Audrey Repetti

We contribute to an uncertainty quantification problem in imaging that evaluates a hypothesis test questioning the existence of local “artifacts” appearing in the maximum a posteriori (MAP) estimate (obtained from standard numerical tools). Such a method, called Bayesian uncertainty quantification by optimization (BUQO), was introduced a few years ago as an efficient and scalable alternative to sampling methods when per-pixel error bars are not needed. BUQO formulates a hypothesis test for probing the existence of local structures in the MAP estimate as a minimization problem that can be solved efficiently with standard optimization algorithms. In this context, BUQO requires a “mathematical” definition of the “local artifact.” This definition can be interpreted as an inpainting of the structure. However, only simple handcrafted techniques have been proposed so far due to the complexity of the problem. In this work, we propose a data-driven alternative to BUQO where the inpainting procedure in the algorithm is performed using a convolutional inpainting neural network (NN). This results in a plug-and-play algorithm, based on the primal-dual Condat–Vũ iterations, where the inpainting procedure is performed with a NN. The proposed approach is assessed on two image reconstruction problems inspired by medicine. We specifically perform simulations on two Fourier undersampling problems (discrete and non-uniform) encountered in magnetic resonance imaging, as well as a computed tomography problem using the Radon measurement operator.

Markov Chain Monte Carlo (MCMC) algorithms are standard approaches to solve imaging inverse problems and quantify estimation uncertainties, a key requirement in absence of ground-truth data. To improve estimation quality, Plug-and- Play MCMC algorithms, such as PnP-ULA, have been recently developed to accommodate priors encoded by a denoising neural network. Designing scalable samplers for high-dimensional imaging inverse problems remains a challenge: drawing and storing high-dimensional samples can be prohibitive, especially for highresolution images. To address this issue, this work proposes a distributed sampler based on approximate data augmentation and PnP-ULA to solve very large problems. The proposed sampler uses lightweight denoising convolutional neural network, to efficiently exploit multiple GPUs on a Single Program Multiple Data architecture. Reconstruction performance and scalability are evaluated on several imaging problems. Communication and computation overheads due to the denoiser are carefully discussed. The proposed distributed approach noticeably combines three very precious qualities: it is scalable, enables uncertainty quantification, for a reconstruction performance comparable to other PnP methods.

Une approche efficace pour la résolution de problèmes inverses consiste à définir le signal (ou l'image) recherché(e) par minimisation d'un critère pénalisé. Ce dernier s'écrit souvent sous la forme d'une somme de fonctions composées avec des opérateurs linéaires. En pratique, ces fonctions peuvent n'être ni convexes ni différentiables. De plus, les problèmes auxquels on doit faire face sont souvent de grande dimension. L'objectif de cette thèse est de concevoir de nouvelles méthodes pour résoudre de tels problèmes de minimisation, tout en accordant une attention particulière aux coûts de calculs ainsi qu'aux résultats théoriques de convergence. Une première idée pour construire des algorithmes rapides d'optimisation est d'employer une stratégie de préconditionnement, la métrique sous-jacente étant adaptée à chaque itération. Nous appliquons cette technique à l'algorithme explicite-implicite et proposons une méthode, fondée sur le principe de majoration-minimisation, afin de choisir automatiquement les matrices de préconditionnement. L'analyse de la convergence de cet algorithme repose sur l'inégalité de Kurdyka-L ojasiewicz. Une seconde stratégie consiste à découper les données traitées en différents blocs de dimension réduite. Cette approche nous permet de contrôler à la fois le nombre d'opérations s'effectuant à chaque itération de l'algorithme, ainsi que les besoins en mémoire, lors de son implémentation. Nous proposons ainsi des méthodes alternées par bloc dans les contextes de l'optimisation non convexe et convexe. Dans le cadre non convexe, une version alternée par bloc de l'algorithme explicite-implicite préconditionné est proposée. Les blocs sont alors mis à jour suivant une règle déterministe acyclique. Lorsque des hypothèses supplémentaires de convexité peuvent être faites, nous obtenons divers algorithmes proximaux primaux-duaux alternés, permettant l'usage d'une règle aléatoire arbitraire de balayage des blocs. L'analyse théorique de ces algorithmes stochastiques d'optimisation convexe se base sur la théorie des opérateurs monotones. Un élément clé permettant de résoudre des problèmes d'optimisation de grande dimension réside dans la possibilité de mettre en oeuvre en parallèle certaines étapes de calculs. Cette parallélisation est possible pour les algorithmes proximaux primaux-duaux alternés par bloc que nous proposons: les variables primales, ainsi que celles duales, peuvent être mises à jour en parallèle, de manière tout à fait flexible. A partir de ces résultats, nous déduisons de nouvelles méthodes distribuées, où les calculs sont répartis sur différents agents communiquant entre eux suivant une topologie d'hypergraphe. Finalement, nos contributions méthodologiques sont validées sur différentes applications en traitement du signal et des images. Nous nous intéressons dans un premier temps à divers problèmes d'optimisation faisant intervenir des critères non convexes, en particulier en restauration d'images lorsque l'image originale est dégradée par un bruit gaussien dépendant du signal, en démélange spectral, en reconstruction de phase en tomographie, et en déconvolution aveugle pour la reconstruction de signaux sismiques parcimonieux. Puis, dans un second temps, nous abordons des problèmes convexes intervenant dans la reconstruction de maillages 3D et dans l'optimisation de requêtes pour la gestion de bases de données

In this work we study the behavior of the forward-backward (FB) algorithm when the proximity operator is replaced by a subiterative procedure to approximate a Gaussian denoiser, in a Plug-and-Play (PnP) fashion. Specifically, we consider both analysis and synthesis Gaussian denoisers within a dictionary framework, obtained by unrolling dual-FB iterations or FB iterations, respectively. We analyze the associated minimization problems as well as the asymptotic behavior of the resulting FB-PnP iterations. In particular, we show that the synthesis Gaussian denoising problem can be viewed as a proximity operator. For each case, analysis, and synthesis, we show that the FB-PnP algorithms solve the same problem whether we use only one or an infinite number of subiteration to solve the denoising problem at each iteration. To this aim, we show that each “one subiteration” strategy within the FB-PnP can be interpreted as a primal-dual algorithm when a warm-restart strategy is used. We further present similar results when using a Moreau–Yosida smoothing of the global problem, for an arbitrary number of subiterations. Finally, we provide numerical simulations to illustrate our theoretical results. In particular we first consider a toy compressive sensing example, as well as an image restoration problem in a deep dictionary framework.

Markov Chain Monte Carlo (MCMC) algorithms are standard approaches to solve imaging inverse problems and quantify estimation uncertainties, a key requirement in absence of ground-truth data. To improve estimation quality, Plug-and-Play MCMC algorithms, such as PnP-ULA, have been recently developed to accommodate priors encoded by a denoising neural network. Designing scalable samplers for high-dimensional imaging inverse problems remains a challenge: drawing and storing high-dimensional samples can be prohibitive, especially for high-resolution images. To address this issue, this work proposes a distributed sampler based on approximate data augmentation and PnP-ULA to solve very large problems. The proposed sampler uses lightweight denoising convolutional neural network, to efficiently exploit multiple GPUs on a Single Program Multiple Data architecture. Reconstruction performance and scalability are evaluated on several imaging problems. Communication and computation overheads due to the denoiser are carefully discussed. The proposed distributed approach noticeably combines three very precious qualities: it is scalable, enables uncertainty quantification, for a reconstruction performance comparable to other PnP methods.

This article introduces a novel approach to learning monotone neural networks (NNs) through a newly defined penalization loss. The proposed method is particularly effective in solving classes of variational problems, specifically monotone inclusion problems, commonly encountered in image processing tasks. The forward-backward-forward (FBF) algorithm is employed to address these problems, offering a solution even when the Lipschitz constant of the NN is unknown. Notably, the FBF algorithm provides convergence guarantees under the condition that the learned operator is monotone. Building on plug-and-play methodologies, our objective is to apply these newly learned operators to solving nonlinear inverse problems. To achieve this, we initially formulate the problem as a variational inclusion problem. Subsequently, we train a monotone NN to approximate an operator that may not inherently be monotone. Leveraging the FBF algorithm, we then show simulation examples where the nonlinear inverse problem is successfully solved.

In this paper, we present a new edge detection model based on proximal unfolded neural networks. The architecture relies on unfolding proximal Blake–Zisserman iterations, leading to a composition of two blocks: a smoothing block and an edge detection block. We show through simulations that the proposed approach efficiently eliminates irrelevant details while retaining key edges and significantly improves performance with respect to state-of-the-art strategies. Additionally, our architecture is significantly lighter than recent learning models designed for edge detection in terms of number of learnable parameters and inference time.

Asymptotic analysis of generic stochastic algorithms often relies on descent conditions. In a convex setting, some technical shortcuts can be considered to establish asymptotic convergence guarantees of the associated scheme. However, in a non-convex setting, obtaining similar guarantees is usually more complicated, and relies on the use of the Kurdyka-Łojasiewicz (KŁ) property. While this tool has become popular in the field of deterministic optimization, it is much less widespread in the stochastic context and the few works making use of it are essentially based on trajectory-by-trajectory approaches. In this paper, we propose a new framework for using the KŁ property in a non-convex stochastic setting based on conditioning theory. We show that this framework allows for deeper asymptotic investigations on stochastic schemes verifying some generic descent conditions. We further show that our methodology can be used to prove convergence of generic stochastic gradient descent (SGD) schemes, and unifies conditions investigated in multiple articles of the literature.

A common approach to solve inverse imaging problems relies on finding a maximum a posteriori (MAP) estimate of the original unknown image, by solving a minimization problem. In this context, iterative proximal algorithms are widely used, enabling to handle non-smooth functions and linear operators. Recently, these algorithms have been paired with deep learning strategies, to further improve the estimate quality. In particular, proximal neural networks (PNNs) have been introduced, obtained by unrolling a proximal algorithm as for finding a MAP estimate, but over a fixed number of iterations, with learned linear operators and parameters. As PNNs are based on optimization theory, they are very flexible, and can be adapted to any image restoration task, as soon as a proximal algorithm can solve it. They further have much lighter architectures than traditional networks. In this article we propose a unified framework to build PNNs for the Gaussian denoising task, based on both the dual-FB and the primal-dual Chambolle-Pock algorithms. We further show that accelerated inertial versions of these algorithms enable skip connections in the associated NN layers. We propose different learning strategies for our PNN framework, and investigate their robustness (Lipschitz property) and denoising efficiency. Finally, we assess the robustness of our PNNs when plugged in a forward-backward algorithm for an image deblurring problem.