Image recovery from Fourier domain measurements via classification using Bayesian approach and total variation regularization

In this paper we propose a Potts–Markov prior and total variation regularization associated with Bayesian approach to simultaneously reconstruct and segment piecewise homogeneous images in Fourier synthesis inverse problem. When the observed data do not fill uniformly the Fourier domain which is the case in many applications in tomographic imaging, or when the phase of the signal is lacking as in optical interferometry the results obtained by deterministic methods are not satisfactory. Such inverse problem is known to be nonlinear and ill-posed. It then needs to be regularized by introducing prior information. The particular a priori information on which we rely is the fact that the image is composed of a different regions finite known number. Such an appropriate modeling of the image gives the possibility of compensating the lack of information in the data thus giving satisfactory results. We define the appropriate Potts–Markov model to define parameters of label regions for such images and total variation to be used in a Bayesian estimation framework.