of
sheath dynamics, angular distribution of ions and reemission of
neutrals in the trench, etch kinetics, and a level set equation
for tracking a moving front of the feature profile. Sheath
dynamics showed that the damped potential was somewhat shifted
to the right and smaller than the applied potential. Etch
profile simulations were performed for etching of silicon in
inductively coupled plasmas of Cl2 and CF4 under various
conditions. In dry etching of Si with CF4 discharges, polymer
deposition was dominant at p(CFx) > 10mTorr, while surface
fluorination (or ionenhanced etching) was a main mechanism at
p(CFx) < 10 mTorr. The predicted etch profiles showed a slight
bowing on the sidewalls and substantial tapering near the
bottom, depending on the plasma parameters. (C) 2001 American
Vacuum Society.
 Chung, MH, "A level set approach for computing solutions to inviscid compressible flow with moving solid boundary using fixed Cartesian grids," INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, vol. 36, pp. 373389, 2001.
Abstract:
A level set approach for computing solutions to inviscid
compressible flow with moving solid surface is presented. The
solid surface is considered to be sharp and is described as the
zero level set of a smooth explicit function of space and time.
The finite volume TVDMacCormack's twostep procedure is used.
The boundary conditions on the solid surface are easily
implemented by defining the smooth level set function. The
present treatment of the level set method allows the handling
of fluid flows in the presence of irregularly shaped solid
boundaries, escaping from the bookkeeping complexity in the so
called 'surfacetracking' method. Using the proposed numerical
techniques, a twodimensional numerical simulation is made to
investigate the aerodynamic phenomena induced by two highspeed
trains passing by each other in a tunnel. Copyright (C) 2001
John Wiley & Sons, Ltd.
 Debreuve, E, Barlaud, M, Aubert, G, Laurette, I, and Darcourt, J, "Spacetime segmentation using level set active contours applied to myocardial gated SPECT," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 20, pp. 643659, 2001.
Abstract:
This paper presents a new variational method for the
segmentation of a moving object against a still background,
over a sequence of [twodimensional or threedimensional (3D)]
image frames. The method is illustrated in application to
myocardial gated single photon emission computed tomography
(SPECT) data, and incorporates a level set framework to handle
topological changes while providing closed boundaries. The key
innovation is the introduction of a geometrical constraint into
the derivation of the EulerLagrange equations, such that the
segmentation of each individual frame can be interpreted as a
closed boundary of an object (an isolevel of a set of hyper
surfaces) while integrating information over the entire
sequence. This results in the definition of an evolution
velocity normal to the object boundary. Applying this method to
3D myocardial gated SPECT sequences, the left ventricle
endocardial and epicardial limits can be computed in each
frame. This spacetime segmentation method was tested on
simulated and clinical 3D myocardial gated SPECT sequences and
the corresponding ejection fractions were computed.
 Stolarska, M, Chopp, DL, Moes, N, and Belytschko, T, "Modelling crack growth by level sets in the extended finite element method," INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, vol. 51, pp. 943960, 2001.
Abstract:
An algorithm which couples the level set method (LSM)with the
extended finite element method (XFEM) to model crack growth is
described. The level set method is used to represent the crack
location, including the location of crack tips. The extended
finite element method is used to compute the stress and
displacement fields necessary for determining the rate of crack
growth. This combined method requires no remeshing as the crack
progresses, making the algorithm very efficient. The
combination of these methods has a tremendous potential for a
wide range of applications. Numerical examples are presented to
demonstrate the accuracy of the combined methods. Copyright (C)
2001 John Wiley & Sons, Ltd.
 Cao, F, and Moisan, L, "Geometric computation of curvature driven plane curve evolutions," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 39, pp. 624646, 2001.
Abstract:
We present a new numerical scheme for planar curve evolution
with a normal velocity equal to F (k), where k is the curvature
and F is a nondecreasing function such that F (0) = 0 and
either x bar right arrow F (x(3)) is Lipschitz with Lipschitz
constant less than or equal to 1 or F (x) = x(gamma) for gamma
greater than or equal to 1/3. The scheme is completely
geometrical and avoids some drawbacks of finite difference
schemes. In particular, no special parameterization is needed
and the scheme is monotone ( that is, if a curve initially
surrounds another one, then this remains true during their
evolution), which guarantees numerical stability. We prove
consistency and convergence of this scheme in a weak sense.
Finally, we display some numerical experiments on synthetic and
real data.
 Jakobsen, ER, Karlsen, HK, and Risebro, NH, "On the convergence rate of operator splitting for Hamilton Jacobi equations with source terms," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 39, pp. 499518, 2001.
Abstract:
We establish a rate of convergence for a semidiscrete operator
splitting method applied to Hamilton Jacobi equations with
source terms. The method is based on sequentially solving a
Hamilton Jacobi equation and an ordinary differential equation.
The Hamilton Jacobi equation is solved exactly while the
ordinary differential equation is solved exactly or by an
explicit Euler method. We prove that the Linfinity error
associated with the operator splitting method is bounded by
O(Deltat), where Deltat is the splitting (or time) step. This
error bound is an improvement over the existing O (root Deltat)
bound due to Souganidis [Nonlinear Anal., 9 (1985), pp. 217
257]. In the onedimensional case, we present a fully discrete
splitting method based on an unconditionally stable front
tracking method for homogeneous Hamilton Jacobi equations. It
is proved that this fully discrete splitting method possesses a
linear convergence rate. Moreover, numerical results are
presented to illustrate the theoretical convergence results.
 Chopp, DL, "Some improvements of the fast marching method," SIAM JOURNAL ON SCIENTIFIC COMPUTING, vol. 23, pp. 230244, 2001.
Abstract:
The fast marching method published by Sethian [ Proc. Natl.
Acad. Sci. USA, 93 ( 1996), pp. 15911595] is an optimally
efficient algorithm for solving problems of front volution
where the front speed is monotonic. It has been used in a wide
variety of applications such as robotic path planning [R.
Kimmel and J. Sethian, Fast Marching Methods for Computing
Distance Maps and Shortest Paths, Tech. Report 669, CPAM,
University of California, Berkeley, 1996], crack propagation
[M. Stolarska et al., Internat. J. Numer. Methods Engrg., 51 (
2001), pp. 943960; N. Sukumar, D. L. Chopp, and B. Moran,
Extended finite element method and fast marching method for
threedimensional fatigue crack propagation, J. Comput. Phys.,
submitted], seismology [ J. Sethian and A. Popovici,
Geophysics, 64 (1999), pp. 516523], photolithography [ J.
Sethian, Fast marching level set methods for threedimensional
photolithography development, in Proceedings of the SPIE 1996
International Symposium on Microlithography, Santa Clara, CA,
1996], and medical imaging [ R. Malladi and J. Sethian, Proc.
Natl. Acad. Sci. USA, 93 ( 1996), pp. 93899392]. It has also
been a valuable tool for the implementation of modern level set
methods where it is used to efficiently compute the distance to
the front and/or an extended velocity function. In this paper,
we improve upon the second order fast marching method of
Sethian [SIAM Rev., 41 ( 1999), pp. 199235] by constructing a
second order approximation of the interface generated from
local data on the mesh. The data is interpolated on a single
box of the mesh using a bicubic approximation. The distance to
the front is then calculated by using a variant of Newtons
method to solve both the level curve equation and the
orthogonality condition for the nearest point to a given node.
The result is a second order approximation of the distance to
the interface which can then be used to produce second order
accurate initial conditions for the fast marching method and a
third order fast marching method.
 Khenner, M, Averbuch, A, Israeli, M, and Nathan, M, "Numerical simulation of grainboundary grooving by level set method," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 170, pp. 764784, 2001.
Abstract:
A numerical investigation of grainboundary grooving by means
of a level set method is carried out. An idealized
polycrystalline interconnect which consists of grains separated
by parallel grain boundaries aligned normal to the average
orientation of the surface is considered. Initially. the
surface diffusion is the only physical mechanism assumed. The
surface diffusion is driven by surfacecurvature gradients.
while a tired surface slope and zero atomic flux are assumed at
the groove root. The corresponding mathematical system is an
initial boundary value problem For a twodimensional equation
of HamiltonJacobi type. The results obtained are in good
agreement with both Mullins analytical "smallslope" solution
of the linearized problem (W. W. Mullins. 1957. J. Appl. Phys.
28. 333) (for the case of an isolated grain boundary) and with
the solution for a periodic array of grain boundaries (S. A.
Hackney, 1988. Scripta Metall. 22. 1731). Incorporation of an
electric field changes the problem to one of electromigration.
Preliminary results of electromigration drift velocity
simulations in copper lines art: presented and discussed. (C)
2001 Academic Press.
 Burchard, P, Cheng, LT, Merriman, B, and Osher, S, "Motion of curves in three spatial dimensions using a level set approach," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 170, pp. 720741, 2001.
Abstract:
The level set method was originally designed for problems
dealing with codimension one objects. where it has been
extremely succesful. especially when topological changes in the
interface, i.e., merging and breaking, occur. Attempts have
been made to modify it to handle objects of higher codimension,
such as vortex filaments, while preserving the merging and
breaking property. We present numerical simulations of a level
set based method for moving curves in R3. the model problem
for higher codimension, that allows for topological changes. A
vector valued level set function is used with the zero level
set representing the curve. Our results show that this method
can handle many types of curves moving under all types of
geometrically based flows while automatically enforcing merging
and breaking. (C) 2001 Academic Press.
 Tsai, A, Yezzi, A, and Willsky, AS, "Curve evolution implementation of the MumfordShah functional for image segmentation, denoising, interpolation, and magnification," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 10, pp. 11691186, 2001.
Abstract:
In this work, we first address the problem of simultaneous
image segmentation and smoothing by approaching the Mumford
Shah paradigm from a curve evolution perspective. In
particular, we let a set of deformable contours define the
boundaries between regions in an image where we model the data
via piecewise smooth functions and employ a gradient flow to
evolve these contours. Each gradient step involves solving an
optimal estimation problem for the data within each region,
connecting curve evolution and the MumfordShah functional with
the theory of boundaryvalue stochastic processes. The
resulting active contour model offers a tractable
implementation of the original MumfordShah model (i.e.,
without resorting to elliptic approximations which have
traditionally been favored for greater ease in implementation)
to simultaneously segment and smoothly reconstruct the data
within a given image in a coupled manner. Various
implementations of this algorithm are introduced to increase
its speed of convergence. We also outline a hierarchical
implementation of this algorithm to handle important image
features such as triple points and other multiple junctions.
Next, by generalizing the data fidelity term of the original
MumfordShah functional to incorporate a spatially varying
penalty, we extend our method to problems in which data quality
varies across the image and to images in which sets of pixel
measurements are missing. This more general model leads us to a
novel PDEbased approach for simultaneous image magnification,
segmentation, and smoothing, thereby extending the traditional
applications of the MumfordShah functional which only
considers simultaneous segmentation and smoothing.
 Sifakis, E, and Tziritas, G, "Moving object localisation using a multilabel fast marching algorithm," SIGNAL PROCESSINGIMAGE COMMUNICATION, vol. 16, pp. 963976, 2001.
Abstract:
In this paper, we address two problems crucial to motion
analysis: the detection of moving objects and their
localisation. Statistical and level set approaches are adopted
in formulating these problems. For the change detection
problem, the interframe difference is modelled by a mixture of
two zeromean Laplacian distributions. At first, statistical
tests using criteria with negligible error probability are used
for labelling as changed or unchanged as many sites as
possible. All the connected components of the labelled sites
are used thereafter as region seeds, which give the initial
level sets for which velocity fields for label propagation are
provided, We introduce a new multilabel fast marching
algorithm for expanding competitive regions. The solution of
the localisation problem is based on the map of changed pixels
previously extracted. The boundary of the moving object is
determined by a level set algorithm, which is initialised by
two curves evolving in converging opposite directions. The
sites of curve contact determine the position of the object
boundary. Experimental results using real video sequences are
presented, illustrating the efficiency of the proposed
approach. (C) 2001 Elsevier Science B.V. All rights reserved.
 Vvedensky, DD, "Epitaxial phenomena across length and time scales," SURFACE AND INTERFACE ANALYSIS, vol. 31, pp. 627636, 2001.
Abstract:
The morphological evolution of an epitaxial film results from
atomistic processes such as adatom motion and stepadatom
interactions asserting their influence over the macroscopic
length and time scales of the growth front. Modelling epitaxial
phenomena thus necessitates making a compromise between the
detailed information provided by firstprinciples methods and
the computational flexibility afforded by methods such as
MonteCarlo simulations and continuum equations of motion, in
which atomistic processes are replaced by coarsegrained
effective kinetics. We will review the various approaches that
are available for modelling epitaxial phenomena using the (001)
surfaces of IIIV compound semiconductors as a case study, with
a view to making direct comparisons with experimental
measurements and to establishing a methodology that is capable
of incorporating all pertinent length and time scales.
Copyright (C) 2001 John Wiley & Sons, Ltd.
 Suri, JS, "Twodimensional fast magnetic resonance brain segmentation," IEEE ENGINEERING IN MEDICINE AND BIOLOGY MAGAZINE, vol. 20, pp. 8495, 2001.
Abstract:
Many problems in engineering design involve optimizing the
geometry to maximize a certain design objective. Geometrical
constraints are often imposed. In this paper, we use the level
set method devised in (Osher and Sethian, J. Comput. Phys. 79.
12 ( 1988)), the variational level set calculus presented in
(Zhao et al.. J. Comput. Phys. 127, 179 (1996)), and the
projected gradient method. as in (Rudin et al.. Physica D. 60.
259 (1992)), to construct a simple numerical approach for
problems of this type. We apply this technique to a model
problem involving a vibrating system whose resonant frequency
or whose spectral gap is to be optimized subject to constraints
on geometry. Our numerical results are quite promising. We
expect to use this approach to deal with a wide class of
optimal design problems in the future. (C) 2001 Academic Press.
 Osher, SJ, and Santosa, F, "Level set methods for optimization problems involving geometry and constraints I. Frequencies of a twodensity inhomogeneous drum," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 171, pp. 272288, 2001.
Abstract:
Many problems in engineering design involve optimizing the
geometry to maximize a certain design objective. Geometrical
constraints are often imposed. In this paper, we use the level
set method devised in (Osher and Sethian, J. Comput. Phys. 79.
12 ( 1988)), the variational level set calculus presented in
(Zhao et al.. J. Comput. Phys. 127, 179 (1996)), and the
projected gradient method. as in (Rudin et al.. Physica D. 60.
259 (1992)), to construct a simple numerical approach for
problems of this type. We apply this technique to a model
problem involving a vibrating system whose resonant frequency
or whose spectral gap is to be optimized subject to constraints
on geometry. Our numerical results are quite promising. We
expect to use this approach to deal with a wide class of
optimal design problems in the future. (C) 2001 Academic Press.
 Richards, DF, Bloomfield, MO, Sen, S, and Cale, TS, "Extension velocities for level set based surface profile evolution," JOURNAL OF VACUUM SCIENCE & TECHNOLOGY AVACUUM SURFACES AND FILMS, vol. 19, pp. 16301635, 2001.
Abstract:
Topography simulations are widely used in the microelectronics
industry to study the evolution of surface profiles during such
processes as deposition or etching. Comparisons between
simulations and experiments are used to test proposed transport
and chemistry models. The method used to move the surface (the
moving algorithm) should not interfere with this testing
process; i.e., it should not introduce artifacts. The reference
method, shown to be accurate by several groups in many studies,
is conservation law based "front tracking." Level set
approaches are being increasingly used, largely for their
robustness to topological changes. They have not been tested
against front tracking to determine their accuracy. In this
article, we present guidelines on the use of level set methods
for twodimensional surface evolutions as commonly used.
Specifically, we deal with two major issues with level set
algorithms: the need for "extension velocities" and the
rounding of sharp corners due to contouring. We also deal with
a specific approach to velocity extension that is called "fast
marching." Although all methods discussed can provide the same
results within the limit of small enough time steps, we
demonstrate that our proposed "Riemann based" extension
velocities can improve overall simulation efficiency by
approximately a factor of 2, depending upon the complexity of
the process being simulated. We also show that as the grid size
used in the level set method decreases, extracted surface
profiles can approach those calculated by front tracking, and
hence to experiments. (C) 2001 American Vacuum Society.
 Phan, AV, Kaplan, T, Gray, LJ, Adalsteinsson, D, Sethian, JA, BarvosaCarter, W, and Aziz, MJ, "Modelling a growth instability in a stressed solid," MODELLING AND SIMULATION IN MATERIALS SCIENCE AND ENGINEERING, vol. 9, pp. 309325, 2001.
Abstract:
The growth of crystalline silicon from the amorphous phase in
the presence of an applied stress is modelled using advanced
numerical methods. The crystal region is modelled as a linear
elastic solid and the amorphous as a viscous fluid with a time
dependent viscosity to reflect structural relaxation.
Appropriate coupling conditions across the boundary are
defined, and both problems are solved using a symmetric
Galerkin boundary integral method. The interface is advanced in
time using the level set technique. The results match well with
experiments and support the proposed kinetic mechanism for the
observed interface growth instability.
 Ratsch, C, Kang, M, and Caflisch, RE, "Atomic size effects in continuum modeling  art. no. 020601," PHYSICAL REVIEW E, vol. 6402, pp. 0601+, 2001.
Abstract:
Continuum modeling of many physical systems typically assumes
that the spatial extent of an atom is small compared to the
quantities of interest and can therefore be neglected. We show
that this is valid only asymptotically. For many applications
of practical interest, the spatial extent of a discrete atom
cannot be neglected. We have developed a model for the
description of epitaxial growth based on the levelset method,
and find that we can accurately predict quantities such as the
island densities, if we implement boundary conditions in a
region with atomic width, rather than just on a line without
any spatial extent. Only in the limit of very large islands and
island spacings can this be neglected.
 Bartesaghi, A, and Sapiro, G, "A system for the generation of curves on 3D brain images," HUMAN BRAIN MAPPING, vol. 14, pp. 115, 2001.
Abstract:
In this study, a computational optimal system for the
generation of curves on triangulated surfaces representing 3D
brains is described. The algorithm is based on optimally
computing geodesics on the triangulated surfaces following;
Kimmel and Sethian ([1998]: Proc Natl Acad Sci 95:15). The
system can be used to compute geodesic curves for accurate
distance measurements as well as to detect sulci and gyri.
These curves are defined based on local surface curvatures that
are computed following a novel approach presented in this
study. The corresponding software is available to the research
community. Hum. Brain Mapping 14:115, 2001. (C) 2001 Wiley
Liss, Inc.
 Ye, T, Shyy, W, and Chung, JN, "A fixedgrid, sharpinterface method for bubble dynamics and phase change," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 174, pp. 781815, 2001.
Abstract:
A numerical method has been developed for direct simulation of
bubble dynamics with large liquidtovapor density ratio and
phase change. The numerical techniques are based on a fixed
grid, finite volume method capable of treating the interface as
a sharp discontinuity. The unsteady, axisymmetric NavierStokes
equations and energy equation in both liquid and vapor phases
are computed. The mass, momentum, and energy conditions are
explicitly matched at the phase boundary to determine the
interface shape and movement. The cubic Bspline is used in
conjunction with a fairing algorithm to yield smooth and
accurate information of curvatures. Nondimensional parameters
including Reynolds, Weber, and Jakob numbers are varied to
offer insight into the physical and numerical characteristics
of the bubble dynamics. Based on the present sharp interface
approach, bubble dynamics for density ratio of 1600 or higher,
with and without phase change, can be successfully computed.
(C) 2001 Elsevier Science.
 Li, ZL, and Lai, MC, "The immersed interface method for the NavierStokes equations with singular forces," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 171, pp. 822842, 2001.
Abstract:
Peskin's Immersed Boundary Method has been widely used for
simulating many fluid mechanics and biology problems. One of
the essential components of the method is the usage of certain
discrete delta functions to deal with singular forces along one
or several interfaces in the fluid domain, However, the
Immersed Boundary Method is known to be firstorder accurate
and usually smears out the solutions. In this paper, we propose
an immersed interface method for the incompressible Navier
Stokes equations with singular forces along one or several
interfaces in the solution domain. The new method is based on a
secondorder projection method with modifications only at grid
points near or on the interface. From the derivation of the new
method, we expect fully secondorder accuracy for the velocity
and nearly secondorder accuracy for the pressure in the
maximum norm including those grid points near or on the
interface. This has been confirmed in our numerical
experiments. Furthermore, the computed solutions are sharp
across the interface. Nontrivial numerical results are provided
and compared with the Immersed Boundary Method. Meanwhile, a
new version of the Immersed Boundary Method using the level set
representation of the interface is also proposed in this paper.
(C) 2001 Academic Press.
 Cerne, G, Petelin, S, and Tiselj, I, "Coupling of the interface tracking and the twofluid models for the simulation of incompressible twophase flow," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 171, pp. 776804, 2001.
Abstract:
The volume of fluid (VOF) method, which uses an interface
tracking algorithm for the simulation of the twophase flow, is
coupled with the "twofluid" model, which is based on time and
space averaged equations and cannot track the interface
explicitly. The idea of the present work is to use the VOF
method in the parts of the computational domain where the grid
density allows surface tracking, In the pam of the domain where
the flow is too dispersed to be described by the interface
tracking algorithms, the twofluid model is used, The equations
of the twofluid model are less accurate than the VOF model due
to the empirical closures required in the averaged equations.
However, in the case of the sufficiently dispersed flow, the
twofluid model results are still much closer to the real world
than the results of the VOF method, which do not have any
physical meaning when the grid becomes too coarse. Each model
in the present work uses a separate set of equations suitable
for description of twodimensional, incompressible, viscous
twophase flow. Similar discretization techniques are used for
both sets of equations and solved with the same numerical
method. Coupling of both models is achieved via the volume
fraction of one of the fluids, which is used in both models. A
special criterion for the transition between the models is
derived from the interface reconstruction function in the VOF
method. An idealized vortical flow and the RayleighTaylor
instability are used as tests of the coupling. In both cases
the time development causes mixing of the fluids and dispersion
of the interface that is beyond the capabilities of the model
based on the VOF method. Therefore the twofluid model
gradually replaces the interface tracking model. In the final
stages of the RayleighTaylor instability, when both fluids are
approaching their final positions and the tractable interface
appears again, the twofluid model is gradually replaced by the
VOF method. (C) 2001 Academic Press.
 Bertalmio, M, Cheng, LT, Osher, S, and Sapiro, G, "Variational problems and partial differential equations on implicit surfaces," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 174, pp. 759780, 2001.
Abstract:
A novel framework for solving variational problems and partial
differential equations for scalar and vectorvalued data
defined on surfaces is introduced in this paper. The key idea
is to implicitly represent the surface as the level set of a
higher dimensional function and to solve the surface equations
in a fixed Cartesian coordinate system using this new embedding
function. The equations are then both intrinsic to the surface
and defined in the embedding space. This approach thereby
eliminates the need for performing complicated and inaccurate
computations on triangulated surfaces, as is commonly done in
the literature. We describe the framework and present examples
in computer graphics and image processing applications,
including texture synthesis, flow field visualization, and
image and vector field intrinsic regularization for data
defined on 3D surfaces. (C) 2001 Elsevier Science.
 Chan, TF, and Vese, LA, "Active contours without edges," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 10, pp. 266277, 2001.
Abstract:
In this paper, we propose a new model for active contours to
detect objects in a given image, based on techniques of curve
evolution, MumfordShah functional for segmentation and level
sets. Our model can detect objects whose boundaries are not
necessarily defined by gradient. We minimize an energy which
can he seen as a particular case of the minimal partition
problem, In the level set formulation, the problem becomes a
"meancurvature flow"like evolving the active contour, which
will stop on the desired boundary. However, the stopping term
does not depend on the gradient of the. image, as in the
classical active contour models, hut is instead related to a
particular segmentation of the image. We will give a numerical
algorithm using finite differences. Finally, we will present
various experimental results and in particular some examples
for which the classical snakes methods based on the gradient
are not applicable. Also, the initial curve can be anywhere in
the image, and interior contours are automatically detected.
 Ki, H, Mohanty, PS, and Mazumder, J, "Modelling of highdensity lasermaterial interaction using fast level set method," JOURNAL OF PHYSICS DAPPLIED PHYSICS, vol. 34, pp. 364372, 2001.
Abstract:
A highenergydensity laser beammaterial interaction process
has been simulated considering a selfevolving liquidvapour
interface profile. A mathematical scheme called the levelset
technique has been adopted to capture the transient Liquid
vapour interface. Inherent to this technique are: the ability
to simulate merger and splitting of the liquidvapour interface
and the simultaneous updating of the surface normal and the
curvature. Unsteady heat transfer and fluid flow phenomena are
modelled, considering the thermocapillary effect and the
recoil pressure. A kinetic Knudsen layer has been considered to
simulate evaporation phenomena at the liquidvapour interface.
Also, the homogeneous boiling phenomenon near the critical
point is implemented. Energy distribution inside the vapour
cavity is computed considering multiple reflection phenomena.
The effect of laser power on the material removal mode, liquid
layer thickness, surface temperature and the evaporation speed
are presented and discussed.
 Iafrati, A, Di Mascio, A, and Campana, EF, "A level set technique applied to unsteady free surface flows," INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN FLUIDS, vol. 35, pp. 281297, 2001.
Abstract:
An unsteady NavierStokes solver for incompressible fluid is
coupled with a level set approach to describe free surface
motions. The twophase now of air and water is approximated by
the flow of a single fluid whose properties, such as density
and viscosity, change across the interface. The free surface
location is captured as the zero level of a distance function
convected by the flow field. To validate the numerical
procedure, two classical twodimensional free surface problems
in hydrodynamics, namely the oscillating flow in a tank and the
waves generated by the flow over a bottom bump, are studied in
nonbreaking conditions, and the results are compared with
those obtained with other numerical approaches. To check the
capability of the method in dealing with complex free surface
configurations, the breaking regime produced by the flow over a
high bump is analyzed. The analysis covers the successive
stages of the breaking phenomenon: the steep wave evolution,
the falling jet, the splashup and the air entrainment. In all
phases, numerical results qualitatively agree with the
experimental observations. Finally, to investigate a flow in
which viscous effects are relevant, the numerical scheme is
applied to study the wavy flow past a submerged hydrofoil.
Copyright (C) 2001 John Wiley & Sons, Ltd.
 Quecedo, M, and Pastor, M, "Application of the level set method to the finite element solution of twophase flows," INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, vol. 50, pp. 645663, 2001.
Abstract:
This paper presents a method to solve twophase flows using the
finite element method. On one hand, the algorithm used to solve
the NavierStokes equations provides the neccessary
stabilization for using the efficient and accurate threenode
triangles for both the velocity and pressure fields. On the
other hand, the interface position is described by the zero
level set of an indicator function. To maintain accuracy, even
for largedensity ratios, the pseudoconcentration function is
corrected at the end of each time step using an algorithm
successfully used in the finite difference context. Coupling of
both problems is solved in a staggered way. As demonstrated by
the solution of a number of numerical tests, the procedure
allows dealing with problems involving two interacting fluids
with a largedensity ratio. Copyright (C) 2001 John Wiley &
Sons, Ltd.
 Xu, K, "A kinetic method for hyperbolicelliptic equations and its application in twophase flow," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 166, pp. 383399, 2001.
Abstract:
A kinetic method for hyperbolicelliptic equations is presented
in this paper. In the mixed type system, the coexistence of
liquid and gas and the phase transition between them are
described by the van der Waalstype equation of state (EOS).
Because the fluid is unstable in the elliptic region, the
interface between liquid and gas can be captured naturally
through condensation and evaporation processes. which
continuously remove any "averaged" numerical fluid away from
the elliptic region at the interfaces. As a result, a sharp
liquidgas interface can be constructed from the competition
between the numerical diffusion and phase transition. The
numerical examples presented in this paper include both phase
transition and multifluid interface problems. (C) 2001 Academic
Press.
 Frangi, AF, Niessen, WJ, and Viergever, MA, "Threedimensional modeling for functional analysis of cardiac images: A review," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 20, pp. 225, 2001.
Abstract:
Threedimensional (3D) imaging of the heart is a rapidly del
eloping area of research in medical imaging, Advances in
hardware and methods for fast spatiotemporal cardiac imaging
are extending the frontiers of clinical diagnosis and research
on cardiovascular diseases. In the last few Sears, many
approaches hare been proposed to analyze images and extract
parameters of cardiac shape and function from a variety of
cardiac imaging modalities. In particular, techniques based on
spatiotemporal geometric models have received considerable
attention. This paper surveys the literature of tno decades of
research on cardiac modeling. The contribution of the paper is
threefold: 1) to serve as a tutorial of the field for both
clinicians and technologists, 2) to provide an extensive
account of modeling techniques in a comprehensive and
systematic manner, and 3) to critically review these approaches
in terms of their performance and degree of clinical evaluation
with respect to the final goal of cardiac functional analysis,
From this review it is concluded that whereas 3D modelbased
approaches have the capability. to improve the diagnostic value
of cardiac images, issues as robustness, 3D interaction,
computational complexity and clinical validation still require
significant attention.
 Caiden, R, Fedkiw, RP, and Anderson, C, "A numerical method for twophase flow consisting of separate compressible and incompressible regions," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 166, pp. 127, 2001.
Abstract:
We propose a numerical method for modeling twophase flow
consisting of separate compressible and incompressible regions.
This is of interest, for example, when the combustion of fuel
droplets or the shockinduced mixing of liquids is numerically
modeled. We use the level set method to track the interface
between the compressible and incompressible regions, as well as
the Ghost Fluid Method (GFM) to create accurate discretizations
across the interface. The GFM is particularly effective here
since the equations differ in both number and type across the
interface. The numerical method is presented in two spatial
dimensions with numerical examples in both one and two spatial
dimensions, while threedimensional extensions are
straightforward. (C) 2001 Academic Press.
 Chen, S, Merriman, B, Kang, M, Caflisch, RE, Ratsch, C, Cheng, LT, Gyure, M, Fedkiw, RP, Anderson, C, and Osher, S, "A level set method for thin film epitaxial growth," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 167, pp. 475500, 2001.
Abstract:
We present a level set based numerical algorithm for simulating
a model of epitaxial growth. The island dynamics model is a
continuum model for the growth of thin films. In this paper, we
emphasize the details of the numerical method used to simulate
the island dynamics model. (C) 2001 Academic Press.
 Aslam, TD, "A levelset algorithm for tracking discontinuities in hyperbolic conservation laws I. Scalar equations," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 167, pp. 413438, 2001.
Abstract:
A levelset algorithm for tracking discontinuities in
hyperbolic conservation laws is presented. The algorithm uses a
simple finite difference approach, analogous to the method of
lines scheme presented in C.W. Shu and S. Osher (1988, J.
Comput. Phys. 77, 439). The zero of a levelset function is
used to specify the location of the discontinuity. Since a
levelset function is used to describe the front location, no
extra data structures are needed to keep track of the location
of the discontinuity. Also, two solution states are used at all
computational nodes, one corresponding to the ''real" state,
and one corresponding to a "ghost node" state, analogous to the
"Ghost Fluid Method" of R. P. Fedkiw et al. (1999, J. Comput.
phys. 154, 459). Highorder pointwise convergence is
demonstrated for linear and nonlinear conservation laws, even
at discontinuities and in multiple dimensions. The solutions
are compared to standard highorder shockcapturing schemes.
This paper focuses on scalar conservation laws. An example is
given for shock tracking in the onedimensional Euler
equations. Levelset tracking for systems of conservation laws
in multidimensions will be presented in future work.
 Baumann, FH, Chopp, DL, de la Rubia, TD, Gilmer, GH, Greene, JE, Huang, H, Kodambaka, S, O'Sullivan, P, and Petrov, I, "Multiscale modeling of thinfilm deposition: Applications to Si device processing," MRS BULLETIN, vol. 26, pp. 182189, 2001.
Abstract:
A technique to simulate the flow field near a moving material
interface is developed for multimaterial compressible flow, in
particular, for compressible gaswater flow. This technique can
be conveniently applied with a wellestablished conservative
scheme to solve for the regions away from the interface.
Material interfaces are captured using the level set technique
with minimum or no smearing. To treat wave interaction with the
interface, an implicit characteristic method is developed. In
this paper, the method is described in detail and tested
extensively for several onedimensional gasgas and gaswater
cases. Application to multidimensional shockfree surface
interaction and shockgas bubble interaction are presented in
Part II [Liu TG, Khoo BC, Yeo KS. The simulation of
compressible multimedium flow. Part II: Applications to 2D
underwater shuck refraction. Comp. and Fluids 2000;30:31537].
(C) 2001 Elsevier Science Ltd. All rights reserved.
 Liu, TG, Khoo, BC, and Yeo, KS, "The simulation of compressible multimedium flow. I. A new methodology with test applications to 1D gasgas and gas water cases," COMPUTERS & FLUIDS, vol. 30, pp. 291314, 2001.
Abstract:
A technique to simulate the flow field near a moving material
interface is developed for multimaterial compressible flow, in
particular, for compressible gaswater flow. This technique can
be conveniently applied with a wellestablished conservative
scheme to solve for the regions away from the interface.
Material interfaces are captured using the level set technique
with minimum or no smearing. To treat wave interaction with the
interface, an implicit characteristic method is developed. In
this paper, the method is described in detail and tested
extensively for several onedimensional gasgas and gaswater
cases. Application to multidimensional shockfree surface
interaction and shockgas bubble interaction are presented in
Part II [Liu TG, Khoo BC, Yeo KS. The simulation of
compressible multimedium flow. Part II: Applications to 2D
underwater shuck refraction. Comp. and Fluids 2000;30:31537].
(C) 2001 Elsevier Science Ltd. All rights reserved.
 Ida, M, and Yamakoshi, Y, "An Eulerian scheme for direct numerical simulation of multibubble dynamics in an acoustic field," JAPANESE JOURNAL OF APPLIED PHYSICS PART 1REGULAR PAPERS SHORT NOTES & REVIEW PAPERS, vol. 40, pp. 38463851, 2001.
Abstract:
An Eulerian numerical scheme for the direct simulation of
bubble dynamics in an acoustic field is proposed. The
compressible NavierStokes equation with a surface tension term
is used as a governing equation. The convection terms in the
equation are solved using the hybrid interpolation
extrapolation scheme which stably provides a nonsmoothed
solution of density interface between bubbles and a surrounding
material. The acoustic terms in the equation are solved by the
generalized CrankNicholson method which is an implicit method
and has a temporally secondorder accuracy under the low
Courantnumber condition and is applicable to highCourant
number computation. Using this scheme, some typical results of
single and multibubble phenomena are given.
 Nguyen, DQ, Fedkiw, RP, and Kang, M, "A boundary condition capturing method for incompressible flame discontinuities," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 172, pp. 7198, 2001.
Abstract:
In this paper, we propose a new numerical method for treating
twophase incompressible flow where one phase is being
converted into the other, e.g., the vaporization of liquid
water. We consider this numerical method in the context of
treating discontinuously thin flame fronts for incompressible
flow. This method was designed as an extension of the Ghost
Fluid Method (1999, J. Comput. Phys. 152, 457) and relies
heavily on the boundary condition capturing technology
developed in Liu et al. (2000, J. Comput. Phys. 154, 15) for
the variable coefficient Poisson equation and in Kang et al.
(in press J. Comput. Phys.) for multiphase incompressible flow.
Our new numerical method admits a sharp interface
representation similar to the method proposed in Helenbrook et
al. (1999, J. Comput. Phys. 148, 366). Since the interface
boundary conditions are handled in a simple and straightforward
fashion, the code is very robust, e.g. no special treatment is
required to treat the merging of flame fronts. The method is
presented in three spatial dimensions, with numerical examples
in one, two, and three spatial dimensions. (C) 2001 Academic
Press.
 Baillard, C, Hellier, P, and Barillot, C, "Segmentation of brain 3D MR images using level sets and dense registration," MEDICAL IMAGE ANALYSIS, vol. 5, pp. 185194, 2001.
Abstract:
This paper presents a strategy for the segmentation of brain
from volumetric MR images which integrates 3D segmentation and
3D registration processes. The segmentation process is based on
the level set formalism. A closed 3D surface propagates towards
the desired boundaries through the iterative evolution of a 4D
implicit function. In this work, the propagation relies on a
robust evolution model including adaptive parameters. These
depend on the input data and on statistical distribution
models. The main contribution of this paper is the use of an
automatic registration method to initialize the surface, as an
alternative solution to manual initialization, The registration
is achieved through a robust multiresolution and multigrid
minimization scheme. This coupling significantly improves the
quality of the method, since the segmentation is faster, more
reliable and fully automatic. Quantitative and qualitative
results on both synthetic and real volumetric brain MR images
are presented and discussed. (C) 2001 Elsevier Science B.V. All
rights reserved.
 Karma, A, "Phasefield formulation for quantitative modeling of alloy solidification  art. no. 115701," PHYSICAL REVIEW LETTERS, vol. 8711, pp. 5701+, 2001.
Abstract:
A phasefield formulation is introduced to simulate
quantitatively microstructural pattern formation in alloys. The
thininterface limit of this formulation yields a much less
stringent restriction on the choice of interface thickness than
previous formulations and permits one to eliminate
nonequilibrium effects at the interface. Dendrite growth
simulations with vanishing solid diffusivity show that both the
interface evolution and the solute profile in the solid are
accurately modeled by this approach.
 Breen, DE, Mauch, S, Whitaker, RT, and Mao, J, "3D metamorphosis between different types of geometric models," COMPUTER GRAPHICS FORUM, vol. 20, pp. C36+, 2001.
Abstract:
We present a powerful morphing technique based on level set
methods, that can be combined with a variety of scan
conversion/model processing techniques. Bringing these
techniques together creates a general morphing approach that
allows a user to morph a number of geometric model types in a
single animation. We have developed techniques for converting
several types of geometric models (polygonal meshes, CSG models
and MRI scans) into distance volumes, the volumetric
representation required by our level set morphing approach. The
combination of these two capabilities allows a user to create a
morphing sequence regardless of the model type of the source
and target objects, freeing him/her to use whatever model type
is appropriate for a particular animation.
 Goldenberg, R, Kimmel, R, Rivlin, E, and Rudzsky, M, "Fast geodesic active contours," IEEE TRANSACTIONS ON IMAGE PROCESSING, vol. 10, pp. 14671475, 2001.
Abstract:
We use an unconditionally stable numerical scheme to implement
a fast version of the geodesic active contour model. The
proposed scheme is useful for object segmentation in images,
like tracking moving objects in a sequence of images. The
method is based on the WeickertRomeneyViergever (additive
operator splitting) AOS scheme. It is applied at small regions,
motivated by AdalsteinssonSethian level set narrow band
approach, and uses Sethian's fast marching method for re
initialization. Experimental results demonstrate the power of
the new method for tracking in color movies.
 Bernoussi, A, El Jai, A, and Pritchard, AJ, "Spreadability and evolving interfaces," INTERNATIONAL JOURNAL OF SYSTEMS SCIENCE, vol. 32, pp. 12171232, 2001.
Abstract:
The notion of spreadability was introduced by El Jai et al. in
1994 to describe expansion phenomena. The principle consists in
exploring the evolution of sets where a given property is
satisfied. An extension (the A spreadability) was given in 1998
by Bernoussi who considered the measure of such sets. In some
cases, the sets can be described by their boundaries and the
aim of this paper is to explore some interesting connections of
spreadability with wave fronts and level sets. The results are
illustrated by a variety of examples.
 Chopp, DL, "Commentary  Replacing iterative algorithms with singlepass algorithms," PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA, vol. 98, pp. 1099210993, 2001.
Abstract:
A methodology to model arbitrary holes and material interfaces
(inclusions) without meshing the internal boundaries is
proposed. The numerical method couples the level set method (S.
Osher, J.A. Sethian, J. Comput. Phys. 79 (1) (1988) 12) to the
extended finiteelement method (XFEM) (N. Moes, J. Dolbow, T.
Belytschko, Int. J. Numer. Methods Engrg. 46 (1) (1999) 131).
In the XFEM, the finiteelement approximation is enriched by
additional functions through the notion of partition of unity.
The level set method is used for representing the location of
holes and material interfaces, and in addition, the level set
function is used to develop the local enrichment for material
interfaces. Numerical examples in twodimensional linear
elastostatics are presented to demonstrate the accuracy and
potential of the new technique. (C) 2001 Elsevier Science BY.
All rights reserved.
 Sukumar, N, Chopp, DL, Moes, N, and Belytschko, T, "Modeling holes and inclusions by level sets in the extended finiteelement method," COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, vol. 190, pp. 61836200, 2001.
Abstract:
A methodology to model arbitrary holes and material interfaces
(inclusions) without meshing the internal boundaries is
proposed. The numerical method couples the level set method (S.
Osher, J.A. Sethian, J. Comput. Phys. 79 (1) (1988) 12) to the
extended finiteelement method (XFEM) (N. Moes, J. Dolbow, T.
Belytschko, Int. J. Numer. Methods Engrg. 46 (1) (1999) 131).
In the XFEM, the finiteelement approximation is enriched by
additional functions through the notion of partition of unity.
The level set method is used for representing the location of
holes and material interfaces, and in addition, the level set
function is used to develop the local enrichment for material
interfaces. Numerical examples in twodimensional linear
elastostatics are presented to demonstrate the accuracy and
potential of the new technique. (C) 2001 Elsevier Science BY.
All rights reserved.
 Qian, JL, and Symes, WW, "Paraxial eikonal solvers for anisotropic quasiP travel times," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 173, pp. 256278, 2001.
Abstract:
The firstarrival quasiP wave traveltime field in an
anisotropic elastic solid solves a firstorder nonlinear
partial differential equation, the qP eikonal equation, which
is a stationary HamiltonJacobi equation. The solution of the
paraxial quasiP eikonal equation, an evolution HamiltonJacobi
equation in depth. gives the firstarrival travel time along
downward propagating rays. We devise nonlinear numerical
algorithms to compute the paraxial Hamiltonian for quasiP wave
propagation ill general anisotropic media. A secondorder
essentially nonoscillatory (ENO) RungeKutta scheme solves this
paraxial eikonal equation with a point source as an initial
condition in O(N) floating point operations, where N is the
number of grid points, Numerical experiments using 2D
transversely isotropic models with inclined symmetry axes
demonstrate the accuracy of the algorithms. (C) 2001 Academic
Press.
 Oberlack, M, Wenzel, H, and Peters, N, "On symmetries and averaging of the Gequation for premixed combustion," COMBUSTION THEORY AND MODELLING, vol. 5, pp. 363383, 2001.
Abstract:
It is demonstrated that the Gequation for premixed combustion
admits a diversity of symmetries properties, i.e. invariance
characteristics under certain transformations. Included are
those of classical mechanics such as Galilean invariance,
rotation invariance and others. Also a new generalized scaling
symmetry has been established. It is shown that the generalized
scaling symmetry defines the physical property, of the G
equation precisely. That is to say the value of G at a given
flame front is arbitrary. It is proven that beside the
symmetries of classical mechanics, particularly the generalized
scaling symmetry uniquely defines the basic structure of the G
equation. It is also proven that the generalized scaling
symmetry precludes the application of classical Reynolds
ensemble averaging usually employed in statistical turbulence
theory in order to avoid non unique statistical quantities such
as for the mean flame position. Finally, a new averaging scheme
of the Gfield is presented which is fully consistent with all
symmetries of the Gequation. Equations for the mean Gfield
and flame brush thickness are derived and a route to consistent
invariant modelling of other quantities derived from the G
field is illustrated. Examples of statistical quantities
derived from the Gfield both in the context of Reynolds
averaged models as well as subgridscale models for largeeddy
simulations taken from the literature are investigated as to
whether they are compatible with the important generalized
scaling symmetry.
 De Solorzano, CO, Malladi, R, Lelievre, SA, and Lockett, SJ, "Segmentation of nuclei and cells using membrane related protein markers," JOURNAL OF MICROSCOPYOXFORD, vol. 201, pp. 404415, 2001.
Abstract:
Segmenting individual cell nuclei from microscope images
normally involves volume labelling of the nuclei with a DNA
stain. However, this method often fails when the nuclei are
tightly clustered in the tissue, because there is little
evidence from the images on where the borders of the nuclei
are. In this paper we present a method which solves this
limitation and furthermore enables segmentation of whole cells.
Instead of using volume stains, we used stains that
specifically label the surface of nuclei or cells: lamins for
the nuclear envelope and alpha6 or beta1 integrins for the
cellular surface. The segmentation is performed by identifying
unique seeds for each nucleus/cell and expanding the boundaries
of the seeds until they reach the limits of the nucleus/cell,
as delimited by the lamin or integrin staining, using gradient
curvature flow techniques. We tested the algorithm using
computergenerated objects to evaluate its robustness against
noise and applied it to cells in culture and to tissue
specimens. In all the cases that we present the algorithm gave
accurate results.
 Burger, M, "A level set method for inverse problems," INVERSE PROBLEMS, vol. 17, pp. 13271355, 2001.
Abstract:
This paper is devoted to the solution of shape reconstruction
problems by a level set method. The basic motivation for the
setup of this level set algorithm is the wellstudied method of
asymptotic regularization, which has been developed for ill
posed problems in Hilbert spaces. Using analogies to this
method, the convergence analysis of the proposed level set
method is established and it is shown that the evolving level
set converges to a solution in the symmetric difference metric
as the artificial time evolves to infinity. Furthermore, the
regularizing properties of the level set method are shown, if
the discrepancy principle is used as a stopping rule. The
numerical implementation of the level set method is discussed
and applied to some examples in order to compare the numerical
results with theoretical statements. The numerical results
demonstrate the power of the level set method, in particular
for examples where the number of connected components the
solution consists of is not known a priori.
 Ito, K, Kunisch, K, and Li, ZL, "Levelset function approach to an inverse interface problem," INVERSE PROBLEMS, vol. 17, pp. 12251242, 2001.
Abstract:
A model problem in electrical impedance tomography for the
identification of unknown shapes from data in a narrow strip
along the boundary of the domain is investigated. The
representation of the shape of the boundary and its evolution
during an iterative reconstruction process is achieved by the
level set method. The shape derivatives of this problem involve
the normal derivative of the potential along the unknown
boundary. Hence an accurate resolution of its derivatives along
the unknown interface is essential. It is obtained by the
immersed interface method.
 Gibou, FG, Ratsch, C, Gyure, MF, Chen, S, and Caflisch, RE, "Rate equations and capture numbers with implicit islands correlations  art. no. 115401," PHYSICAL REVIEW B, vol. 6311, pp. 5401+, 2001.
Abstract:
We introduce a numerical method based on the levelset
technique to compute capture numbers used in meanfield rate
equations that describe epitaxial growth. In our levelset
approach, islands grow with a velocity that is computed from
solving the diffusion equation for the adatom concentration.
The capture number for each island is then calculated by
integrating the growth velocity of an island around the island
boundary. Thus, our method by construction includes all spatial
correlations between islands. The functional form of the
capture numbers a, is, to first approximation, affinely
dependent on the island sizes. Integration of a completely
deterministic set of meanfield rate equations for the first
time properly reproduces the correct island densities and
cluster size distribution.
 Quek, FKH, and Kirbas, C, "Vessel extraction in medical images by wavepropagation and traceback," IEEE TRANSACTIONS ON MEDICAL IMAGING, vol. 20, pp. 117131, 2001.
Abstract:
This paper presents an approach for the extraction of
vasculature from angiography images by using a wave propagation
and traceback mechanism. We discuss both the theory and the
implementation of the approach. Using a dualsigmoidal filter,
we label each pixel in an angiogram with the likelihood that it
is within a vessel. Representing the reciprocal of this
likelihood image as an array of refractive indexes, we
propagate a digital wave through the image from the base of the
vascular tree. This wave "washes" over the vasculature,
ignoring local noise perturbations. The extraction of the
vasculature becomes that of tracing the wave along the local
normals to the waveform, While the approach is inherently
single instruction stream multiple data stream (SIMD), we
present an efficient sequential algorithm for the wave
propagation and discuss the traceback algorithm, We demonstrate
the effectiveness of our integer image neighborhoodbased
algorithm and its robustness to image noise.
 Mikula, K, and Sevcovic, D, "Evolution of plane curves driven by a nonlinear function of curvature and anisotropy," SIAM JOURNAL ON APPLIED MATHEMATICS, vol. 61, pp. 14731501, 2001.
Abstract:
In this paper we study evolution of plane curves satisfying a
geometric equation v = beta (k,v), where v is the normal
velocity and k and are the curvature and tangential angle of a
plane curve. We follow the direct approach and we analyze the
socalled intrinsic heat equation governing the motion of plane
curves obeying such a geometric equation. The intrinsic heat
equation is modi ed to include an appropriate nontrivial
tangential velocity functional. We show how the presence of a
nontrivial tangential velocity can prevent numerical solutions
from forming various instabilities. From an analytical point of
view we present some new results on short time existence of a
regular family of evolving curves in the degenerate case when
beta (k,v) = gamma (v)k(m), 0 < m 2,
and the governing system of equations includes a nontrivial
tangential velocity functional.
 Giga, MH, and Giga, Y, "Generalized motion by nonlocal curvature in the plane," ARCHIVE FOR RATIONAL MECHANICS AND ANALYSIS, vol. 159, pp. 295333, 2001.
Abstract:
This paper considers the levelset equation for a general
planar anisotropic curvature flow equation when the interfacial
energy is very singular so that the anisotropic curvature
effect is nonlocal. A new notion of solutions is introduced to
establish an analytic foundation of the levelset method
including a comparison principle and stability results. The
main idea behind the proofs is to convert the levelsets of
solutions into graphlike functions. This new procedure is
called slicing and it is not limited to nonlocal curvature flow
equations. Our theory is useful for establishing the
convergence of a crystalline algorithm as well as for
justifying the crystalline flow as a limit of anisotropic
curvature flow with smooth interfacial energy.
 Vemuri, BC, Guo, YL, and Wang, ZZ, "Deformable pedal curves and surfaces: Hybrid geometric active models for shape recovery," INTERNATIONAL JOURNAL OF COMPUTER VISION, vol. 44, pp. 137155, 2001.
Abstract:
In this paper, we propose significant extensions to the "snake
pedal" model, a powerful geometric shape modeling scheme
introduced in (Vemuri and Guo, 1998). The extension allows the
model to automatically cope with topological changes and for
the first time, introduces the concept of a compact global
shape into geometric active models. The ability to characterize
global shape of an object using very few parameters facilitates
shape learning and recognition. In this new modeling scheme,
object shapes are represented using a parameterized function
called the generatorwhich accounts for the global shape of an
object and the pedal curve (surface) of this global shape with
respect to a geometric snake to represent any local detail.
Traditionally, pedal curves (surfaces) are defined as the loci
of the feet of perpendiculars to the tangents of the generator
from a fixed point called the pedal point. Local shape control
is achieved by introducing a set of pedal pointslying on a
snakefor each point on the generator. The model dubbed as a
"snake pedal" allows for interactive manipulation via forces
applied to the snake. In this work, we replace the snake by a
geometric snake and derive all the necessary mathematics for
evolving the geometric snake when the snake pedal is assumed to
evolve as a function of its curvature. Automatic topological
changes of the model may be achieved by implementing the
geometric snake in a levelset framework. We demonstrate the
applicability of this modeling scheme via examples of shape
recovery from a variety of 2D and 3D image data.
 Memoli, F, and Sapiro, G, "Fast computation of weighted distance functions and geodesics on implicit hypersurfaces," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 173, pp. 730764, 2001.
Abstract:
An algorithm for the computationally optimal construction of
intrinsic weighted distance functions on implicit hyper
surfaces is introduced in this paper. The basic idea is to
approximate the intrinsic weighted distance by the Euclidean
weighted distance computed in a band surrounding the implicit
hypersurface in the embedding space, thereby performing all
the computations in a Cartesian grid with classical and
efficient numerics. Based on work on geodesics on Riemannian
manifolds with boundaries, we bound the error between the two
distance functions. We show that this error is of the same
order as the theoretical numerical error in computationally
optimal, HamiltonJacobibased, algorithms for computing
distance functions in Cartesian grids. Therefore, we can use
these algorithms, modified to deal with spaces with boundaries,
and obtain also for the case of intrinsic distance functions on
implicit hypersurfaces a computationally efficient technique.
The approach can be extended to solve a more general class of
HamiltonJacobi equations defined on the implicit surface,
following the same idea of approximating their solutions by the
solutions in the embedding Euclidean space. The framework here
introduced thereby allows for the computation, to be performed
on a Cartesian grid with computationally optimal algorithms, in
spite of the fact that the distance and HamiltonJacobi
equations are intrinsic to the implicit hypersurface. (C) 2001
Academic Press.
 Salden, AH, Romeny, BMT, and Viergever, MA, "A dynamic scalespace paradigm," JOURNAL OF MATHEMATICAL IMAGING AND VISION, vol. 15, pp. 127168, 2001.
Abstract:
We present a novel mathematical, physical and logical framework
for describing an input image of the dynamics of physical
fields, in particular the optic field dynamics. Our framework
is required to be invariant under a particular gauge group,
i.e., a group or set of transformations consistent with the
symmetries of that physical field dynamics enveloping
renormalisation groups. It has to yield a most concise field
description in terms of a complete and irreducible set of
equivalences or invariants. Furthermore, it should be robust to
noise, i.e., unresolvable perturbations (morphisms) of the
physical field dynamics present below a specific dynamic scale,
possibly not covered by the gauge group, do not affect Lyapunov
or structural stability measures expressed in equivalences
above that dynamic scale. The related dynamic scale symmetry
encompasses then a gauge invariant similarity operator with
which similarly prepared ensembles of physical field dynamics
are probed and searched for partial equivalences coming about
at higher scales. The framework of our dynamic scalespace
paradigm is partly based on the initialisation of joint
(non)local equivalences for the physical field dynamics
external to, induced on and stored in a vision system and
represented by an image, possibly at various scales. These
equivalences are consistent with the scalespace paradigm
considered and permit a faithful segmentation and
interpretation of the dynamic scalespace at initial scale.
Among the equivalences are differential invariants, integral
invariants and topological invariants not affected by the
considered gauge group. These equivalences form a quantisation
of the external, induced and stored physical field dynamics,
and are associated to a frame field, coframe field, metric
and/or connection invariant under the gauge group. Examples of
these equivalences are the curvature and torsion twoforms of
general relativity, the Burgers and Frank vector density fields
of crystal theory (in both disciplines these equivalences
measure the inhomogeneity of translational and (affine)
rotation groups over spacetime), and the winding numbers and
other topological charges popping up in electromagnetism and
chromodynamics. Besides based on a gauge invariant
initialisation of equivalences the framework of our dynamic
scalespace paradigm assumes that a robust, i.e. stable and
reproducible, partially equivalent representation of the
physical field dynamics is acquired by a multiscale filtering
technique adapted to those initial equivalences. Effectively,
the hierarchy of nested structures of equivalences, by
definition too invariant under the gauge group, is obtained by
applying an exchange principle for a free energy of the
physical field dynamics (represented through the equivalences)
that in turn is linked to a statistical partition function.
This principle is operationalised as a topological current of
free energy between different regions of the physical field
dynamics. It translates for each equivalence into a process
governed by a system of integral and/or partial differential
equations (PDES) with local and global initialboundary
conditions (IBC). The scaled physical field dynamics is
concisely classified in terms of local and nonlocal
equivalences, conserved densities or curvatures of the dynamic
scalespace paradigm that in generally are not coinciding with
all initial equivalences. Our dynamic scalespace paradigm
distinguishes itself intrinsically from the standard ones that
are mainly developed for scalar fields. A dynamic scalespace
paradigm is also operationalised for nonscalar fields like
curvature and torsion tensor fields and even more complex
nonlocal and global topological fields supported by the
physical field dynamics. The description of the dynamic scale
spaces are given in terms of again equivalences, and the
paradigms, in terms, of symmetries, curvatures and conservation
laws. The topological characteristics of the paradigm form then
a representation of the logical framework. A simple example of
a dynamic scalespace paradigm is presented for a timesequence
of twodimensional satellite images in the visual spectrum. The
segmentation of the sequence in fore and background dynamics
at various scales is demonstrated together with a detection of
ridges, courses and inflection lines allowing a concise
triangulation of the image. Furthermore, the segmentation
procedure of a dynamic scalespace is made explicit allowing a
true hierarchically description in terms of nested
equivalences. How to unify all the existing scalespace
paradigms using our frame work is illustrated. This unification
comes about by a choice of gauge and renormalisation group, and
setting up a suitable scalespace paradigm that might be user
defined. How to extend and to generalise the existing scale
space paradigm is elaborated on. This is illustrated by
pointing out how to retain a pure topological or covariant
scalespace paradigm from an initially segmented image that
instead of a scalar field also can represent a density field
coinciding with dislocation and disclination fields capturing
the cutting and pasting procedures underlying the image
formation.
 Hartmann, E, "The normalform of a space curve and its application to surface design," VISUAL COMPUTER, vol. 17, pp. 445456, 2001.
Abstract:
The normalform of a space curve is introduced analogously to
the normalform of a plane curve and a surface, i.e. an implicit
representation h (x) = 0 with \ \ delh \ \ = 1. The normalform
function h is (unlike the latter cases) not differentiable at
curve points. Despite of this disadvantage the normalform is a
suitable tool for designing surfaces which can be treated as
common implicit surfaces. Many examples (bisector surfaces,
constant distance sum/product surfaces, metamorphoses, blending
surfaces, smooth approximation surfaces) demonstrate
applications of the normalform to surface design.
 Elmoataz, A, Schupp, S, and Bloyet, D, "Fast and simple discrete approach for active contours for biomedical applications," INTERNATIONAL JOURNAL OF PATTERN RECOGNITION AND ARTIFICIAL INTELLIGENCE, vol. 15, pp. 12011212, 2001.
Abstract:
In this paper, we present a fast and simple discrete approach
for active contours. It is based on discrete contour evolution,
which operates on the boundary of digital shape, by iterative
growth processes on the boundary of the shape. We consider a
curve to be the boundary of a discrete shape, We attach at each
point of the boundary a cost function and deform this shape
according to that cost function. The method presents some
advantages. It is a discrete method, which takes an implicit
representation and uses discrete algorithm with a simple data
structure.
 Udaykumar, HS, Mittal, R, Rampunggoon, P, and Khanna, A, "A sharp interface cartesian grid method for simulating flows with complex moving boundaries," JOURNAL OF COMPUTATIONAL PHYSICS, vol. 174, pp. 345380, 2001.
Abstract:
A Cartesian grid method for computing flows with complex
immersed, moving boundaries is presented. The flow is computed
on a fixed Cartesian mesh and the solid boundaries are allowed
to move freely through the mesh. A mixed EulerianLagrangian
framework is employed, which allows us to treat the immersed
moving boundary as a sharp interface. The incompressible
NavierStokes equations are discretized using a secondorder
accurate finitevolume technique, and a secondorderaccurate
fractionalstep scheme is employed for time advancement. The
fractionalstep method and associated boundary conditions are
formulated in a manner that property accounts for the boundary
motion. A unique problem with sharp interface methods is the
temporal discretization of what are termed "freshly cleared"
cells, i.e., cells that are inside the solid at one time step
and emerge into the fluid at the next time step. A simple and
consistent remedy for this problem is also presented. The
solution of the pressure Poisson equation is usually the most
timeconsuming step in a fractional step scheme and this is
even more so for moving boundary problems where the flow domain
changes constantly. A multigrid method is presented and is
shown to accelerate the convergence significantly even in the
presence of complex immersed boundaries. The methodology is
validated by comparing it with experimental data on two cases:
(1) the flow in a channel with a moving indentation on one wall
and (2) vortex shedding from a cylinder oscillating in a
uniform freestream. Finally, the application of the current
method to a more complicated moving boundary situation is also
demonstrated by computing the flow inside a diaphragmdriven
micropump with moving valves. (C) 2001 Elsevier Science.
 Petersen, M, Ratsch, C, Caflisch, RE, and Zangwill, A, "Level set approach to reversible epitaxial growth  art. no. 061602," PHYSICAL REVIEW E, vol. 6406, pp. 1602+, 2001.
Abstract:
We generalize the level set approach to model epitaxial growth
to include thermal detachment of atoms from island edges. This
means that islands do not always grow and island dissociation
can occur. We make no assumptions about a critical nucleus.
Excellent quantitative agreement is obtained with kinetic Monte
Carlo simulations for island densities and island size
distributions in the submonolayer regime.
 Montagnat, J, Delingette, H, and Ayache, N, "A review of deformable surfaces: topology, geometry and deformation," IMAGE AND VISION COMPUTING, vol. 19, pp. 10231040, 2001.
Abstract:
Deformable models have raised much interest and found various
applications in the fields of computer vision and medical
imaging. They provide an extensible framework to reconstruct
shapes. Deformable surfaces, in particular, are used to
represent 3D objects. They have been used for pattern
recognition [Computer Vision and Image Understanding 69(2)
(1998) 201; IEEE Transactions on Pattern Analysis and Machine
Intelligence 19(10) (1997) 1115], computer animation [ACM
Computer Graphics (SIGGRAPH'87) 21(4) (1987) 205], geometric
modelling [Computer Aided Design (CAD) 24(4) (1992) 178],
simulation [Visual Computer 16(8) (2000) 437], boundary
tracking [ACM Computer Graphics (SIGGRAPH'94) (1994) 185],
image segmentation [Computer Integrated Surgery, Technology and
Clinical Applications (1996) 59; IEEE Transactions on Medical
Imaging 14 (1995) 442; Joint Conference on Computer Vision,
Virtual Reality and Robotics in Medicine (CVRMedMRCAS'97) 1205
(1997) 13; Medical Image Computing and ComputerAssisted
Intervention (MICCAI'99) 1679 (1999) 176; Medical Image
Analysis 1(1) (1996) 19], etc. In this paper we propose a
survey on deformable surfaces. Many surface representations
have been proposed to meet different 3D reconstruction problem
requirements. We classify the main representations proposed in
the literature and we study the influence of the representation
on the model evolution behavior, revealing some similarities
between different approaches. (C) 2001 Elsevier Science B.V.
All rights reserved.
 Leoni, F, "Convergence of an approximation scheme for curvaturedependent motions of sets," SIAM JOURNAL ON NUMERICAL ANALYSIS, vol. 39, pp. 11151131, 2001.
Abstract:
We prove the convergence of an approximation scheme for
computing evolutions of sets having various normal velocities
depending on the curvature. This scheme is an extension of the
socalled BenceMerrimanOsher scheme for computing mean
curvature motions. In order to obtain the convergence, we make
use of the new weak notion of fronts motion introduced by
Barles and Souganidis [Arch. Ration. Mech. Anal., 141 (1998),
pp. 237296], which is equivalent, under the nointerior
condition, to the wellknown levelset evolution.
