Sparsity-based estimation bounds with corrupted measurements

• Published in: Signal processing Vol. 143; pp. 86 - 93
• Main Authors: Boyer, Rémy, Larzabal, Pascal
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 01.02.2018; Elsevier
• Subjects: Applications; Compressed sensing; Corrupted measurements; Cramér–Rao Bound; Engineering Sciences; Gaussian measurement matrix; Signal and Image processing; Statistical priors for support sets of random cardinalities; Statistics; Cramér–Rao Bound; Gaussian measurement matrix; Statistical priors for support sets of random cardinalities; Corrupted measurements; Compressed sensing; corrupted measurements; Cramér-Rao
• ISSN: 0165-1684; 1872-7557
• DOI: 10.1016/j.sigpro.2017.08.004

•Hierarchical Bernoulli-based prior for generation of corrupted measurements and sparse entries of the vector of interest.•Lower bound for random patterns and cardinalities of corrupted measurements and non-zero entries in the vector of interest of random.•Practical importance of the lower bound in the context of sparse-based estimation corrupted by an impulsive (sparse) noise. 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 Iuc containing the indexes of the uncorrupted measures. As Iuc and its cardinality |Iuc|<N are unknown, each sample of vector y follows an i.i.d. Bernoulli prior of probability Puc, 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|||Iuc| follows a hierarchical Binomial distribution on set {1,…,|Iuc|−1}. The derived lower bounds do not belong to the family of “oracle” or “genie-aided” bounds since our a priori knowledge on support Iuc and its cardinality is limited to probability Puc. In this context, very compact and simple expressions of the Expected Cramér–Rao Bound (ECRB) are proposed. Finally, the proposed lower bounds are compared to standard estimation strategies robust to an impulsive (sparse) noise.