Guaranteed sparse signal recovery with highly coherent sensing matrices

Compressive sensing is a methodology for the reconstruction of sparse or compressible signals using far fewer samples than required by the Nyquist criterion. However, many of the results in compressive sensing concern random sampling matrices such as Gaussian and Bernoulli matrices. In common physically feasible signal acquisition and reconstruction scenarios such as superresolution of images, the sensing matrix has a non-random structure with highly correlated columns. Here we present a compressive sensing recovery algorithm, called Partial Inversion (PartInv), that shows better performance than existing greedy methods for random matrices, and is especially suitable for matrices that have subsets of highly correlated columns. We provide theoretical justification as well as empirical comparisons.