International Journal of Computer Vision

, Volume 84, Issue 2, pp 220–236 | Cite as

Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing

  • Sébastien BougleuxEmail author
  • Abderrahim Elmoataz
  • Mahmoud Melkemi


We propose a discrete regularization framework on weighted graphs of arbitrary topology, which unifies local and nonlocal processing of images, meshes, and more generally discrete data. The approach considers the problem as a variational one, which consists in minimizing a weighted sum of two energy terms: a regularization one that uses the discrete p-Dirichlet form, and an approximation one. The proposed model is parametrized by the degree p of regularity, by the graph structure and by the weight function. The minimization solution leads to a family of simple linear and nonlinear processing methods. In particular, this family includes the exact expression or the discrete version of several neighborhood filters, such as the bilateral and the nonlocal means filter. In the context of images, local and nonlocal regularizations, based on the total variation models, are the continuous analog of the proposed model. Indirectly and naturally, it provides a discrete extension of these regularization methods for any discrete data or functions.


Discrete variational problems on graphs Discrete diffusion processes Smoothing Denoising Simplification 


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Copyright information

© Springer Science+Business Media, LLC 2008

Authors and Affiliations

  • Sébastien Bougleux
    • 1
    Email author
  • Abderrahim Elmoataz
    • 2
  • Mahmoud Melkemi
    • 3
  1. 1.GREYC CNRS UMR 6072, Équipe ImageENSICAENCaen CedexFrance
  2. 2.GREYC CNRS UMR 6072, Équipe ImageUniversité de CaenCaen CedexFrance
  3. 3.LMIA, Équipe MAGEUniversité de Haute-AlsaceMulhouse CedexFrance

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