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A review of level set and fast marching methods for image processing

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Modern Methods in Scientific Computing and Applications

Part of the book series: NATO Science Series ((NAII,volume 75))

Abstract

In recent years, a considerable amount of work has been developed concerning a partial differential equations-based approach to image processing. This work has been focussed on the interplay between geometric motion and non-linear evolution equations, and its applicability to problems in image enhancement and denoising, image segmentation, and shape recognition and classification. In this review article, we provide an overview for some of these ideas.

This work was supported in part by the Applied Mathematical Science subprogram of the Office of Energy Research, U.S. Department of Energy, under Contract Number DE-AC03-76SF00098, and the Division of Mathematical Sciences of the National Science Foundation.

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References

  1. D. Adalsteinsson and J. A. Sethian, A fast level set method for propagating interfaces, J. Comp. Phys. 118 (2) (1995), 269–277.

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Adalsteinsson and J.A. Sethian, The fast construction of extension velocities in level set methods, J. Comp. Phys. 148 (1999), 2–22.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Alvarez, P. L. Lions, and M. Morel, Image selective smoothing and edge detection bynonlinear diffusion. II, SIAM J. Numer. Anal. 29 (3) (1992), 845–866.

    Article  MathSciNet  MATH  Google Scholar 

  4. T. J. Barth and J. A. Sethian, Numerical schemes for the Hamilton-Jacobi and level set equations on triangulated domains, J. Comp. Phys. 145 (1) (1998), 1–40.

    Article  MathSciNet  MATH  Google Scholar 

  5. M. R. Brewer, R. Malladi, G. Pankiewicz, B. Conway, and L. Tarassenko, Methods for large-scale segmentation of cloud images, in: Proc. 1997 EUMETSAT Meteorological Satellite Data Users’ Conference, Brussels.

    Google Scholar 

  6. V. Caselles, F. Catte, T. Coll, and F. Dibos, A geometric model for active contours in image processing recovery, Numer. Math. 66 (1) (1993), 1–31.

    Article  MathSciNet  MATH  Google Scholar 

  7. V. Caselles, R. Kimmel, and G. Sapiro, Geodesic active contours, in: Proc. Fifth IEEE Internat. Conf. on Computer Vision, ICCV ’95, Cambridge, MA, 694–699.

    Google Scholar 

  8. D. L. Chopp, Computing minimal surfaces via level set curvature flow, J. Comput. Phys. 106 (1993), 77–91.

    Article  MathSciNet  MATH  Google Scholar 

  9. E. W. Dijkstra, A note on two problems in connection with graphs, Numer. Math. 1 (1959), 269–271.

    Article  MathSciNet  MATH  Google Scholar 

  10. L. C. Evans, Partial Differential Equations, Berkeley Math. Lecture Notes Ser. 3A, 3B, Center for Pure and Applied Mathematics, University of California, Berkeley, CA, 1994.

    Google Scholar 

  11. R. Goldenberg, R. Kimmel, E. Rivlin, and M. Rudzsky, Fast geodesic active contours, IEEE Trans. Image Processing 10 (10) (2001), 1467–1475.

    Article  MathSciNet  Google Scholar 

  12. M. Grayson, The heat equation shrinks embedded plane curves to round points, J. Differential Geom. 26 (1987), 285.

    MathSciNet  MATH  Google Scholar 

  13. M. Grayson, A short note on the evolution of surfaces via mean curvatures, J. Differential Geom. 58 (1989), 555.

    MathSciNet  MATH  Google Scholar 

  14. J. Helmsen, E. G. Puckett, P. Colella, and M. Dorr, Two new methods for simulating photolithography development, in: Proc. SPIE 1996 Internat. Sympos. on Microlithography, SPIE, 2726, June, 1996.

    Google Scholar 

  15. M. Kass, A. Witkin, and D. Terzopoulos, Snakes: active contour models, Internat. J. Computer Vision 1 (1988), 321–331.

    Article  Google Scholar 

  16. R. Kimmel and J. A. Sethian, Fast marching methods on triangulated domains, Proc. Nat. Acad. Sci. 95 (1998), 8341–8435.

    Article  MathSciNet  Google Scholar 

  17. R. Malladi and J. A. Sethian, A unified approach for shape segmentation, representation, and recognition, Center for Pure and Applied Mathematics, Report 614, University of California, Berkeley, 1994.

    Google Scholar 

  18. R. Malladi and J. A. Sethian, Image processing via level set curvature flow, Proc. Nat. Acad. Sci. 92 (15) (1995), 7046–7050.

    Article  MathSciNet  MATH  Google Scholar 

  19. R. Malladi and J. A. Sethian, Level set methods for curvature flow, image enhancement, and shape recovery in medical images, Visualization and Mathematics (H.-C. Hege and K. Polthier, eds.), Springer-Verlag, Berlin-Heidelberg-New York, 1997, 329–345.

    Chapter  Google Scholar 

  20. R. Malladi and J. A. Sethian, A unified approach to noise removal, image enhancement, and shape recovery, IEEE Trans. Image Processing 5 (11) (1996), 1554–1568.

    Article  Google Scholar 

  21. R. Malladi, and J. A. Sethian, An O(N log N) algorithm for shape modeling, Proc. Nat. Acad. Sci. U.S.A. 93 (1996), 9389–9392.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Malladi and J. A. Sethian, Image processing: flows under min/max curvature and mean curvature, Graphical Models and Image Processing 58 (2) (1996), 127–141.

    Article  MathSciNet  Google Scholar 

  23. R. Malladi and J. A. Sethian, Shape modeling in medical imaging with marching methods, Report LBNL-39541, LBNL, University of California, Berkeley, 1996.

    Google Scholar 

  24. R. Malladi, and J. A. Sethian, Level set and fast marching methods in image processing and computer vision, in: Proc. IEEE Internat. Conf, on Image Processing, Lausanne, 1996.

    Google Scholar 

  25. R. Malladi, J. A. Sethian, and B. Vemuri, A topology independent shape modeling scheme, in: Proc. SPIE Conf, on Geometric Methods in Computer Vision II, Vol. 2031, San Diego, CA, 1993, 246–258.

    Google Scholar 

  26. R. Malladi, J. A. Sethian, and B.C. Vemuri, A fast level set based algorithm for topology-independent shape modeling, J. Math. Imaging and Vision 6 (2/3) (1996), 269–290.

    Article  MathSciNet  Google Scholar 

  27. R. Malladi, Geometric Methods in Biomedical Image Analysis (R. Malladi, ed.), Springer Verlag, March 2002.

    Book  Google Scholar 

  28. S. Osher and J. A. Sethian, Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi Formulations, J. Comput. Phys. 79 (1988), 12–49.

    Article  MathSciNet  MATH  Google Scholar 

  29. P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Machine Intelligence 12 (7) (1990), 629–639.

    Article  Google Scholar 

  30. E. Rouy and A. Tourin, A viscosity solutions approach to shape-from-shading, SIAM J. Numer. Anal 29 (3) (1992), 867–884.

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Sapiro, Geometric Partial Differential Equations and Image Analysis, Cambridge University Press, 2001.

    Book  MATH  Google Scholar 

  32. A. Sarti and R. Malladi, A geometric level set model for ultrasound analysis, LBNL-44442, October 1999, to appear in: Geometric Methods in Biomedical Image Analysis (R. Malladi, ed.), Springer Verlag, March 2002.

    Google Scholar 

  33. A. Sarti, R. Malladi, and J. A. Sethian, Subjective surfaces: a method for completing missing boundaries, Proc. Nat. Acad. Sci. U.S.A. 97 (2000), 6258–6263.

    Article  MathSciNet  MATH  Google Scholar 

  34. A. Sarti, R. Malladi, and J. A. Sethian, Subjective surfaces: a geometric model for boundary completion, IEEE Trans. Pattern Anal. Machine Intelligence (2001).

    Google Scholar 

  35. J. A. Sethian, An Analysis of Flame Propagation, Ph.D. Dissertation, Dept. of Mathematics, University of California, Berkeley, CA, 1982.

    Google Scholar 

  36. J. A. Sethian, Curvature and the evolution of fronts, Comm. Math. Phys. 101 (1985), 487–499.

    Article  MathSciNet  MATH  Google Scholar 

  37. J. A. Sethian, Numerical methods for propagating fronts, in: Variational Methods for Free Surface Interfaces, Menlo Park, Calif., 1985 (P. Concus and R. Finn, eds.), Springer-Verlag, New York, 1987, 155–164.

    Chapter  Google Scholar 

  38. J. A. Sethian, A fast marching level set method for monotonically advancing fronts, Proc. Nat. Acad. Sci. U.S.A. 93 (4) (1996), 1591–1595.

    Article  MathSciNet  MATH  Google Scholar 

  39. J. A. Sethian, A Review of the Theory, Algorithms, and Applications of Level Set Methods for Propagating Interfaces, Acta Numerica, Cambridge University Press, 1996.

    Google Scholar 

  40. J. A. Sethian, Fast marching level set methods for three-dimensional photolithography development, in: Proc. SPIE 1996 Internat. Sympos. on Microlithography, Santa Clara, CA.

    Google Scholar 

  41. J. A. Sethian, Fast marching methods, SIAM Rev. 41 (1999), 199–235.

    Article  MathSciNet  MATH  Google Scholar 

  42. J. A. Sethian, Level Set Methods and Fast Marching Methods, Cambridge Monogr. Appl. Comput. Math. 3, Cambridge University Press, 1999.

    MATH  Google Scholar 

  43. J. A. Sethian, Evolution, implementation, and application of level set and fast marching methods for advancing fronts, J. Comp. Phys. 169 (2) (2001), 503–555.

    Article  MathSciNet  MATH  Google Scholar 

  44. J. A. Sethian and M. Popovici, Fast marching methods applied to computation of seismic travel times, Geophysics 64 (2) (1999).

    Google Scholar 

  45. J. A. Sethian and A. Vladimirsky, Fast methods for the eikonal and related Hamilton-Jacobi equations on unstructured meshes, Proc. Nat. Acad. Sci. U.S.A. 97 (11) (2000), 5699–5703.

    Article  MathSciNet  MATH  Google Scholar 

  46. J. A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations, Proc. Nat. Acad. Sci. U.S.A. 98 (20) (2001), 11069–11074.

    Article  MathSciNet  MATH  Google Scholar 

  47. J. A. Sethian and A. Vladimirsky, Ordered upwind methods for static Hamilton-Jacobi equations: theory and algorithms, Center for Pure and Applied Mathematics Technical Report PAM-792, University of California, Berkeley, 2001.

    Google Scholar 

  48. N. Sochen, R. Kimmel, and R. Malladi, A general framework for low level vision, IEEE Trans. Image Proc. 7 (3) (1998), 310–318, 1998.

    Article  MathSciNet  MATH  Google Scholar 

  49. J. N. Tsitsiklis, Efficient algorithms for globally optimal trajectories, IEEE Trans. Automatic Control 40 (1995), 1528–1538.

    Article  MathSciNet  MATH  Google Scholar 

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Sethian, J.A. (2002). A review of level set and fast marching methods for image processing. In: Bourlioux, A., Gander, M.J., Sabidussi, G. (eds) Modern Methods in Scientific Computing and Applications. NATO Science Series, vol 75. Springer, Dordrecht. https://doi.org/10.1007/978-94-010-0510-4_10

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  • DOI: https://doi.org/10.1007/978-94-010-0510-4_10

  • Publisher Name: Springer, Dordrecht

  • Print ISBN: 978-1-4020-0782-8

  • Online ISBN: 978-94-010-0510-4

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