A Message-Passing Approach to Phase Retrieval of Sparse Signals

Abstract

In phase retrieval, the goal is to recover a signal \(\boldsymbol{x} \in \mathbb{C}^{N}\) from the magnitudes of linear measurements \(\boldsymbol{Ax} \in \mathbb{C}^{M}\). While recent theory has established that M ≈ 4 N intensity measurements are necessary and sufficient to recover generic \(\boldsymbol{x}\), there is great interest in reducing the number of measurements through the exploitation of sparse \(\boldsymbol{x}\), which is known as compressive phase retrieval. In this work, we detail a novel, probabilistic approach to compressive phase retrieval based on the generalized approximate message passing (GAMP) algorithm. We then present a numerical study of the proposed PR-GAMP algorithm, demonstrating its excellent phase-transition behavior, robustness to noise, and runtime. For example, to successfully recover K-sparse signals, approximately \(M \geq 2K\log _{2}(N/K)\) intensity measurements suffice when K ≪ N and \(\boldsymbol{A}\) has i.i.d Gaussian entries. When recovering a 6k-sparse 65k-pixel grayscale image from 32k randomly masked and blurred Fourier intensity measurements, PR-GAMP achieved 99% success rate with a median runtime of only 12. 6 seconds. Compared to the recently proposed CPRL, sparse-Fienup, and GESPAR algorithms, experiments show that PR-GAMP has a superior phase transition and orders-of-magnitude faster runtimes as the problem dimensions increase.