Efficient Algorithms for Image and High Dimensional Data Processing Using Eikonal Equation on Graphs

  • Xavier Desquesnes
  • Abderrahim Elmoataz
  • Olivier Lézoray
  • Vinh-Thong Ta
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6454)


In this paper we propose an adaptation of the static eikonal equation over weighted graphs of arbitrary structure using a framework of discrete operators. Based on this formulation, we provide explicit solutions for the \(\mathcal{L}_1, \mathcal{L}_2\) and \(\mathcal{L}_\infty\) norms. Efficient algorithms to compute the explicit solution of the eikonal equation on graphs are also described. We then present several applications of our methodology for image processing such as superpixels decomposition, region based segmentation or patch-based segmentation using non-local configurations. By working on graphs, our formulation provides an unified approach for the processing of any data that can be represented by a graph such as high-dimensional data.


Weighted Graph Eikonal Equation Arbitrary Graph Discrete Operator Region Base Segmentation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Kimmel, R., Sethian, J.A.: Computing geodesic paths on manifolds. Proc. Natl. Acad. Sci. USA, 8431–8435 (1998)Google Scholar
  2. 2.
    Bronstein, A.M., Bronstein, M.M., Kimmel, R.: Weighted distance maps computation on parametric three-dimensional manifolds. J. Comput. Phys. 225, 771–784 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Rouy, E., Tourin, A.: A viscosity solutions approach to shape-from-shading. SIAM J. Numer. Anal. 29, 867–884 (1992)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Bruss, A.R.: The eikonal equation: some results applicable to computer vision, pp. 69–87 (1989)Google Scholar
  5. 5.
    Siddiqi, K., Bouix, S., Tannenbaum, A., Zucker, S.W.: The hamilton-jacobi skeleton. In: ICCV 1999: Proceedings of the International Conference on Computer Vision, Washington, DC, USA, vol. 2, p. 828. IEEE Computer Society, Los Alamitos (1999)Google Scholar
  6. 6.
    Sethian, J.A.: Level set methods and fast marching methods - evolving interfaces in computational geometry, fluid mechanics, computer vision, and materials science. In: Interfaces in Computational Geometry, Fluid Mechanics, Computer Vision, and Materials Science. Cambridge University Press, Cambridge (1998)Google Scholar
  7. 7.
    Malladi, R., Sethian, J.A.: A unified approach to noise removal, image enhancement, and shape recovery. IEEE Trans. On Image Processing 5, 1554–1568 (1996)CrossRefGoogle Scholar
  8. 8.
    Zhang, Y.T., Shu, C.W.: High-order weno schemes for hamilton-jacobi equations on triangular meshes. SIAM J. Sci. Comput. 24, 1005–1030 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Zhao, H.: A fast sweeping method for eikonal equations. Mathematics of Computation 74, 603–627 (1999)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Tsitsiklis, J.N.: Efficient algorithms for globally optimal trajectories. IEEE Transactions on Automatic Control 40, 1528–1538 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Elmoataz, A., Lézoray, O., Bougleux, S., Ta, V.T.: Unifying local and nonlocal processing with partial difference operators on weighted graphs. In: Proc. of LNLA, vol. 44, pp. 11–26 (2008)Google Scholar
  12. 12.
    Bougleux, S., Elmoataz, A., Melkemi, M.: Local and nonlocal discrete regularization on weighted graphs for image and mesh processing. Int. J. Comput. Vision 84, 220–236 (2009)CrossRefGoogle Scholar
  13. 13.
    Ta, V.T., Elmoataz, A., Lézoray, O.: Adaptation of eikonal equation over weighted graph. In: Tai, X.-C., Mørken, K., Lysaker, M., Lie, K.-A. (eds.) Scale Space and Variational Methods in Computer Vision. LNCS, vol. 5567, pp. 187–199. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  14. 14.
    Boykov, Y., Kolmogorov, V.: Computing geodesics and minimal surfaces via graph cuts. In: ICCV 2003: Proceedings of the Ninth IEEE International Conference on Computer Vision, Washington, DC, USA, p. 26. IEEE Computer Society, Los Alamitos (2003)Google Scholar
  15. 15.
    Jeong, W.K., Whitaker, R.T.: A fast iterative method for eikonal equations. SIAM J. Sci. Comput. 30, 2512–2534 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Ren, X., Malik, J.: Learning a classification model for segmentation. In: ICCV 2003: Proceedings of the Ninth IEEE International Conference on Computer Vision, Washington, DC, USA, p. 10. IEEE Computer Society, Los Alamitos (2003)Google Scholar
  17. 17.
    Levinshtein, A., Stere, A., Kutulakos, K.N., Fleet, D.J., Dickinson, S.J., Siddiqi, K.: Turbopixels: Fast superpixels using geometric flows. IEEE Trans. Pattern Anal. Mach. Intell. 31, 2290–2297 (2009)CrossRefGoogle Scholar
  18. 18.
    Grady, L.: Minimal surfaces extend shortest path segmentation methods to 3D. IEEE Trans. on Pattern Analysis and Machine Intelligence 32, 321–334 (2010)CrossRefGoogle Scholar
  19. 19.
    Boykov, Y., Funka-Lea, G.: Graph cuts and efficient nd image segmentation. International Journal of Computer Vision 70, 109–131 (2006)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2010

Authors and Affiliations

  • Xavier Desquesnes
    • 1
  • Abderrahim Elmoataz
    • 1
  • Olivier Lézoray
    • 1
  • Vinh-Thong Ta
    • 2
  1. 1.GREYC Image TeamUniversité de Caen Basse-Normandie, ENSICAEN, CNRSFrance
  2. 2.LaBRI(Université de Bordeaux CNRS) IPBFrance

Personalised recommendations