Proximal methods for convex minimization of Phi-divergences.

Application to computer vision.

Oct. 2013 - May 2014: PhD in image processing using convex variational approaches, at Université Paris-Est (in collaboration with the Lebanese University).

Defense date: May 27th, 2014

Supervisors: Jean-Christophe Pesquet and Joumana Farah.

Jury: Michèle Basseville, Amel Benazza, Caroline Chaux, Bilal Chebaro, Alfred Hero and François Malgouyres.

Convex optimization aims at searching for the minimum of a convex function over a convex set. While the theory of convex optimization has been largely explored for about a century, several related developments have stimulated a new interest in the topic. The first one is the emergence of efficient optimization algorithms, such as proximal methods, which allow one to easily solve large-size nonsmooth convex problems in a parallel manner. The second development is the discovery of the fact that convex optimization problems are more ubiquitous in practice than was thought previously.

In this thesis, we address two different problems within the framework of convex optimization. The first one is an application to computer stereo vision, where the goal is to recover the depth information of a scene from a pair of images taken from the left and right positions. The second one is the proposition of new mathematical tools to deal with convex optimization problems involving information measures, where the objective is to minimize the divergence between two statistical objects such as random variables or probability distributions.

We propose a convex approach to address the problem of dense disparity estimation under varying illumination conditions. A convex energy function is derived for jointly estimating the disparity and the illumination variation. The resulting problem is tackled in a set theoretic framework and solved using proximal tools. It is worth emphasizing the ability of this method to process multicomponent images under illumination variation. The conducted experiments indicate that this approach can effectively deal with the local illumination changes and yields better results compared with existing methods.

We then extend the previous approach to the problem of multi-view disparity estimation. Rather than estimating a single depth map, we estimate a sequence of disparity maps, one for each input image. We address this problem by adopting a discrete reformulation that can be efficiently solved through a convex relaxation. This approach offers the advantage of handling both convex and nonconvex similarity measures within the same framework. We have shown that the additional complexity required by the application of our method to the multi-view case is small with respect to the stereo case.

Finally, we have proposed a novel approach to handle a broad class of statistical distances, called Phi-divergences, within the framework of proximal algorithms. In particular, we have developed the expression of the proximity operators of several Phi-divergences, such as Kulback-Leibler, Jeffrey-Kulback, Hellinger, Chi-Square, I-alpha, and Renyi divergences. This allows proximal algorithms to deal with problems involving such divergences, thus overcoming the limitations of current state-of-the-art approaches for similar problems. The proposed approach is validated in two different contexts. The first is an application to image restoration that illustrates how to employ divergences as a regularization term, while the second is an application to image registration that employs divergences as a data fidelity term.