Beyond coherence: Recovering structured time–frequency representations

We consider the problem of recovering a structured sparse representation of a signal in an overcomplete time–frequency dictionary with a particular structure. For infinite dictionaries that are the union of a nice wavelet basis and a Wilson basis, sufficient conditions are given for the basis pursuit and (orthogonal) matching pursuit algorithms to recover a structured representation of an admissible signal. The sufficient conditions take into account the structure of the wavelet/Wilson dictionary and allow very large (even infinite) support sets to be recovered even though the dictionary is highly coherent.