Bayesian inference with hierarchical prior models for inverse problems in imaging systems

Bayesian approach is nowadays commonly used for inverse problems. Simple prior laws (Gaussian, Generalized Gaussian, Gauss-Markov and more general Markovian priors) are common in modeling and in their use in Bayesian inference methods. But, we need still more appropriate prior models which can account for non station-narities in signals and for the presence of the contours and homogeneous regions in images. Recently, we proposed a family of hierarchical prior models, called Gauss-Markov-Potts, which seems to be more appropriate for many applications in Imaging systems such as X ray Computed Tomography (CT) or Microwave imaging in Non Destructive Testing (NDT). In this tutorial paper, first some backgrounds on the Bayesian inference and the tools for assignment of priors and doing efficiently the Bayesian computation is presented. Then, more specifically hiearachical models and particularly the Gauss-Markov-Potts family of prior models are presented. Finally, their real applications in image restoration, in different practical Computed Tomography (CT) or other imaging systems are presented.