  This paper discusses algorithms for phase retrieval where the measurements
follow independent Poisson distributions, using maximum likelihood (ML)
estimation. To optimize the log-likelihood for the Poisson phase retrieval
model, we developed and compared several algorithms including Wirtinger flow
(WF), Gerchberg Saxton (GS), majorize minimize (MM) and alternating direction
method of multipliers (ADMM). Simulation results using random Gaussian sensing
matrix, discrete Fourier transform (DFT) matrix and an empirical transmission
matrix under Poisson measurement noise demonstrated that algorithms based on
the Poisson model consistently produced higher quality reconstructions than
algorithms (WF, GS) derived from Gaussian noise models when applied to such
data. Moreover, the reconstruction quality can be further improved by adding
regularizers that exploit assumed properties of the latent signal/image, such
as sparsity of finite differences (anisotropic total variation) or of the
coefficients of a discrete wavelet transform.
