ED Mathématiques et Informatique
Regularizing and structuring deep projective priors for imaging inverse problems
by Ali JOUNDI (IMB - Institut de Mathématiques de Bordeaux)
The defense will take place at 10h00 - Salle de conférences Institut de Mathématiques de Bordeaux UMR 5251 Université de Bordeaux 351, Cours de la Libération F-33405 TALENCE
in front of the jury composed of
- Yann TRAONMILIN - Chargé de recherche - IMB, CNRS, Bordeaux - CoDirecteur de these
- Jalal FADILI - Professor - GREYC , ENSICAEN - Rapporteur
- Nelly PUSTELNIK - Directrice de recherche - Laboratoire de Physique,CNRS, ENS de Lyon - Rapporteur
- Audrey GIREMUS - Professeure - IMS, Université de Bordeaux - Examinateur
- Christophe KERVAZO - Maître de conférences - LTCI, Telecom Paris - Examinateur
- Alasdair NEWSON - Maître de conférences - ISIR, Sorbonne université - CoDirecteur de these
Many crucial tasks in image processing and computer vision are formulated as inverse problems. Therefore, it is of great importance to design efficient and robust algorithms to solve these problems. In this work, we focus on generalized projected gradient descent (GPGD) algorithms where generalized projections are implemented with learned neural networks, as they provide state-of-the-art results for imaging inverse problems. Indeed, neural networks enable projections onto unknown low-dimensional sets that model complex data, such as images. We call these projections deep projective priors. The choice of these to use in a GPGD framework is critical for the recovery of the original image. We propose in this work to constrain and regularize two deep projective priors: autoencoders and denoisers, to solve imaging inverse problems. First, we consider an improved version of the atomic autoencoder, a neural network architecture that initially decomposes an image as a sum of low-dimensional atoms. Specifically, we define a new atomic model, the max-sparsity model, to better represent images that are not well modeled by the classical summation model. We study experimentally how the input images are decomposed, and implement these atomic autoencoders in a GPGD algorithm for solving a super-resolution inverse problem, illustrating the benefit of a more accurate prior. Next, we focus on the influence of the deep projective prior on the convergence of the GPGD. It was shown that under a restricted isometry assumption and a restricted Lipschitz condition on the used projections, this algorithm converges with a linear rate, and near-optimal convergence for the orthogonal one (within the class of GPGD methods) in the classical case of sparse recovery. However, for deep projective priors trained with classical mean squared error losses, there is little guarantee that the hypotheses for linear convergence are satisfied. Thus, we propose a stochastic orthogonal regularization of the training loss for deep projective priors. This regularization is motivated by our theoretical results: a sufficiently good approximation of the orthogonal projection guarantees linear stable recovery with performance close to orthogonal PGD. We demonstrate that our regularization yields projections that improve convergence speed and robustness of GPGD in challenging inverse problem settings, in accordance with our theoretical findings. Lastly, we extend the previous convergence results to include robustness to model and projection errors. We leverage these results to explore ways to better control stability and robustness constants. To reduce recovery errors due to measurement noise, we consider generalized back-projection strategies to adapt GPGD to structured noise, such as sparse outliers. To improve the stability of GPGD, we propose a normalized idempotent regularization for the learning of deep projective priors. These contributions highlight the different trade-offs between identifiability and stability.