Closed-form mmse estimator for denoising signals under sparse representation modelling

This paper deals with the signal denoising problem, assuming a prior based on a sparse representation with respect to a unitary dictionary. It is well known that the maximum a posteriori probability (MAP) estimator in such a case has a closed-form solution based on shrinkage. The focus in this paper is on the better performing and less familiar minimum-mean-squared-error (MMSE) estimator. We show that this estimator also leads also to a simple closed-form formula, in the form of a plain recursive expression for evaluating the contribution of every atom in the solution. We demonstrate this formula, and compare it to the MAP and the random-OMP method devised for approximating the MMSE result.