30 Septembre – Thesis defense - Benjamin Harroué

14 h Amphi J.P. Dom - laboratory IMS / building A31 (Talence campus)

Bayesian approach for model selection : image restauration application.

This thesis studies methodology for image restoration problems. We focus on image deconvolution problems where one seeks to analyze a blurred and noisy observation in order to recover a high-resolution image with fine details. This is an ill-conditioned estimation problem that is frequently formulated in the Bayesian statistical framework, which relies on probabilistic models to describe the data observation process as well as the prior knowledge available and where solutions are derived by using Bayesian decision theory. A natural
question in this setting is how to objectively compare different Bayesian models to analyse the observed data: what criteria should we use? which is the best model? what quantities or statistics can we trust? This work studies methodology for objectively comparing alternative Bayesian models to perform image deconvolution, with a focus on automatic Bayesian model selection in the absence of ground truth. Adopting a Bayesian decision-theoretic approach, we select the model with the highest posterior probability given the observed data. These posterior probabilities are determined by the so-called evidence or marginal likelihood of the models, obtained by marginalizing the unknown image and the unknown model hyperparameters. This marginal likelihood is typically intractable, a difficulty that we address by using carefully designed numerical methods. More precisely, we consider the circular Gaussian case, which allows to analytically marginalize the unknown image conditionally to the data and the model hyperparameters, and which simplifies the manipulation of covariances matrices. We compare several numerical methods to calculate model evidences and perform model selection in image deconvolution problems, including the Chib algorithm coupled with a Gibbs sampler, the power posterior algorithm, and the harmonic mean algorithm.

Event localization