NDLI: Some Improvements of the Fast Marching Method

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Chopp, David L.
Copyright Year	2001
Abstract	The fast marching method published by Sethian [Proc. Natl. Acad. Sci. USA, 93 (1996), pp. 1591--1595] is an optimally efficient algorithm for solving problems of front evolution 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., Modelling crack growth by level sets in the extended finite element method, Comput. Methods Appl. Mech. Engrg., to appear; Internat. J. Numer. Methods Engrg., 51 (2001), pp. 943--960; N. Sukumar, D. L. Chopp, and B. Moran, Extended finite element method and fast marching method for three-dimensional fatigue crack propagation, J. Comput. Phys., submitted], seismology [J. Sethian and A. Popovici, Geophysics, 64 (1999), pp. 516--523], photolithography [J. Sethian, Fast marching level set methods for three-dimensional 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. 9389--9392]. 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. 199--235] 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 Newton's 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.
Starting Page	230
Ending Page	244
Page Count	15
File Format	PDF
ISSN	10648275
DOI	10.1137/S106482750037617X
e-ISSN	10957197
Issue Number	1
Volume Number	23
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-25
Access Restriction	Subscribed
Subject Keyword	fast marching method Interpolation bicubic interpolation Smoothing, curve fitting level set method reinitialization Finite difference methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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