Partial Difference Equations over Graphs: Morphological Processing of Arbitrary Discrete Data

  • Vinh-Thong Ta
  • Abderrahim Elmoataz
  • Olivier Lézoray
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5304)


Mathematical Morphology (MM) offers a wide range of operators to address various image processing problems. These processing can be defined in terms of algebraic set or as partial differential equations (PDEs). In this paper, a novel approach is formalized as a framework of partial difference equations (PdEs) on weighted graphs. We introduce and analyze morphological operators in local and nonlocal configurations. Our framework recovers classical local algebraic and PDEs-based morphological methods in image processing context; generalizes them for nonlocal configurations and extends them to the treatment of any arbitrary discrete data that can be represented by a graph. It leads to considering a new field of application of MM processing: the case of high-dimensional multivariate unorganized data.


Weight Function Texture Image Weighted Graph Mathematical Morphology Graph Topology 
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.


  1. 1.
    Soille, P.: Morphological Image Analysis, Principles and Applications, 2nd edn. Springer, Heidelberg (2002)zbMATHGoogle Scholar
  2. 2.
    Brockett, R., Maragos, P.: Evolution equations for continuous-scale morphology. In: IEEE International Conference on Acoustics, Speech, and Signal Processing, vol. 3, pp. 125–128 (1992)Google Scholar
  3. 3.
    Sapiro, G., Kimmel, R., Shaked, D., Kimia, B., Bruckstein, A.: Implementing continuous-scale morphology by curve evolution. Pattern Recognition 26(9), 1363–1372 (1993)CrossRefGoogle Scholar
  4. 4.
    Maragos, P.: PDEs for morphology scale-spaces and eikonal applications. In: Bovik, A. (ed.) The Image and Video Processing Handbook, 2nd edn., pp. 587–612. Elsevier Academic Press, Amsterdam (2004)Google Scholar
  5. 5.
    Breuß, M., Burgeth, B., Weickert, J.: Anisotropic continuous-scale morphology. In: Martí, J., Benedí, J.M., Mendonça, A.M., Serrat, J. (eds.) IbPRIA 2007. LNCS, vol. 4478, pp. 512–522. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  6. 6.
    Buades, A., Coll, B., Morel, J.: Nonlocal image and movie denoising. International Journal of Computer Vision 76(2), 123–139 (2008)CrossRefGoogle Scholar
  7. 7.
    Gilboa, G., Osher, S.: Nonlocal operators with applications to image processing. Report 07-23, UCLA, Los Angeles (July 2007)Google Scholar
  8. 8.
    Peyré, G.: Manifold models for signals and images. Technical report, CEREMADE, Université Paris Dauphine (2007)Google Scholar
  9. 9.
    Elmoataz, A., Lézoray, O., Bougleux, S.: Nonlocal discrete regularization an weighted graphs: a framework for image and manifolds processing. IEEE Transactions on Image Processing 17(7), 1047–1060 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Burgeth, B., Bruhn, A., Didas, S., Weickert, J., Welk, M.: Morphology for matrix data: Ordering versus pde-based approach. Image and Vision Computing 25(4), 496–511 (2007)CrossRefGoogle Scholar
  11. 11.
    Postaire, J., Zhang, R., Lecocq-Botte, C.: Cluster analysis by binary morphology. IEEE Trans. Patt. Anal. Machine Intell. 15(2), 170–180 (1993)CrossRefGoogle Scholar
  12. 12.
    Heijmans, H., Nacken, P., Toet, A., Vincent, L.: Graph morphology. Journal of Visual Communication and Image Representation 3(1), 24–38 (1992)CrossRefGoogle Scholar
  13. 13.
    Meyer, F., Lerallut, R.: Morphological operators for flooding, leveling and filtering images using grpahs. In: Escolano, F., Vento, M. (eds.) GbRPR. LNCS, vol. 4538, pp. 158–167. Springer, Heidelberg (2007)CrossRefGoogle Scholar
  14. 14.
    Diestel, R.: Graph Theory. Graduate Texts in Mathematics, vol. 173. Springer, Heidelberg (2005)zbMATHGoogle Scholar
  15. 15.
    Chan, T., Osher, S., Shen, J.: The digital TV filter and nonlinear denoising. IEEE Transactions on Image Processing 10(2), 231–241 (2001)CrossRefzbMATHGoogle Scholar
  16. 16.
    Trémeau, A., Colantoni, P.: Regions adjacency graph applied to color image segmentation. IEEE Transactions on Image Processing 9(4), 735–744 (2000)CrossRefGoogle Scholar
  17. 17.
    Von Luxburg, U.: A tutorial on spectral clustering. Statistics and Computing 17(4), 395–416 (2007)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Osher, S., Sethian, J.: Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations. Journal of Computational Physics 79, 12–49 (1988)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Vinh-Thong Ta
    • 1
  • Abderrahim Elmoataz
    • 1
  • Olivier Lézoray
    • 1
  1. 1.University of Caen Basse-Normandie, GREYC CNRS UMR 6072, Image TeamCaen CedexFrance

Personalised recommendations