Fixed Point Analysis of Douglas-Rachford Splitting for Ptychography and Phase Retrieval

Douglas-Rachford Splitting (DRS) methods based on the proximal point algorithms for the Poisson and Gaussian log-likelihood functions are proposed for ptychography and phase retrieval. Fixed point analysis shows that the DRS iterated sequences are always bounded explicitly in terms of the step size and that the fixed points are attracting if and only if the fixed points are regular solutions. This alleviates two major drawbacks of the classical Douglas-Rachford algorithm: slow convergence when the feasibility problem is consistent and divergent behavior when the feasibility problem is inconsistent. Fixed point analysis also leads to a simple, explicit expression for the optimal step size in terms of the spectral gap of an underlying matrix. When applied to the challenging problem of blind ptychography, which seeks to recover both the object and the probe simultaneously, Alternating Minimization with the DRS inner loops, even with a far from optimal step size, converges geometrically under the nearly minimum conditions established in the uniqueness theory.