Bayesian inference tools for inverse problems

In this paper, first the basics of Bayesian inference with a parametric model of the data is presented. Then, the needed extensions are given when dealing with inverse problems and in particular the linear models such as Deconvolution or image reconstruction in Computed Tomography (CT). The main point to discuss then is the prior modeling of signals and images. A classification of these priors is presented, first in separable and Markovien models and then in simple or hierarchical with hidden variables. For practical applications, we need also to consider the estimation of the hyper parameters. Finally, we see that we have to infer simultaneously on the unknowns, the hidden variables and the hyper parameters. Very often, the expression of this joint posterior law is too complex to be handled directly. Indeed, rarely we can obtain analytical solutions to any point estimators such the Maximum A posteriori (MAP) or Posterior Mean (PM). Three main tools are then can be used: Laplace approximation (LAP), Marko...