Compressive Phase Retrieval via Generalized Approximate Message Passing

In phase retrieval, the goal is to recover a signal x ∈ C
^N
from the magnitudes of linear measurements Ax ∈ C
^M
. While recent theory has established that M ≈ 4N intensity measurements are necessary and sufficient to recover generic x, there is great interest in reducing the number of measurements through the exploitation of sparse \mbi 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. Our experiments suggest that approximately M ≥ 2 Klog
₂
(N/K) intensity measurements suffice to recover K-sparse Bernoulli-Gaussian signals for \mbi A with i.i.d Gaussian entries and K ≪ N. Meanwhile, when recovering a 6678-sparse 65536-pixel grayscale image from 32768 randomly masked and blurred Fourier intensity measurements at 30 dB measurement SNR, PR-GAMP achieved an output SNR of no less than 28 dB in all of 100 random trials, with a median runtime of only 7.3 seconds. Compared to the recently proposed CPRL, sparse-Fienup, and GESPAR algorithms, our experiments suggest that PR-GAMP has a superior phase transition and orders-of-magnitude faster runtimes as the sparsity and problem dimensions increase.