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

In a non supervised Bayesian estimation approach for inverse problems in imaging systems, one tries to estimate jointly the unknown image pixels $f$ and the hyperparameters $\theta$ given the observed data $g$ and a model $M$ linking these quantities. This is, in general, done through the joint posterior law $p(f,\theta|g;M)$. 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 or integration of multivariate probability laws. In any of these cases, we need to do approximations. Laplace approximation and sampling by MCMC are two approximation methods, respectively analytical and numerical, which have been used before with success for this task. In this paper, we explore the possibility of approximating this joint law by a separable one in $f$ and in $\theta$. 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. To illustrate more in detail this approach, we consider the case of image restoration by simple or myopic deconvolution with separable, simple markovian or hidden markovian models.