Fast Marching Methods

Fast Marching Methods are numerical schemes for computing solutions to the nonlinear Eikonal equation and related static Hamilton--Jacobi equations. Based on entropy-satisfying 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, shape-from-shading, 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.

  • [1]  David Adalsteinsson and , James Sethian, A fast level set method for propagating interfaces, J. Comput. Phys., 118 (1995), 269–277 10.1006/jcph.1995.1098 96a:65154 CrossrefISIGoogle Scholar

  • [2]  Timothy Barth and , James Sethian, Numerical schemes for the Hamilton‐Jacobi and level set equations on triangulated domains, J. Comput. Phys., 145 (1998), 1–40 10.1006/jcph.1998.6007 99d:65277 CrossrefISIGoogle Scholar

  • [3]  David Chopp, Computing minimal surfaces via level set curvature flow, J. Comput. Phys., 106 (1993), 77–91 10.1006/jcph.1993.1092 94f:53007 CrossrefISIGoogle Scholar

  • [4]  Colella, P., and Puckett, E. G., Modern Numerical Methods for Fluid Flow, Lecture Notes, Department of Mechanical Engineering, University of California, Berkeley, CA, 1994. Google Scholar

  • [5]  Udi Manber, Introduction to algorithms, Addison‐Wesley Publishing Company Advanced Book Program, 1989xiv+478, A creative approach 93a:68002 Google Scholar

  • [6]  M. Crandall, L. Evans and , P.‐L. Lions, Some properties of viscosity solutions of Hamilton‐Jacobi equations, Trans. Amer. Math. Soc., 282 (1984), 487–502 86a:35031 CrossrefISIGoogle Scholar

  • [7]  M. G. Crandall, H. Ishii and , and P‐L. Lions, User’s Guide to Viscosity Solutions of Second Order Partial Differential Equations, Bull. AMS, 27/1, pp. 1–67, 1992. CrossrefISIGoogle Scholar

  • [8]  Panagiotis Souganidis, Approximation schemes for viscosity solutions of Hamilton‐Jacobi equations, J. Differential Equations, 59 (1985), 1–43 86k:35028 CrossrefISIGoogle Scholar

  • [9]  Adil Bagirov, Numerical methods for minimizing quasidifferentiable functions: a survey and comparison, Nonconvex Optim. Appl., Vol. 43, Kluwer Acad. Publ., Dordrecht, 2000, 33–71 1766792 Google Scholar

  • [10]  Dijkstra, E. W., A Note on Two Problems in Connection with Graphs, Numerische Mathematic, 1:269–271, 1959. Google Scholar

  • [11]  Garabedian, P., Partial Differential Equations, Wiley, New York, 1964. Google Scholar

  • [12]  Kimmel, R., and Sethian, J. A., Fast Marching Methods for Robotic Navigation with Constraints, Center for Pure and Applied Mathematics Report, Univ. of California, Berkeley, May 1996, submitted for publication, Int. Journal Robotics Research, 1998. Google Scholar

  • [13]  R. Kimmel and , and J. A. Sethian, Fast Marching Methods on Triangulated Domains, Proc. Nat. Acad. Sci., 95, pp. 8341–8435, 1998. CrossrefISIGoogle Scholar

  • [14]  S. LaValle, Robot motion planning: a game‐theoretic foundation, Algorithmica, 26 (2000), 430–465, Algorithmic foundations of robotics 1752366 CrossrefISIGoogle Scholar

  • [15]  Randall LeVeque, Numerical methods for conservation laws, Lectures in Mathematics ETH Zürich, Birkhäuser Verlag, 1992x+214 92m:65106 CrossrefGoogle Scholar

  • [16]  Malladi, R., Sethian, J. A., and Vemuri, B. C., Evolutionary Fronts for Topology‐independent Shape Modeling and Recovery, in Proceedings of Third European Conference on Computer Vision, Stockholm, Sweden, Lecture Notes in Computer Science, 800:3–13, 1994. Google Scholar

  • [17]  Malladi, R., and Sethian, J. A., An O(NlogN) Algorithm for Shape Modeling, Proc. Nat. Acad. Sci., Vol. 93, 1996. Google Scholar

  • [18]  Stanley Osher and , James Sethian, Fronts propagating with curvature‐dependent speed: algorithms based on Hamilton‐Jacobi formulations, J. Comput. Phys., 79 (1988), 12–49 89h:80012 CrossrefISIGoogle Scholar

  • [19]  M. Popovici, Prestack migration by split‐step DSR, Geophysics, 61, 5, pp. 1412‐16, 1996. gpy GPYSA7 0016-8033 Geophysics CrossrefISIGoogle Scholar

  • [20]  Elisabeth Rouy and , Agnès Tourin, A viscosity solutions approach to shape‐from‐shading, SIAM J. Numer. Anal., 29 (1992), 867–884 93d:65019 LinkISIGoogle Scholar

  • [21]  W. A. Schneider, Robust and efficient upwind finite‐difference traveltime calculations in three dimensions, Geophysics, 60, pp. 1108–1117, 1995. gpy GPYSA7 0016-8033 Geophysics CrossrefISIGoogle Scholar

  • [22]  Jordi Castro, A specialized interior‐point algorithm for multicommodity network flows, SIAM J. Optim., 10 (2000), 852–877 1774776 LinkISIGoogle Scholar

  • [23]  Sethian, J. A., An Analysis of Flame Propagation, Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, CA, 1982. Google Scholar

  • [24]  J. Sethian, Curvature and the evolution of fronts, Comm. Math. Phys., 101 (1985), 487–499 87d:58032 CrossrefISIGoogle Scholar

  • [25]  James Sethian, Numerical methods for propagating fronts, Springer, New York, 1987, 155–164 872900 Google Scholar

  • [26]  J. Sethian, Numerical algorithms for propagating interfaces: Hamilton‐Jacobi equations and conservation laws, J. Differential Geom., 31 (1990), 131–161 91d:65206 CrossrefISIGoogle Scholar

  • [27]  J. A. Sethian, A Marching Level Set Method for Monotonically Advancing Fronts, Proc. Nat. Acad. Sci., 93(4): 1591–1595, 1996. CrossrefISIGoogle Scholar

  • [28]  James Sethian, Theory, algorithms, and applications of level set methods for propagating interfaces, Acta Numer., Vol. 5, Cambridge Univ. Press, Cambridge, 1996, 309–395 99d:65397 Google Scholar

  • [29]  Sethian, J. A., Fast Marching Level Set Methods for Three‐Dimensional Photolithography Development, Proceedings, SPIE 1996 International Symposium on Microlithography, Santa Clara, California, March, 1996. Google Scholar

  • [30]  Sethian, J. A., Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision and Material Science, Cambridge University Press, 1996. Google Scholar

  • [31]  Sethian, J. A., Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Materials Sciences, Cambridge University Press, 1998. Google Scholar

  • [32]  Sethian, J. A., and Popovici, A. M., Three dimensional traveltimes computation using the Fast Marching Method, Geophysics, March‐April, 1999. Google Scholar

  • [33]  Sethian, J.A,. and Vladimirsky, A., Extensions to Triangulated Fast Marching Methods, to be submitted for publication, 1998. Google Scholar

  • [34]  , Sixteenth International Conference on Numerical Methods in Fluid Dynamics, Proceedings of the conference held in Arcachon, July 6–10, 1998, Lecture Notes in Physics, Vol. 515, Springer‐Verlag, 1998, 0–0, xvi+568 2000h:76001 Google Scholar

  • [35]  3DGeo Corporation, Computing and Imaging using Fast Marching Methods, 3DGeo Corporation, Internal Report, June, 1998. Google Scholar

  • [36]  Technology Modeling Associates, Three‐Dimensional Photolithography Simulation with Depict 4.0”, Technology Modeling Associates, Internal Documentation, January 1996. Google Scholar

  • [37]  J. van Trier and , and W. W. Symes, Upwind Finite‐difference Calculations of Traveltimes, Geophysics, 56, 6, pp. 812–821, 1991. gpy GPYSA7 0016-8033 Geophysics CrossrefISIGoogle Scholar

  • [38]  J. Vidale, Finite‐Difference Calculation of Travel Times, Bull. of Seism. Soc. of Amer., 78, 6, pp. 2062–2076, 1988. bss BSSAAP 0037-1106 Bull. Seismol. Soc. Am. ISIGoogle Scholar

  • [39]  J. Vidale, Finite‐difference calculation of traveltimes in three dimensions, Geophysics, 55, 521–526, 1990. gpy GPYSA7 0016-8033 Geophysics CrossrefISIGoogle Scholar