A Cumulant Expansion Technique for Simultaneous Markov Random Field Image Restoration and Hyperparameter Estimation

Abstract

We investigate hyperparameter estimation for incomplete data in Markov random Field image restoration. Assuming linear dependence of energies with respect to hyperparameters framework, we use a cumulant expansion technique widely known in Statistical Physics and Signal Processing. New insight is given on Maximum Likelihood estimation of hyperparameters of the prior, regularization and contour probability distribution functions (pdfs) for an explicit joint boundary-pixel process aimed to preserve discontinuities. In particular the case where the prior regularization potential is an homogeneous function of pixels is fully analyzed. A Generalized Stochastic Gradient (GSG) algorithm with a fast sampling technique is devised aiming to achieve simultaneous hyperparameter estimation and pixel restoration. Image restoration performances of Posterior Mean performed during GSG convergence and of Simulated Annealing performed after GSG convergence are compared experimentally. Results and perspectives are given.