Algorithms for Poisson Phase Retrieval

This paper discusses algorithms based on maximum likelihood (ML) estimation for phase retrieval where the measurements follow independent Poisson distributions. To optimize the log-likelihood for the Poisson ML model, we investigated and implemented several algorithms including a modified Wirtinger flow (WF), majorize minimize (MM) and alternating direction method of multipliers (ADMM), and compared them to the classical WF and Gerchberg Saxton (GS) methods for phase retrieval. Our modified WF approach uses a step size based on the observed Fisher information, eliminating all parameter tuning except the number of iterations. Simulation results using random Gaussian sensing matrix and discrete Fourier transform (DFT) matrix under Poisson measurement noise demonstrated that algorithms based on the Poisson ML model consistently produced higher quality reconstructions than algorithms (WF, GS) derived from Gaussian noise ML 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 (TV)) or of the coefficients of a discrete wavelet transform. In terms of the convergence speed, the WF using observed Fisher information for step size decreased NRMSE the fastest among all unregularized algorithms; the regularized WF approach also converged the fastest among all regularized algorithms with the TV regularizer approximated by the Huber function.