In this paper, nonlocal mathematical morphology operators are introduced as a natural extension of nonlocal-means in the max-plus algebra. Firstly, we show that nonlocal morphology is a particular case of adaptive morphology. Secondly, we present the necessary properties to have algebraic properties on the associated pair of transformations. Finally, we recommend a sparse version to introduce an efficient algorithm that computes these operators in reasonable computational time.


Complete Lattice Mathematical Morphology Reasonable Computational Time Adaptive Neighbourhood Patch Distance 
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.
    Baudes, A., Coll, B., Morel, J.M.: A review of image denoising algorithms with a new one. Multiscale Modeling and Simulation 4(2), 490–530 (2005)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Angulo, J., Velasco-Forero, S.: Structurally adaptive math. morph. based on nonlinear scale-space decomp. Image Analysis & Stereology 30(2), 111–122 (2011)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Angulo, J.: Morphological bilateral filtering and spatially-variant adaptive structuring functions. In: Soille, P., Pesaresi, M., Ouzounis, G.K. (eds.) ISMM 2011. LNCS, vol. 6671, pp. 212–223. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Angulo, J., Velasco-Forero, S.: Stochastic morphological filtering and bellman-maslov chains. In: Luengo, C., Borgefors, G. (eds.) ISMM 2013. LNCS, vol. 7883, pp. 171–182. Springer, Heidelberg (2013)Google Scholar
  5. 5.
    Beucher, S., Blosseville, J.M., Lenoir, F.: Traffic spatial measurements using video image processing. In: Proc. Intelligent Robots and Computer Vision. SPIE (1988)Google Scholar
  6. 6.
    Bouaynaya, N., Charif-Chefchaouni, M., Schonfeld, D.: Theoretical Foundations of Spatially-Variant Mathematical Morphology Part I: Binary Images. IEEE Trans. Patt. Ana. Mach. Lear. 30(5), 823–836 (2008)CrossRefGoogle Scholar
  7. 7.
    Bresson, X., Chan, T.F.: Non-local unsupervised variational image segmentation models. Tech. rep., UCLA CAM (2008)Google Scholar
  8. 8.
    Buades, A., Coll, B., Morel, J.M.: Image denoising methods. SIAM Review 52(1), 113–147 (2010)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Burgeth, B., Weickert, J.: An explanation for the logarithmic connection between linear and morph. system theory. Int. J. Comp. Vision 64(2-3), 157–169 (2005)CrossRefGoogle Scholar
  10. 10.
    Cuisenaire, O.: Locally adaptable mathematical morphology using distance transformations. Pattern Recognition 39(3), 405–416 (2006)zbMATHCrossRefGoogle Scholar
  11. 11.
    Debayle, J., Pinoli, J.C.: Spatially adaptive morphological image filtering using intrinsic structuring elements. Image Analysis and Stereology 39(3), 145–158 (2005)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Gilboa, G., Osher, S.: Nonlocal Linear Image Regularization and Supervised Segmentation. Multiscale Modeling & Simulation 6(2), 595–630 (2007)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Grazzini, J., Soille, P.: Edge-preserving smoothing using a similarity measure in adaptive geodesic neighbourhoods. Pattern Recognition 42(10), 2306–2316 (2009)zbMATHCrossRefGoogle Scholar
  14. 14.
    Heijmans, H.: Theoretical aspects of gray-level morphology. IEEE Trans. Patt. Ana. Mach. Lear. 13(6), 568–582 (1991)CrossRefGoogle Scholar
  15. 15.
    Katkovnik, V., Foi, A., Egiazarian, K., Astola, J.: From local kernel to nonlocal multiple-model image denoising. Inte. Journal of Comp. Vision 86, 1–32 (2010)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Lerallut, R., Decencière, E., Meyer, F.: Image filtering using morphological amoebas. Image and Vision Computing 4(25), 395–404 (2007)CrossRefGoogle Scholar
  17. 17.
    Lézoray, O., Elmoataz, A.: Nonlocal and multivariate mathematical morphology. In: International Conference on Image Processing. IEEE (2012)Google Scholar
  18. 18.
    Maragos, P.: Slope transforms: theory and application to nonlinear signal processing. IEEE Transactions on Signal Processing 43(4), 864–877 (1995)CrossRefGoogle Scholar
  19. 19.
    Milanfar, P.: Symmetrizing smoothing filters. SIAM Journal on Imaging Sciences (to appear 2013)Google Scholar
  20. 20.
    Morard, V., Decencière, E., Dokládal, P.: Region growing structuring elements and new operators based on their shape. In: Signal and Image Proc. ACTA Press (2011)Google Scholar
  21. 21.
    Roerdink, J.B.T.M.: Adaptivity and group invariance in mathematical morphology. In: ICIP, pp. 2253–2256 (2009)Google Scholar
  22. 22.
    Salembier, P.: Study on nonlocal morph. operators. In: ICIP, pp. 2269–2272 (2009)Google Scholar
  23. 23.
    Serra, J.: Image Analysis and Mathematical Morphology. Academic Press, Inc., Orlando (1982)Google Scholar
  24. 24.
    Serra, J.: Image Analysis and Mathematical Morphology vol. 2: Theoretical Advances. Academic Press (1988)Google Scholar
  25. 25.
    Ta, V., Elmoataz, A., Lezoray, O.: Nonlocal PDEs-based morphology on weighted graphs for image and data proc. IEEE Trans. Im. Proc. 20(6), 1504–1516 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Santiago Velasco-Forero
    • 1
  • Jesús Angulo
    • 2
  1. 1.ITWM - Fraunhofer InstituteKaiserslauternGermany
  2. 2.CMM-Centre de Morphologie Mathématique, Mathématiques et SystèmesMINES ParisTechFrance

Personalised recommendations