Séminaire des doctorants Orléans
Emile Pierret: Stochastic Superresolution and Inverse Problems: From Gaussian Conditional Sampling to Diffusion Models .Emile Pierret (IDP-Orléans)
Monday 16 June 2025 14:30 - IDP-Orléans - salle de séminaire
Résumé :
This thesis investigates the stochastic resolution of inverse problems in imaging, through Gaussian conditional sampling and diffusion models. It primarily focuses on super-resolution, which aims to enhance the resolution of an image from a low-resolution version, and deblurring, which seeks to restore the sharpness of a blurred image. In contrast to classical deterministic approaches, we adopt a probabilistic perspective, with the goal of modeling and sampling the set of plausible images consistent with a degraded observation. The objective is to study this paradigm within a mathematically rigorous and well-founded framework. To this end, we examine Gaussian image distributions, including Gaussian microtextures associated with the Asymptotic Discrete Spot Noise (ADSN) model. We formulate the problem in a Bayesian framework, emphasizing the challenges related to conditional sampling in the Gaussian setting. We then discuss recent advances in generative modeling, particularly diffusion probabilistic models (DDPMs), which involve reversing a progressive noising process using a learned model. More specifically, we first study, via a kriging-based reasoning, the exact sampling of inverse problems for Gaussian microtextures, with a particular focus on super-resolution. This leads to the construction of an efficient and exact sampler within this restricted setting. Next, we investigate diffusion models applied to Gaussian distributions in a generative context, in order to assess their ability to provide a meaningful prior distribution over images. By leveraging the reformulation of diffusion models as stochastic differential equations (SDEs), we derive the governing equations in the linear and Gaussian case, and analyze their properties. This allows us to establish explicit results on sampling errors, notably in terms of Wasserstein distances, and to gain a better understanding of the nature of the approximations induced by the models used. Finally, we study algorithms for solving inverse problems based on diffusion models in the context of Gaussian microtexture deblurring. This enables us to highlight undesired behaviors of a sampler intended to approximate the theoretical posterior distribution.
Keywords: Gaussian sampling, Diffusion models, Inverse problems, Generative models
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