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

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Abstract

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.

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Authors and Affiliations

  1. GREYC CNRS UMR 6072, Équipe Image, ENSICAEN, 6 Bd du Maréchal Juin, 14050, Caen Cedex, France

    Sébastien Bougleux

  2. GREYC CNRS UMR 6072, Équipe Image, Université de Caen, Bd du Maréchal Juin, 14050, Caen Cedex, France

    Abderrahim Elmoataz

  3. LMIA, Équipe MAGE, Université de Haute-Alsace, 4 rue des Frères Lumière, 68093, Mulhouse Cedex, France

    Mahmoud Melkemi

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  1. Sébastien Bougleux
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  2. Abderrahim Elmoataz
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  3. Mahmoud Melkemi
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Correspondence to Sébastien Bougleux.

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Bougleux, S., Elmoataz, A. & Melkemi, M. Local and Nonlocal Discrete Regularization on Weighted Graphs for Image and Mesh Processing. Int J Comput Vis 84, 220–236 (2009). https://doi.org/10.1007/s11263-008-0159-z

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  • DOI: https://doi.org/10.1007/s11263-008-0159-z

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