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.

翻译：本文讨论基于最大可能性( ML) 估算的算法, 以便根据独立 Poisson 分布分布的测量结果进行阶段检索。 为了优化 Poisson ML 模型的日志比值, 我们调查并实施了几项算法, 包括修改 Wirtinger 流(WF), 大规模最小化(MMM) 和乘数交替方向法(ADMMM ), 并将其与古典 WFS 和 Gerchberg Saxton 电视台(GS) 的阶段检索方法相比较。 我们修改的 WFS 方法使用基于所观测到的Fisherish 信息, 消除除迭代数之外的所有参数调试。 使用随机高斯感测矩阵和离散的 Fourier(DFourier) 矩阵的模拟结果显示, 以Poisson ML模型为基础的算法持续产生比运算法(WFIS, GS) 和 古尔茨 噪音 MLTL 模型(G) 使用这些数据所观测到的最快速度速度变异。