  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.
