Phase Retrieval with One or Two Diffraction Patterns by Alternating Projection with Null Initialization
This work addresses the practical challenge of reducing the number of measurements in phase retrieval, which is critical for applications like X-ray crystallography and coherent diffractive imaging.
The paper proposes alternating projection methods for phase retrieval using only one or two coded diffraction patterns, proving geometric convergence with sharp bounds and introducing null initialization that yields asymptotically accurate initial guesses. Numerical experiments show faster convergence and more accurate reconstruction compared to Wirtinger Flow and spectral 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 for initial guess and proved to produce asymptotically accurate initialization for the case of Gaussian random measurement. Numerical experiments show that the null initialization produces more accurate initial guess than the spectral initialization and that AP converges faster to the true object than other iterative schemes for non-convex optimization such as the Wirtinger Flow. In numerical experiments, AP with the null initialization converges globally to the true object.