Phase retrieval with sparse phase constraint

For the first time, this paper investigates the phase retrieval problem with the assumption that the phase (of the complex signal) is sparse in contrast to the sparsity assumption on the signal itself as considered in the literature of sparse signal processing. The intended application of this new problem model, which will be conducted in a follow-up paper, is to practical phase retrieval problems where the aberration phase is sparse with respect to the orthogonal basis of Zernike polynomials. Such a problem is called sparse phase retrieval (SPR) problem in this paper. When the amplitude modulation at the exit pupil is uniform, a new scheme of sparsity regularization on phase is proposed to capture the sparsity property of the SPR problem. Based on this regularization scheme, we design and analyze an efficient solution method, named SROP algorithm, for solving SPR given only a single intensity point-spread-function image. The algorithm is a combination of the Gerchberg-Saxton algorithm with the newly proposed sparsity regularization on the phase. The latter regularization step is mathematically a rotation but with direction varying in iterations. Surprisingly, this rotation is shown to be a metric projection on an auxiliary set which is independent of iterations. As a consequence, SROP algorithm is proved to be the cyclic projections algorithm for solving a feasibility problem involving three auxiliary sets. Analyzing regularity properties of the latter auxiliary sets, we obtain convergence results for SROP algorithm based on recent convergence theory for the cyclic projections algorithm. Numerical results show clear effectiveness of the new regularization scheme for solving the SPR problem.