  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.
