Phase Retrieval with One or Two Diffraction Patterns by Alternating Projections with the Null Initialization

Alternating projection (AP) of various forms, including the parallel AP (PAP), real-constrained AP (RAP) and the serial AP (SAP), are proposed to solve phase retrieval with at most two coded diffraction patterns. The proofs of geometric convergence are given with sharp bounds on the rates of convergence in terms of a spectral gap condition. To compensate for the local nature of convergence, the null initialization is proposed to produce good-quality initial guess. Numerical experiments show that the null initialization is more accurate than the spectral initialization and that AP converges faster to the true object than other iterative schemes such as the Wirtinger flow (WF). In numerical experiments AP with the null initialization converges globally to the true object.