Restoration and Segmentation Using Gauss - Markov - Potts Prior Models and Variational Bayesian Computation : Technical Details

We propose a method to restore and to segment simultaneously images degraded by a known point spread function (PSF) and additive white noise. For this purpose, we propose a joint Bayesian estimation framework, where a family of non-homogeneous Gauss-Markov fields with Potts region labels models are chosen to serve as priors for images. Since neither the joint maximum a posteriori estimator nor posterior mean one are tractable, the joint posterior law of the image, its segmentation and all the hyper-parameters, is approximated by a separable probability laws using the Variational Bayes technique. This yields a known probability laws of the posterior with mutually dependent shaping parameter, which aims to enhance the convergence speed of the estimator compared to stochastic sampling based estimator. The main work is description is given in [1], while technical details of the variational calculations are presented in the current paper.