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Image Restoration with Discrete Constrained Total Variation Part II: Levelable Functions, Convex Priors and Non-Convex Cases

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Abstract

In Part II of this paper we extend the results obtained in Part I for total variation minimization in image restoration towards the following directions: first we investigate the decomposability property of energies on levels, which leads us to introduce the concept of levelable regularization functions (which TV is the paradigm of). We show that convex levelable posterior energies can be minimized exactly using the level-independant cut optimization scheme seen in Part I. Next we extend this graph cut scheme to the case of non-convex levelable energies.We present convincing restoration results for images corrupted with impulsive noise. We also provide a minimum-cost based algorithm which computes a global minimizer for Markov Random Field with convex priors. Last we show that non-levelable models with convex local conditional posterior energies such as the class of generalized Gaussian models can be exactly minimized with a generalized coupled Simulated Annealing.

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KAP:

Kluwer Academic Publishers

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Electronically submitted article

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Authors and Affiliations

  1. EPITA Research and Development Laboratory (LRDE), 14-16 rue Voltaire, F-94276, Le Kremlin-Bicêtre, France

    Jérôme Darbon

  2. École Nationale Supérieure des Télécommunications (ENST), 46 rue Barrault, F-75013, Paris, France

    Jérôme Darbon

  3. École Nationale Supérieure des Télécommunications (ENST) / LTCI CNRS UMR 5141, 46 rue Barrault, F-75013, Paris, France

    Marc Sigelle

Authors
  1. Jérôme Darbon
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  2. Marc Sigelle
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Correspondence to Jérôme Darbon.

Additional information

Jérôme Darbon was born in Chenôve, France in 1978. From 1998 to 2001, he studied computer science at Ecole Pour l’Informatique et les Techniques Avancées (EPITA), France. He received the M.Sc. degree in applied mathematics from E.N.S. de Cachan, France, in 2001. In 2005, he received the Ph.D. degree from Ecole Nationale des Télécommunications (ENST), Paris, France. He is currently a postdoc at the Department of Mathematics, University of California, Los Angeles, hosted by Prof. T.F. Chan. His main research interests include fast algorithms for exact energy minimization and mathematical morphology.

Marc Sigelle was born in Paris on 18th March 1954. He received an engeneer diploma from Ecole Polytechnique Paris 1975 and from Ecole Nationale Supérieure des Télécommunications Paris in 1977. In 1993 he received a Ph.D. from Ecole Nationale Supérieure des Télécommunications. He worked first at Centre National d’Etudes des Télécommunications in Physics and Computer algorithms. Since 1989 he has been working at Ecole Nationale Supérieure des Télécommunications in image and more recently in speech processing. His main subjects of interests are restoration and segmentation of signals and images with Markov Random Fields (MRF’s), hyperparameter estimation methods and relationships with Statistical Physics. His interests concerned first reconstruction in angiographic medical imaging and processing of remote sensed satellital and synthetic aperture radar images, then speech and character recognition using MRF’s and bayesian networks. His most recent interests concern a MRF approach to image restoration with Total Variation and its extensions.

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Darbon, J., Sigelle, M. Image Restoration with Discrete Constrained Total Variation Part II: Levelable Functions, Convex Priors and Non-Convex Cases. J Math Imaging Vis 26, 277–291 (2006). https://doi.org/10.1007/s10851-006-0644-3

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