Robust Phase Retrieval with the Swept Approximate Message Passing (prSAMP) Algorithm

In phase retrieval, the goal is to recover a complex signal from the magnitude of its linear measurements. While many well-known algorithms guarantee deterministic recovery of the unknown signal using i.i.d. random measurement matrices, they suffer serious convergence issues for some ill-conditioned measurement matrices. As an example, this happens in optical imagers using binary intensity-only spatial light modulators to shape the input wavefront. The problem of ill-conditioned measurement matrices has also been a topic of interest for compressed sensing researchers during the past decade. In this paper, using recent advances in generic compressed sensing, we propose a new phase retrieval algorithm that well-behaves for a large class of measurement matrices, including Gaussian and Bernoulli binary i.i.d. random matrices, using both sparse and dense input signals. This algorithm is also robust to the strong noise levels found in some imaging applications