Poisson Phase Retrieval in Very Low-count Regimes

TL;DR

This paper introduces a modified Wirtinger flow algorithm with Fisher information-based step size and a novel curvature for MM algorithms, significantly improving phase retrieval performance in very low-count Poisson noise regimes.

Contribution

It proposes a new step size scheme for Wirtinger flow and a sharper curvature for MM algorithms, enhancing convergence and accuracy in low-photon-count phase retrieval.

Findings

Poisson ML models outperform Gaussian noise models in low-count data.

Proposed WF algorithm converges faster than existing methods in unregularized cases.

In regularized cases, the new WF converges faster than other algorithms like LBFGS, MM, and ADMM.

Abstract

This paper discusses phase retrieval algorithms for maximum likelihood (ML) estimation from measurements following independent Poisson distributions in very low-count regimes, e.g., 0.25 photon per pixel. To maximize the log-likelihood of the Poisson ML model, we propose a modified Wirtinger flow (WF) algorithm using a step size based on the observed Fisher information. This approach eliminates all parameter tuning except the number of iterations. We also propose a novel curvature for majorize-minimize (MM) algorithms with a quadratic majorizer. We show theoretically that our proposed curvature is sharper than the curvature derived from the supremum of the second derivative of the Poisson ML cost function. We compare the proposed algorithms (WF, MM) with existing optimization methods, including WF using other step-size schemes, quasi-Newton methods such as LBFGS and alternating direction method of multipliers (ADMM) algorithms, under a variety of experimental settings. Simulation experiments with a random Gaussian matrix, a canonical DFT matrix, a masked DFT matrix and an empirical transmission matrix demonstrate the following. 1) As expected, algorithms based on the Poisson ML model consistently produce higher quality reconstructions than algorithms derived from Gaussian noise ML models when applied to low-count data. 2) For unregularized cases, our proposed WF algorithm with Fisher information for step size converges faster than other WF methods, e.g., WF with empirical step size, backtracking line search, and optimal step size for the Gaussian noise model; it also converges faster than the LBFGS quasi-Newton method. 3) In regularized cases, our proposed WF algorithm converges faster than WF with backtracking line search, LBFGS, MM and ADMM.