Sparsity-based estimation bounds with corrupted measurements

In typical Compressed Sensing operational contexts, the measurement vector y is often partially corrupted. The estimation of a sparse vector acting on the entire support set exhibits very poor estimation performance. It is crucial to estimate set I-uc containing the indexes of the uncorrupted measures. As I-uc and its cardinality |I-uc| < N are unknown, each sample of vector y follows an i.i.d. Bernoulli prior of probability P-uc, leading to a Binomial-distributed cardinality. In this context, we derive and analyze the performance lower bound on the Bayesian Mean Square Error (BMSE) on a |S|-sparse vector where each random entry is the product of a continuous variable and a Bernoulli variable of probability P and |S|||I-uc| follows a hierarchical Binomial distribution on set (1,...,|I-uc| - 1}. The derived lower bounds do not belong to the family of "oracle" or "genie-aided" bounds since our a priori knowledge on support I-uc and its cardinality is limited to probability Pm. In this context, very compact and simple expressions of the Expected Cramer-Rao Bound (ECRB) are proposed. Finally, the proposed lower bounds are compared to standard estimation strategies robust to an impulsive (sparse) noise. (C) 2017 Elsevier B.V. All rights reserved.