Statistiques non locales dans les images : modélisation, estimation et échantillonnage

In this thesis we study two non-localstatistics in images from a probabilistic point of view: spatialredundancy and convolutional neural network features. Moreprecisely, we are interested in the estimation and detection ofspatial redundancy in naturalimages. We also aim at sampling images with neural network constraints.We start by giving a definition of spatial redundancy in naturalimages. This definition relies on two concepts: a Gestalt analysisof the notion of similarity in images, and a hypothesis testingframework (the a contrario method). We propose an algorithm toidentify this redundancy in natural images. Using this methodologywe can detect similar patches in images and, with this information,we propose new algorithms for diverse image processing tasks(denoising, periodicity analysis).The rest of this thesis deals with sampling images with non-localconstraints. The image models we consider are obtained via themaximum entropy principle. The target distribution is then obtainedby minimizing an energy functional. We use tools from stochasticoptimization to tackle thisproblem.More precisely, we propose and analyze a new algorithm: the SOUL(Stochastic Optimization with Unadjusted Langevin) algorithm. Inthis methodology, the gradient is estimated using Monte Carlo MarkovChains methods. In the case of the SOUL algorithm we use an unadjustedLangevin algorithm. The efficiency of the SOUL algorithm is relatedto the ergodic properties of the underlying Markov chains. Thereforewe are interested in the convergence properties of certain class offunctional autoregressive models. We characterize precisely thedependency of the convergence rates of these models with respect totheir parameters (dimension, smoothness,convexity).Finally, we apply the SOUL algorithm to the problem ofexamplar-based texture synthesis with a maximum entropy approach. Wedraw links between our model and other entropy maximizationprocedures (macrocanonical models, microcanonical models). Usingconvolutional neural network constraints we obtain state-of-the artvisual results.