Compressive Phase Retrieval via Generalized Approximate Message Passing

In this paper, we propose a novel approach to compressive phase retrieval based on loopy belief propagation and, in particular, on the generalized approximate message passing (GAMP) algorithm. Numerical results show that the proposed PR-GAMP algorithm has excellent phase-transition behavior, noise robustness, and runtime. In particular, for successful recovery of synthetic Bernoulli-circular-Gaussian signals, PR-GAMP requires approximate to 4 times the number of measurements as a phase-oracle version of GAMP and, at moderate to large SNR, the NMSE of PR-GAMP is only approximate to 3 dB worse than that of phase-oracle GAMP. A comparison to the recently proposed convex-relation approach known as "CPRL" reveals PR-GAMP's superior phase transition and orders-of-magnitude faster runtimes, especially as the problem dimensions increase. When applied to the recovery of a 65k-pixel grayscale image from 32k randomly masked magnitude measurements, numerical results show a median PR-GAMP runtime of only 13.4 seconds.