This thesis aims to study the incorporation of deep neural networks into the mathematical analysis of images, specifically within variational methods used in constrained optimization algorithms. These algorithms are commonly employed to address challenges in image restoration, formulated as ill-posed inverse problems. The goal of integrating neural networks into these methods, rather than relying solely on neural networks for image processing tasks, is to preserve the stability, comprehensibility, and interpretability of algorithms grounded in well-defined mathematical frameworks. On the other hand, the interest in using neural networks is to more finely model the relevant information contained in images, such as through learning a prior probability distribution.

The thesis proposes to examine both theoretical aspects, such as formalizing latent spaces present in generative models, and practical aspects, focusing on the development of innovative algorithms. Among the applications to be particularly studied are medical imaging, including spectral tomography, as well as the problem of fusion of hyperspectral and panchromatic images encountered in satellite imaging.

The thesis will be conducted within the Laboratoire de Mathématiques de Bretagne Atlantique (LMBA UMR CNRS 6205) at the Université Bretagne Sud (Vannes campus, France), with co-direction from the Université de Bretagne Occidentale (Brest, France). Collaborations with external laboratories such as the Laboratoire de Traitement de l'Information Médicale (LaTIM UMR INSERM 1101, Brest, France) and the research group Mathematical Processing and Analysis of Images (TAMI, University of the Balearic Islands, Spain) are planned in connection with the applications.