Skip to main content

Generalized fast marching method: applications to image segmentation


In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan–Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266–277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm.

This is a preview of subscription content, access via your institution.


  1. Aubert, G., Kornprobst, P.: Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations. Springer Verlag (2002)

  2. Baerentzen, J.A.: On the implementation of fast marching methods for 3D lattices. Technical Report, Informatics and Mathematical Modelling, Technical University of Denmark, DTU (2001)

  3. Barles, G.: Solutions de Viscosité des Équations de Hamilton-Jacobi, vol. 17 of Mathématiques & Applications (Berlin) [Mathematics & Applications]. Springer-Verlag, Paris (1994)

    Google Scholar 

  4. Bresson, X., Esedoglu, S., Vandergheynst, P., Thiran, J., Osher, S.: Fast global minimization of the active contour/snake models. J. Math. Imaging Vis. 28(2), 151–167 (2007)

    Article  MathSciNet  Google Scholar 

  5. Carlini, E., Cristiani, E., Forcadel, N.: A non-monotone fast marching scheme for a Hamilton-jacobi equation modeling dislocation dynamics. ENUMATH 2005, Santiago de Compostela (Spain) (2007)

  6. Caselles, V., Kimmel, R., Sapiro, G.: Geodesic active contours. Int. J. Comput. Vis. 22(1), 61–87 (1993)

    Article  Google Scholar 

  7. Chan, T., Vese, L.: Active contours without edges. IEEE Trans. Image Process. 10(2), 266–277 (2001)

    Article  MATH  Google Scholar 

  8. Chan, T.F., Esedoglu, S., Nikolova, M.: Algorithms for finding global minimizers of image segmentation and denoising models. SIAM J. Appl. Math. 66(5), 1632–1648 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chung, J.T., Vese, L.: Image segmentation using a multilayer level-set approach. EMMCVPR 2005, 439–455 (2005)

  10. Cohen, L.D.: On active contour models and balloons. Comput. Vision Graphics Image Processing: Image Understanding 52(2), 211–218 (1991)

    Google Scholar 

  11. Epstein, C.L., Gage, M.: The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation (1997)

  12. Gout, C., Le Guyader, C.: Segmentation of complex geophysical structures with well data. Comput. Geosci. 10(4), 361–372 (2006)

    Article  MathSciNet  Google Scholar 

  13. Gout, C., Le Guyader, C., Vese, L.: Segmentation under geometrical conditions using geodesic active contours and interpolation using level set methods. Numer. Algoirthms 39(1,3), 155–173 (2005)

    Article  MATH  Google Scholar 

  14. Kass, M., Witkin, A., Terzopoulos, D.: Snakes: active contour models in proceeding. First International Conference on Computer Vision, London, England, IEEE, Piscataway NJ, pp. 259–268 (1987)

  15. Kichenassamy, S., Kumar, A., Olver, P., Tannenbaum, A., Yezzi, A.: Gradient flows and geometric active contour models. In: Proceedings, Fifth International Conference on Computer Vision, pp. 810–815 (1995)

  16. Le Guyader, C., Apprato, D., Gout, C.: The level set methods and image segmentation under interpolation conditions. Numer. Algorithms 39(1–3), 221–235 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  17. Malladi, R., Sethian, J.A.: An \(\mathcal{O}\left( {n\,\text{log}n} \right)\) algorithm for shape modeling. Proc. Natl. Acad. Sci. 93, 9389–9392 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  18. Malladi, R., Sethian, J.A., Vemuri, B.: Evolutionary fronts for topology-independent shape modeling and recovery. In: Proceedings of Third European Conference on Computer Vision, pp. 3–13 (1994)

  19. Merriman, B., Bence, J., Osher, S.: Diffusion generated motion by mean curvature. In: Taylor, J.E. (ed.) Proceedings of the Computational Crystal Growers Workshop, American Mathematical Society, Providence, Rhode Island, pp. 73–83 (1992)

  20. Mumford, D., Shah, J.: Optimal approximation by piecewise smooth functions and associated variational problems. Commun. Pure Appl. Math. 42, 577–685 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  21. Osher, S., Fedkiw, R.: Level Set Methods and Dynamic Implicit Surfaces. Springer Verlag (2003)

  22. Osher, S., Paragios, N.: Geometric Level Set Methods in Imaging, Vision, and Graphics. Springer Verlag (2003)

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

    Article  MathSciNet  MATH  Google Scholar 

  24. Rouy, V., Tourin, F.: A viscosity solutions approach to shape-from-shading. SIAM J. Num. Anal. 29(3), 867–884 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sethian, J.A.: A review of recent numerical algorithms for hypersurfaces moving with curvature dependent flows. J. Differ. Geom. 31, 131–161 (1989)

    MathSciNet  Google Scholar 

  26. Sethian, J.A.: A fast marching level set method for monotonically advancing fronts. Proc. Natl. Acad. Sci. 93, 1591–1595 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  27. Sethian, J.A.: Level Set Methods and Fast Marching Methods: Evolving Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision and Material Science. Cambridge University Press, Londres (1999)

    MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  29. Tsai, Y.R., Cheng, L.T., Osher, S., Zhao, H.K.: Fast sweeping algorithms for a class of Hamilton-Jacobi equations. SIAM J. Numer. Anal. 41(2), 673–694 (2002)

    Article  MathSciNet  Google Scholar 

  30. Tsitsiklis, N.: Efficient algorithms for globally optimal trajectories. IEEE Tran. Automatic. Control 40, 1528–1538 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. Vese, L., Chan, T.: A Multiphase Level Set Framework for Image Segmentation Using the Mumford and Shah Model. Int J. Comput. Vis. 50(3), 271–293 (2002)

    Article  MATH  Google Scholar 

  32. Xu, C., Prince, J.L.: Snakes, shapes, and gradient vector flow. IEEE Trans. Image Process. 7(3), 359–369 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  33. Yatziv, L., Bartesaghi, A., Sapiro, G.: \(\mathcal{O}\left( n \right)\) implementation of the fast marching algorithm. J. Comput. Phys. 212(2), 393–399 (2006)

    Article  MATH  Google Scholar 

  34. Zhao, H.-K.: Fast sweeping method for Eikonal equations. Math. Comput. 74, 603–627 (2004)

    Article  Google Scholar 

  35. Zhao, H.-K., Chan, T., Merriman, B., Osher, S.: A variational level set approach to multiphase motion. J. Comput. Phys. 127, 179–195 (1996)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations


Corresponding author

Correspondence to Carole Le Guyader.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Forcadel, N., Le Guyader, C. & Gout, C. Generalized fast marching method: applications to image segmentation. Numer Algor 48, 189–211 (2008).

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI:


  • Fast marching method
  • Level set methods
  • Chan–Vese model for segmentation

Mathematics Subject Classifications (2000)

  • 65M06
  • 68U10
  • 49Lxx