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Issue No.02 - February (2010 vol.32)
pp: 231-241
Song-Gang Xu , Tsinghua University, Beijing
Yun-Xiang Zhang , Tsinghua University, Beijing
Jun-Hai Yong , Tsinghua University, Beijing
ABSTRACT
A wide range of applications in computer intelligence and computer graphics require computing geodesics accurately and efficiently. The fast marching method (FMM) is widely used to solve this problem, of which the complexity is O(N\log N), where N is the total number of nodes on the manifold. A fast sweeping method (FSM) is proposed and applied on arbitrary triangular manifolds of which the complexity is reduced to O(N). By traversing the undigraph, four orderings are built to produce two groups of interfering waves, which cover all directions of characteristics. The correctness of this method is proved by analyzing the coverage of characteristics. The convergence and error estimation are also presented.
INDEX TERMS
Geodesics, fast sweeping methods, fast marching methods, Eikonal equation, triangular manifold.
CITATION
Song-Gang Xu, Yun-Xiang Zhang, Jun-Hai Yong, "A Fast Sweeping Method for Computing Geodesics on Triangular Manifolds", IEEE Transactions on Pattern Analysis & Machine Intelligence, vol.32, no. 2, pp. 231-241, February 2010, doi:10.1109/TPAMI.2008.272
REFERENCES
 [1] J. Sethian, Level Set Methods and Fast Marching Methods, second ed. Cambridge Univ. Press, 1999. [2] R. Kimmel and J. Sethian, “Optimal Algorithm for Shape from Shading and Path Planning,” J. Math. Imaging and Vision, vol. 14, no. 3, pp. 237-244, 2001. [3] M. Hassouna, A. Abdel-Hakim, and F. Farag, “Robust Robotic Path Planning Using Level Sets,” Proc. IEEE Int'l Conf. Image Processing, Sept. 2005. [4] R. Kimmel, D. Shaked, N. Kiryati, and A. Bruckstein, “Skeletonization via Distance Maps and Level Sets,” Computer Vision and Image Understanding, vol. 62, no. 3, pp. 382-391, 1995. [5] V. Kirshnamurthy and M. Levoy, “Fitting Smooth Surfaces to Dense Polygon Meshes,” Proc. ACM SIGGRAPH, pp. 313-324, 1996. [6] J. van Trier and W. Symes, “Upwind Finite-Difference Calculation of Traveltimes,” J. Geophysics, vol. 56, no. 6, pp. 812-821, 1991. [7] R. Rawlinson and M. Sambridge, “Wave Front Evolution in Strongly Heterogeneous Layered Media Using the Fast Marching Method,” Geophysical J. Int'l, vol. 156, no. 3, pp. 631-647, 2004. [8] J. Sethian, “A Fast Marching Level Set Method for Monotonically Advancing Fronts,” Proc. Nat'l Academy of Sciences USA, vol. 93, no. 4, pp. 1591-1595, 1996. [9] J. Tsitsiklis, “Efficient Algorithms for Globally Optimal Trajectories,” IEEE Trans. Automatic Control, vol. 40, no. 9, pp. 1528-1538, Sept. 1995. [10] J. Sethian, “Fast Marching Methods and Level Set Methods for Propagating Interfaces,” technical report, Von Karman Inst. Lecture Series on Computational Fluid Mechanics, 1998. [11] R. Kimmel and J. Sethian, “Computing Geodesic Paths on Manifolds,” Proc. Nat'l Academy of Sciences USA, vol. 95, no. 15, pp. 8431-8435, 1998. [12] H. Zhao, “Fast Sweeping Method for Eikonal Equations,” J. Math. and Computing, vol. 74, no. 250, pp. 603-627, 2005. [13] J. Qian, Y. Zhang, and H. Zhao, “Fast Sweeping Methods for Eikonal Equations on Triangular Meshes,” SIAM J. Numerical Analysis, vol. 45, no. 1, pp. 83-107, 2005. [14] J. Sethian and A. Vladimirsky, “Ordered Upwind Methods for Static Hamilton-Jacobi Equations: Theory and Algorithms,” SIAM J. Numerical Analysis, vol. 41, no. 1, pp. 325-363, 2003. [15] S. Kim and D. Folie, “The Group Marching Method: An ${\rm O}(N)$ Level Set Eikonal Solver,” research report, Dept. of Math., Univ. of Kentucky, http://www.ms.uky.edu/~math/MAreport/PDF 00-03.pdf, 2000. [16] J. Sethian, “Fast Marching Methods,” SIAM J. Rev., vol. 41, no. 2, pp. 199-235, 1999. [17] J. Rickett and S. Fomel, “A Second-Order Fast Marching Eikonal Solver,” 2000. [18] A.B.L. Yatziv and G. Sapiro, “O($N$ ) Implementation of the Fast Marching Algorithm,” J. Computational Physics, vol. 212, no. 2, pp.393-399, 2006. [19] A. Faraq and M.S. Hassouna, “Multistencils Fast Marching Methods: A Highly Accurate Solution to the Eikonal Equation on Cartesian Domains,” IEEE Trans. Pattern Analysis and Machine Intelligence, vol. 29, no. 9, pp. 1563-1574, June 2007. [20] M. Bardi and L. Evans, “On Hopf's Formulas for Solutions of Hamilton-Jacobi Equations,” J. Nonlinear Analysis: Theory, Methods and Application, vol. 8, no. 11, pp. 1373-1381, 1984. [21] R. Abgrall, “Numerical Discretization of the First-Order Hamilton-Jacobi Equations on Triangular Meshes,” Comm. Pure and Applied Math., vol. 9, no. 12, pp. 1339-1373, 1998. [22] S. Augoula and R. Abgrall, “High Order Numerical Discretization for Hamilton-Jacobi Equations on Triangular Meshes,” J. Scientific Computing, vol. 15, no. 2, pp. 197-229, 2000. [23] A. Bronstein, M. Bronstein, and R. Kimmel, “Weighted Distance Maps Computation on Parametric Three-Dimensional Manifolds,” J. Computational Physics, vol. 225, no. 1, pp. 771-784, 2007. [24] R. Tsai, H. Zhao, and S. Osher, “Fast Sweeping Algorithms for a Class of Hamilton-Jacobi Equations,” SIAM J. Numerical Analysis, vol. 41, no. 2, pp. 673-694, 2003. [25] C. Kao, S. Osher, and J. Qian, “Lax-Friedrichs Sweeping Scheme for Static Hamilton-Jacobi Equations,” J. Computational Physics, vol. 196, no. 1, pp. 367-391, 2004. [26] C. Kao, S. Osher, and R. Tsai, “Fast Sweeping Methods for Hamilton-Jacobi Equations,” SIAM J. Numerical Analysis, vol. 42, no. 1, pp. 2612-2632, 2005. [27] H.Z.Y. Zhang and J. Qian, “High Order Fast Sweeping Methods for Static Hamilton-Jacobi Equations,” J. Computational Physics, vol. 29, no. 1, pp. 25-56, 2006. [28] J. Qian, Y. Zhang, and H. Zhao, “A Fast Sweeping Method for Static Convex Hamilton-Jacobi Equations,” SIAM J. Scientific Computing, vol. 31, no. 2, pp. 237-271, 2007. [29] H. Zhao, S. Osher, B. Merrimanb, and M. Kang, “Implicit and Non-Parametric Shape Reconstruction from Unorganized Points Using Variational Level Set Method,” Computer Vision and Image Understanding, vol. 80, no. 3, pp. 295-314, 2000. [30] P.E. Danielsson, “Euclidean Distance Mapping,” Computer Graphics and Image Processing, vol. 14, no. 3, pp. 227-248, 1980. [31] A. Bronstein, M. Bronstein, Y. Dvir, R. Kimmel, and O. Weber, “Parallel Algorithms for Approximation of Distance Maps on Parametric Surfaces,” technical report, Dept. of Computer Science, Technion, 2007.