On MAP and MMSE estimators for the co-sparse analysis model

The sparse synthesis model for signals has become very popular in the last decade, leading to improved performance in many signal processing applications. This model assumes that a signal may be described as a linear combination of few columns (atoms) of a given synthesis matrix (dictionary). The Co-Sparse Analysis model is a recently introduced counterpart, whereby signals are assumed to be orthogonal to many rows of a given analysis dictionary. These rows are called the co-support. The Analysis model has already led to a series of contributions that address the pursuit problem: identifying the co-support of a corrupted signal in order to restore it. While all the existing work adopts a deterministic point of view towards the design of such pursuit algorithms, this paper introduces a Bayesian estimation point of view, starting with a random generative model for the co-sparse analysis signals. This is followed by a derivation of Oracle, Minimum-Mean-Squared-Error (MMSE), and Maximum-A-posteriori-Probability (MAP) based estimators. We present a comparison between the deterministic formulations and these estimators, drawing some connections between the two. We develop practical approximations to the MAP and MMSE estimators, and demonstrate the proposed reconstruction algorithms in several synthetic and real image experiments, showing their potential and applicability.