Space adaptive and hierarchical Bayesian variational models for image restoration

The main contribution of this thesis is the proposal of novel space-variant regularization or penalty terms motivated by a strong statistical rational. In light of the connection between the classical variational framework and the Bayesian formulation, we will focus on the design of highly flexible priors characterized by a large number of unknown parameters. The latter will be automatically estimated by setting up a hierarchical modeling framework, i.e. introducing informative or non-informative hyperpriors depending on the information at hand on the parameters. More specifically, in the first part of the thesis we will focus on the restoration of natural images, by introducing highly parametrized distribution to model the local behavior of the gradients in the image. The resulting regularizers hold the potential to adapt to the local smoothness, directionality and sparsity in the data. The estimation of the unknown parameters will be addressed by means of non-informative hyperpriors, namely uniform distributions over the parameter domain, thus leading to the classical Maximum Likelihood approach. In the second part of the thesis, we will address the problem of designing suitable penalty terms for the recovery of sparse signals. The space-variance in the proposed penalties, corresponding to a family of informative hyperpriors, namely generalized gamma hyperpriors, will follow directly from the assumption of the independence of the components in the signal. The study of the properties of the resulting energy functionals will thus lead to the introduction of two hybrid algorithms, aimed at combining the strong sparsity promotion characterizing non-convex penalty terms with the desirable guarantees of convex optimization.