SOPR for sparse phase retrieval

We propose an algorithm, named SOPR, which is able to locally solve the sparse phase retrieval (SPR) problem given only one intensity point spread function (PSF) image. In abstract, the algorithm is a combination of the Gerchberg-Saxton method with the sparsity constraint but applied to the phase. The latter idea is newly developed in this paper. Convergence results for SOPR are rigorously established by showing that SOPR is equivalent to the cyclic projections method for solving an auxiliary possibly inconsistent feasibility problem involving three (locally) prox-regular sets and hence recent convergence theory for the cyclic projections method can be applied to SOPR. Extensions of SOPR for solving sparse phase retrieval given multiple intensity images can also be obtained naturally. Our numerical experiments for SPR show that our new regularization scheme which is implicitly incorporated in SOPR algorithm has clear advantages on solving SPR.