A Unified View of Schulz-Snyder Phase Retrieval Algorithm

Phase retrieval is the recovery of signals from Fourier transform magnitude with wide applications in crystallography, microscopy, optics, and astronomy. Although a unique solution almost always exists for two or higher dimensional signals, there is no known algorithm with guaranteed recovery. An iterative algorithm for recovering nonnegative real signals, based on minimizing the Csiszars distance, was developed by Schulz and Snyder. We study the convergence properties of this algorithm from a unified viewpoint. We establish its relation to several well-known algorithms, including blind Richardson-Lucy, expectation-maximization, and alternating-minimization algorithms, and gradient-descent methods. These connections allow the algorithm to be seen in a new light, making many of its convergence properties, advantages and drawbacks almost obvious. The gained understanding yields new insights to improve the algorithm in terms of reliability and speed.