1 Advancing Interfaces : Level Set and Fast Marching Methods

In this article, we review past work on Level Set Methods, introduced by Osher and Sethian in [20], and Fast Marching Methods, introduced by Sethian in [25], for tracking propagating interfaces in two and three space dimensions. Both sets of techniques are based on a partial differential equations view of interface motion, and rely on the use of the theory of viscosity solutions, upwind finite difference schemes for hyperbolic conservation laws, and the theory of curve and surface evolution developed in [23]. Both sets of techniques require an adaptive methodology to obtain computational efficiency. We briefly review some of these methods, and show some examples of some applications.

[1]  G. M.,et al.  Partial Differential Equations I , 2023, Applied Mathematical Sciences.

[2]  Edsger W. Dijkstra,et al.  A note on two problems in connexion with graphs , 1959, Numerische Mathematik.

[3]  Paul Garabedian,et al.  Lectures on partial differential equations , 1964 .

[4]  J. Sethian.  AN ANALYSIS OF FLAME PROPAGATION , 1982 .

[5]  P. Lions,et al.  Viscosity solutions of Hamilton-Jacobi equations , 1983 .

[6]  J. Sethian.  Curvature and the evolution of fronts , 1985 .

[7]  J. Sethian,et al.  Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations , 1988 .

[8]  James A. Sethian,et al.  Numerical Methods for Propagating Fronts , 1987 .

[9]  S. Osher,et al.  Uniformly high order accurate essentially non-oscillatory schemes, 111 , 1987 .

[10]  P. Concus,et al.  Variational Methods for Free Surface Interfaces , 1987 .

[11]  J. Sethian,et al.  Crystal growth and dendritic solidification , 1992 .

[12]  E. Rouy,et al.  A viscosity solutions approach to shape-from-shading , 1992 .

[13]  Baba C. Vemuri,et al.  Topology-independent shape modeling scheme , 1993, Optics & Photonics.

[14]  James A. Sethian,et al.  Flow under Curvature: Singularity Formation, Minimal Surfaces, and Geodesics , 1993, Exp. Math..

[15]  D. Chopp.  Computing Minimal Surfaces via Level Set Curvature Flow , 1993 .

[16]  V. Caselles,et al.  A geometric model for active contours in image processing , 1993 .

[17]  J. Tsitsiklis,et al.  Efficient algorithms for globally optimal trajectories , 1994, Proceedings of 1994 33rd IEEE Conference on Decision and Control.

[18]  J. Sethian,et al.  A Fast Level Set Method for Propagating Interfaces , 1995 .

[19]  J. Sethian,et al.  A level set approach to a unified model for etching, deposition, and lithography II: three-dimensional simulations , 1995 .

[20]  T. Chan,et al.  A Variational Level Set Approach to Multiphase Motion , 1996 .

[21]  James A. Sethian,et al.  Fast-marching level-set methods for three-dimensional photolithography development , 1996, Advanced Lithography.

[22]  James A. Sethian,et al.  Theory, algorithms, and applications of level set methods for propagating interfaces , 1996, Acta Numerica.

[23]  J. Sethian,et al.  An O(N log N) algorithm for shape modeling. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[24]  S. Osher,et al.  A Level Set Formulation of Eulerian Interface Capturing Methods for Incompressible Fluid Flows , 1996 .

[25]  J A Sethian,et al.  A fast marching level set method for monotonically advancing fronts. , 1996, Proceedings of the National Academy of Sciences of the United States of America.

[26]  Elbridge Gerry Puckett,et al.  Two new methods for simulating photolithography development in 3D , 1997 .

[27]  S. Osher,et al.  A Simple Level Set Method for Solving Stefan Problems , 1997, Journal of Computational Physics.

[28]  J. Sethian,et al.  Numerical Schemes for the Hamilton-Jacobi and Level Set Equations on Triangulated Domains , 1998 .

[29]  J. Sethian,et al.  traveltime computation using the fast marching method , 1999 .

[30]  James A. Sethian,et al.  The Fast Construction of Extension Velocities in Level Set Methods , 1999 .

[31]  J. A. Sethian,et al.  Fast Marching Methods , 1999, SIAM Rev..

[32]  Alex M. Andrew,et al.  Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science (2nd edition) , 2000 .