Sparsity-Based Estimation Bounds With Corrupted Measurements

International audienceIn 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 car-dinality. 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| |Iuc| 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 P uc. 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