Highly coherent sensing matrices arise  in discretization of continuum problems such as radar  and medical imaging when the grid spacing is below the Rayleigh threshold as well as in using highly coherent, redundant dictionaries as sparsifying operators.

We propose new algorithms (BLOOMP, BP-BLOT)  based on techniques of band exclusion and local optimization to enhance existing compressed sensing algorithms (OMP, BP) and deal with such coherent sensing matrices.

 BLOOMP has provably performance guarantee of reconstructing sparse, widely separated objects independent  of the redundancy and have a sparsity constraint and computational
cost similar to OMP's.

We demonstrate the effectiveness of our schemes in various compressed sensing  problems with highly coherent, redundant sensing matrices.