Closed-Form MMSE Estimation for Signal Denoising Under Sparse Representation Modeling Over a Unitary Dictionary

Matan Protter, Irad Yavneh, Michael Elad
<span title="">2010</span> <i title="Institute of Electrical and Electronics Engineers (IEEE)"> <a target="_blank" rel="noopener" href="https://fatcat.wiki/container/gkn2pu46ozb4tmkxczacnmtvkq" style="color: black;">IEEE Transactions on Signal Processing</a> </i> &nbsp;
This paper deals with the Bayesian signal denoising problem, assuming a prior based on a sparse representation modeling over 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 a simple 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 to a simple formula, in the form of a plain
more &raquo; ... expression for evaluating the contribution of every atom in the solution. An extension of the model to real-world signals is also offered, considering heterosedastic non-zero entries in the representation, and allowing varying probabilities for the chosen atoms and the overall cardinality of the sparse representation. The MAP and MMSE estimators are re-developed for this extended model, again resulting in closed-form simple algorithms. Finally, the superiority of the MMSE estimator is demonstrated both on synthetically generated signals and on real-world signals (image patches).
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