Bayesian Blind Deconvolution of Images Comparing Jmap, Em and Bva with a Student-t a Priori Model

Blind image deconvolution consists in restoring a blurred and noisy image when the point spread function of the blurring system is not known a priori. This inverse problem is ill-posed and need prior information to obtain a satisfactory solution. Regularization methods, well known, for simple image deconvolution is not enough. Bayesian inference approach with appropriate priors on the image as well as on the PSF has been used successfully, in particular with a Gaussian prior on the PSF and a sparsity enforcing prior on the image. Joint Maximum A posteriori (JMAP), ExpectationMaximization (EM) algorithm for marginalized MAP and Variational Bayesian Approximation (VBA) are the methods which have been considered recently with some advantages for the last one. In this paper, first we review these methods and give some original insights by comparing them, in particular for their respective properties, advantages and drawbacks and their computational complexity. Then we propose to look at these methods in two cases: A simple one which is using Gaussian priors for both the PSF and the image and a more appropriate case which is a Student-t prior for the image to enhance the sharpness (sparsity) of the image while keeping Gaussian prior for the PSF. We take advantages of the Infinite Gaussian Mixture (IGM) property of the Student-t to consider a hierarchical Gaussian-Inverse Gamma prior model for the image. We give detailed comparison of these three methods for this case.