Algorithms for Poisson Phase Retrieval

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 and discrete Fourier transform (DFT) 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. The proposed regularized MM algorithm decreased NRMSE much faster than the regularized ADMM algorithm. We also proposed a variation of the WF approach that uses a step size based on the Fisher information and converges faster than previous WF approaches.