Greedy Algorithm for the Analysis Transform Domain

Many signal and image processing applications have benefited remarkably from the theory of sparse representations. In the classical synthesis model, the signal is assumed to have a sparse representation under a given known dictionary. The algorithms developed for this framework mainly operate in the representation domain. Recently, a new model has been introduced, the cosparse analysis one, in which the signal is treated directly. The (co)sparsity is expressed by the number of zeros in the coefficients of the signal after applying a given transform, the analysis dictionary, on it. Recently it has been shown that using l1-minimization one can stably recover a cosparse signal from a small set of random linear measurements if the operator is a frame. Another effort has provided guarantees for dictionaries that have a near optimal projection procedure using greedy-like algorithms. However, no claims have been given for frames. In this work we propose a greedy-like technique that operates in the transform domain and provide guarantees that close the gap between greedy and relaxation techniques showing that the proposed method achieves a stable recovery for frames as operators. In addition, we treat the case where the noise is random and provide denoising guarantees for it, closing a gap between the synthesis and analysis frameworks.