ProSparse extension: Prony's based sparse pattern recovery with extended dictionaries
ProSparse is a Prony's based method that solves the sparse representation problem of signals in the union of Fourier and canonical bases. By exploiting the structure of the dictionary, ProSparse is able to reconstruct sparse signals beyond the recovery bound of Basis Pursuit. We generalize this framework for a broader class of dictionaries which are still formed from the union of two bases. The proposed algorithm achieves perfect reconstruction over a lower sparsity level than Basis Pursuit in noiseless cases. In the presence of noise, we extend the ProSparse Denoise algorithm to the generalized dictionaries by considering their intrinsic structure. The original ProSparse can be viewed as a special case of our proposed algorithm. From simulation results, our approach outperforms state-of-the-art algorithms.