A Two Stage Generalized Block Orthogonal Matching Pursuit (TSGBOMP) Algorithm

Recovery of an unknown sparse signal from a few of its projections is the key objective of compressed sensing. Often one comes across signals that are not ordinarily sparse but are sparse blockwise. Existing block sparse recovery algorithms like BOMP make the assumption of uniform block size and known block boundaries, which are, however, not very practical in many applications. This paper addresses this problem and proposes a two step procedure, where the first stage is a coarse block location identification stage while the second stage carries out finer localization of a non-zero cluster within the window selected in the first stage. A detailed convergence analysis of the proposed algorithm is carried out by first defining the so-called pseudoblock-interleaved block RIP of the given generalized block sparse signal and then imposing upper bounds on the corresponding RIC. We also extend the analysis for complex vector as well as matrix entries where it turns out that the extension is non-trivial and requires special care. Furthermore, assuming real Gaussian sensing matrix entries, we find a lower bound on the probability that the derived recovery bounds are satisfied. The lower bound suggests that there are sets of parameters such that the derived bound is satisfied with high probability. Simulation results confirm significantly improved performance of the proposed algorithm as compared to BOMP.