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Accurate Tracking Of Monotonically Advancing Fronts

  • M. Sabry Hassouna
  • Aly A. Farag
Part of the Topics in Biomedical Engineering. International Book Series book series (ITBE)

A wide range of computer vision applications—such as distance field computation, shape from shading, shape representation, skeletonization, and optimal path planning — require an accurate solution of a particular Hamilton-Jacobi (HJ) equation, known as the eikonal equation. Although the fast marching method (FMM) is the most stable and consistent method among existing techniques for solving such an equation, it suffers from a large numerical error along diagonal directions, and its computational complexity is not optimal. In this chapter, we propose an improved version of the FMM that is both highly accurate and computationally efficient for Cartesian domains. The new method is called the multistencils fast marching (MSFM) method, which computes a solution at each grid point by solving the eikonal equation along several stencils and then picks the solution that satisfies the fast marching causality relationship. The stencils are centered at each grid point x and cover its entire nearest neighbors. In a 2D space, two stencils cover the eight neighbors of x, while in a 3D space six stencils cover its 26 neighbors. For those stencils that do not coincide with the natural coordinate system, the eikonal equation is derived using directional derivatives and then solved using a higher-order finite-difference scheme.

Keywords

Source Point Directional Derivative Neighbor Point Eikonal Equation Unit Speed 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. 1.
    Cerveny V. 1985. Ray synthetic seismograms for complex two- and three-dimensional structures. J Geophys 58:2-26.Google Scholar
  2. 2.
    Vidale J. 1990. Finite-difference calculation of travel times in three dimensions. J Geophys 55:521-526.CrossRefGoogle Scholar
  3. 3.
    van Trier J, Symes W. 1991. Upwind finite-difference calculation of travel times. J Geophys 56:812-821.CrossRefGoogle Scholar
  4. 4.
    Podvin P, Lecomte I. 1991. Finite-difference computation of travel times in very contrasted velocity models: a massively parallel approach and its associated tools. Geophys J Int 105:271-284.CrossRefGoogle Scholar
  5. 5.
    Adalsteinsson D, Sethian J. 1995. A fast level set method for propagating interfaces. J Comput Phys 118:269-277.MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Kim S. 1999. Eno-dno-ps: a stable, second-order accuracy eikonal solver. Soc Explor Geophys 69:1747-1750.Google Scholar
  7. 7.
    Sethian J. 1999. Level Sets methods and fast marching methods, 2nd ed. Cambridge: Cambridge UP.Google Scholar
  8. 8.
    Kimmel R, Sethian J. 1998. Fast marching methods on triangulated domains. PNAS 95(11):8341-8435.MathSciNetGoogle Scholar
  9. 9.
    Sethian JA, Vladimirsky A. 2000. Fast methods for the eikonal and related Hamilton-Jacobi equations on unstructured meshes. PNAS 97(11):5699-5703.MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Kim S. 2001. An o(n) level set method for eikonal equations. SIAM J Sci Comput 22(6):2178-2193.MATHCrossRefGoogle Scholar
  11. 11.
    Yatziv L, BartesaghiA, Sapiro G. 2006.A fast o(n) implementation of the fast marching algorithm. J Comput Phys 212:393-399.MATHGoogle Scholar
  12. 12.
    Danielsson P-EE, Lin Q. 2003. A modified fast marching method. In Proceedings of the 13th Scandinavian conference, SCIA 2003. Lecture notes in computer science, Vol. 2749, pp. 1154-1161. Berlin: Springer.Google Scholar
  13. 13.
    Tsitsiklis J. 1995. Efficient algorithms for globally optimal trajectories. IEEE Trans Auto Control 40(9):1528-1538.MATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Godunov S. 1959. Finite-difference method for numerical computation of discontinuous solutions of the equations of fluid dynamics. Mat Sbornik 47:271-306. [Trans. from the Russian by I Bohachevsky.]Google Scholar
  15. 15.
    Dijkstra EW. 1959. A note on two problems in connexion with graphs. Numer Mat 1:269-271.MATHCrossRefMathSciNetGoogle Scholar
  16. 16.

Copyright information

© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  • M. Sabry Hassouna
    • 1
  • Aly A. Farag
    • 1
  1. 1.Computer Vision and Image Processing LaboratoryUniversity of LouisvilleLouisvilleUSA

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