Epigraphical splitting of convex constraints. Application to image recovery, supervised classification, and image forgery detection.

In this thesis, we present a convex optimization approach to address three problems arising in multicomponent image recovery, supervised classification, and image forgery detection. The common thread among these problems is the presence of nonlinear convex constraints difficult to handle with state-of-the-art methods. Therefore, we present a novel splitting technique to simplify the management of such constraints. Relying on this approach, we also propose some contributions that are tailored to the aforementioned applications. The first part of the thesis presents the epigraphical splitting of nonlinear convex constraints. The principle is to decompose the sublevel set of a block-separable function into a collection of epigraphs. So doing, we reduce the complexity of optimization algorithms when the above constraint involves the sum of absolute values, distance functions to a convex set, Euclidean norms, infinity norms, or max functions. We demonstrate through numerical simulations that the proposed method can efficiently handle constraints based on functions commonly used in image restoration or supervised learning, such as nonlocal total variation, Kullback-Leibler divergence, and logistic regression. The second part of the thesis presents three contributions grounded on the epigraphical splitting. The first one is a novel regularization for multicomponent images that extends the nonlocal total variation by taking advantage of the structure tensor. The second one is a learning algorithm for efficiently and exactly training a multiclass support vector machine with sparse regularization. The third one is a variational approach to detect image forgeries by using the photo response non-uniformity (a deterministic pattern noise that uniquely identifies each individual camera). We carried out numerical experiments for each application in order to illustrate the efficiency and the performance of the proposed approaches with respect to state-of-the-art solutions.