On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary

Abstract

Among the many ways to model signals, a recent approach that draws considerable attention is sparse representation modeling. In this model, the signal is assumed to be generated as a random linear combination of a few atoms from a pre-specified dictionary. In this work we analyze two Bayesian denoising algorithms -- the Maximum-Aposteriori Probability (MAP) and the Minimum-Mean-Squared-Error (MMSE) estimators, under the assumption that the dictionary is unitary. It is well known that both these estimators lead to a scalar shrinkage on the transformed coefficients, albeit with a different response curve. In this work we start by deriving closed-form expressions for these shrinkage curves and then analyze their performance. Upper bounds on the MAP and the MMSE estimation errors are derived. We tie these to the error obtained by a so-called oracle estimator, where the support is given, establishing a worst-case gain-factor between the MAP/MMSE estimation errors and the oracle's performance. These denoising algorithms are demonstrated on synthetic signals and on true data (images).

29 pages, 10 figures

References in corpus (2)

• On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
• RIP-Based Near-Oracle Performance Guarantees for Subspace-Pursuit, CoSaMP, and Iterative Hard-Thresholding

Cited by in corpus (4)

• Exploiting Statistical Dependencies in Sparse Representations for Signal Recovery
• On MMSE and MAP Denoising Under Sparse Representation Modeling Over a Unitary Dictionary
• MMSE Approximation For Sparse Coding Algorithms Using Stochastic Resonance
• Power-Constrained Sparse Gaussian Linear Dimensionality Reduction over Noisy Channels