Approche variationnelle pour le calcul bay\'esien dans les probl\`emes inverses en imagerie

TL;DR

This paper introduces a variational approach for Bayesian inverse problems in imaging, proposing separable approximations of the joint posterior to develop computationally efficient iterative algorithms.

Contribution

It presents a novel variational method that approximates the complex joint posterior by a separable law, enabling efficient algorithms in Bayesian imaging inverse problems.

Findings

Development of iterative algorithms with reduced computational cost

Use of exponential conjugate families for approximation

Comparison of different approximation strategies

Abstract

In a non supervised Bayesian estimation approach for inverse problems in imaging systems, one tries to estimate jointly the unknown image pixels $\fb$ and the hyperparameters $\thetab$. This is, in general, done through the joint posterior law $p(\fb,\thetab|\gb)$. The expression of this joint law is often very complex and its exploration through sampling and computation of the point estimators such as MAP and posterior means need either optimization of non convex criteria or int\'egration of non Gaussian and multi variate probability laws. In any of these cases, we need to do approximations. We had explored before the possibilities of Laplace approximation and sampling by MCMC. In this paper, we explore the possibility of approximating this joint law by a separable one in $\fb$ and in $\thetab$. This gives the possibility of developing iterative algorithms with more reasonable computational cost, in particular, if the approximating laws are choosed in the exponential conjugate families. The main objective of this paper is to give details of different algorithms we obtain with different choices of these families.