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Image Restoration with Discrete Constrained Total Variation Part I: Fast and Exact Optimization

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Abstract

This paper deals with the total variation minimization problem in image restoration for convex data fidelity functionals. We propose a new and fast algorithm which computes an exact solution in the discrete framework. Our method relies on the decomposition of an image into its level sets. It maps the original problems into independent binary Markov Random Field optimization problems at each level. Exact solutions of these binary problems are found thanks to minimum cost cut techniques in graphs. These binary solutions are proved to be monotone increasing with levels and yield thus an exact solution of the discrete original problem. Furthermore we show that minimization of total variation under L 1 data fidelity term yields a self-dual contrast invariant filter. Finally we present some results.

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Author information

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 PhD 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 I: Fast and Exact Optimization. J Math Imaging Vis 26, 261–276 (2006). https://doi.org/10.1007/s10851-006-8803-0

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