Jérôme Darbon

Image restoration with discrete constrained Total Variation—part II: Levelable functions, convex priors and non-convex case

By Jérôme Darbon, Marc Sigelle

2006-03-24

In Journal of Mathematical Imaging and Vision

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 optimization 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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Fast and exact discrete image restoration based on total variation and on its extensions to levelable potentials

By Jérôme Darbon, Marc Sigelle

2006-02-22

In SIAM conference on imaging sciences

Abstract

We investigate the decomposition property of posterior restoration energies on level sets in a discrete Markov Random Field framework. This leads us to the concept of ’levelable’ potentials (which TV is shown to be the paradigm of). We prove that convex levelable posterior energies can be minimized exactly with level-independant binary graph cuts. We extend this scheme to the case of non-convex levelable energies, and present convincing restoration results for images degraded by impulsive noise.

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Composants logiciels et algorithmes de minimisation exacte d’énergies dédidées au traitement d’images

By Jérôme Darbon

2005-10-01

In

Abstract

Dans cette thèse nous étudions la minimisation d’énergies markoviennes rencontrées dans les domaines du traitement des images et de la vision par ordinateur. Nous proposons des algorithmes de minimisation exacte pour différents types d’énergies. Ces algorithmes ont l’intérêt de fournir un minimum global quand bien même l’énergie n’est pas convexe. Enfin, nous mettons en évidence quelques liens entre les champs de Markov binaires et la morphologie mathématique. La version finale de ce manuscrit suit les recommandations des rapporteurs.

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A vectorial self-dual morphological filter based on total variation minimization

By Jérôme Darbon, Sylvain Peyronnet

2005-08-20

In Proceedings of the first international conference on visual computing

Abstract

We present a vectorial self dual morphological filter. Contrary to many methods, our approach does not require the use of an ordering on vectors. It relies on the minimization of the total variation with $L^1$ norm as data fidelity on each channel. We further constraint this minimization in order not to create new values. It is shown that this minimization yields a self-dual and contrast invariant filter. Although the above minimization is not a convex problem, we propose an algorithm which computes a global minimizer. This algorithm relies on minimum cost cut-based optimizations.

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Total variation minimization with $L^1$ data fidelity as a contrast invariant filter

By Jérôme Darbon

2005-04-27

In Proceedings of the 4th international symposium on image and signal processing and analysis (ISPA 2005)

Abstract

This paper sheds new light on minimization of the total variation under the $L^1$-norm as data fidelity term ($L^1+TV$) and its link with mathematical morphology. It is well known that morphological filters enjoy the property of being invariant with respect to any change of contrast. First, we show that minimization of $L^1+TV$ yields a self-dual and contrast invariant filter. Then, we further constrain the minimization process by only optimizing the grey levels of level sets of the image while keeping their boundaries fixed. This new constraint is maintained thanks to the Fast Level Set Transform which yields a complete representation of the image as a tree. We show that this filter can be expressed as a Markov Random Field on this tree. Finally, we present some results which demonstrate that these new filters can be particularly useful as a preprocessing stage before segmentation.

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An efficient algorithm for attribute openings and closings

By Jérôme Darbon, Ceyhun Burak Akgül

2005-04-14

In Proceedings of the 13th european signal processing conference (EUSIPCO)

Abstract

In this paper, we present fast algorithms for area opening and closing on grayscale images. Salembier’s max-tree based algorithm is one of the well known methods to perform area opening. It makes use of a special representation where each node in the tree stands for a flat region and the tree itself is oriented towards the maxima of the grayscale image. Pruning the tree with respect to some attribute, e.g., the area, boils down to attribute opening. Following the same approach, we propose an algorithm for area opening (closing) without building the max-tree (min-tree). Our algorithm exhibit considerable performance compared to the state-of-the art in this domain.

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A fast and exact algorithm for total variation minimization

By Jérôme Darbon, Marc Sigelle

2005-01-18

In Proceedings of the 2nd iberian conference on pattern recognition and image analysis (IbPRIA)

Abstract

This paper deals with the minimization of the total variation under a convex data fidelity term. We propose an algorithm which computes an exact minimizer of this problem. The method relies on the decomposition of an image into its level sets. Using these level sets, we map the problem into optimizations of independent binary Markov Random Fields. Binary solutions are found thanks to graph-cut techniques and we show how to derive a fast algorithm. We also study the special case when the fidelity term is the $L^1$-norm. Finally we provide some experiments.

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A fast and exact algorithm for total variation minimization

By Jérôme Darbon, Marc Sigelle

2005-01-01

In

Abstract

This paper deals with the minimization of the total variation under a convex data fidelity term. We propose an algorithm which computes an exact minimizer of this problem. The method relies on the decomposition of an image into its level sets. Using these level sets, we map the problem into optimizations of independent binary Markov Random Fields. Binary solutions are found thanks to graph-cut techniques and we show how to derive a fast algorithm. We also study the special case when the fidelity term is the $L^1$-norm. Finally we provide some experiments.

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Exact optimization of discrete constrained total variation minimization problems

By Jérôme Darbon, Marc Sigelle

2004-10-01

In

Abstract

This paper deals with the total variation minimization problem when the fidelity is either the $L^2$-norm or the $L^1$-norm. We propose an algorithm which computes the exact solution of these two problems after discretization. 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 associated with each level set. Exact solutions of these binary problems are found thanks to minimum-cut techniques. We prove that these binary solutions are increasing and thus allow to reconstruct the solution of the original problems.

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Exact optimization of discrete constrained total variation minimization problems

By Jérôme Darbon, Marc Sigelle

2004-09-01

In Proceedings of the 10th international workshop on combinatorial image analysis (IWCIA)

Abstract

This paper deals with the total variation minimization problem when the fidelity is either the $L^2$-norm or the $L^1$-norm. We propose an algorithm which computes the exact solution of these two problems after discretization. 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 associated with each level set. Exact solutions of these binary problems are found thanks to minimum-cut techniques. We prove that these binary solutions are increasing and thus allow to reconstruct the solution of the original problems.

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