MMSE Approximation for Denoising Using Several Sparse Representations

Cleaning of noise from signals is a classical and long-studied problem in signal processing. For signals that admit sparse representations over a known dictionary, MAP-based denoising seeks the sparsest representation that synthesizes a signal close to the corrupted one. While this task is NP-hard, it can usually be approximated quite well by a greedy method, such as the Orthogonal Matching Pursuit (OMP). In this work we consider a Minimum-Mean-Squared-Error (MMSE) denoising algorithm, superior to the above MAP approach. We show that this estimator amounts to a weighted averaging of many sparse representation solutions. As its deployment is also NP-hard, we propose a practical randomized version of the OMP algorithm for generating such a group of representations. Simulations of the proposed algorithm are provided and its superiority over plain OMP is demonstrated.