FASPR: A fast sparse phase retrieval algorithm via the epigraph concept

In this paper, we propose a fast sparsity-based approach to phase retrieval (PR) that refers to recovering an image of interest from the recorded magnitudes or intensities of its linear transform. The corresponding non-linear sampling operator makes the recovery task a challenging one. Recent efforts to solve this problem exploit sparse priors via fusing a data fidelity term with an elaborate sparsity-induced regularization term. However, two drawbacks exist in these sparsity-based PR algorithms. One is high computation complexity, and the other is the issue of many finely-tuned parameters, i.e., regular parameters and thresholding values. To address these issues, we propose a fast sparse PR algorithm (FASPR) based on the epigraph concept of the total variation (TV) function. We exploit the constraint based on the TV model, namely that the underlying image should be sparse in the gradient domain, to formulate a PR problem, and then solve the formulated problem in two steps: a phase-recovering step and an image-updating step. The fast iterative shrinkage-thresholding algorithm (FISTA) is utilized for solving the corresponding sub-problem in the image-updating step. The step size in FISTA is selected in an adaptive manner by solving a step size optimization problem. In the projection step of FISTA, the epigraph set of the TV function is defined, and the projection onto this set is calculated. For both real-valued images and complex-valued images, diffraction imaging from under-sampled and noisy coded diffraction patterns is simulated. Experimental results validate the effectiveness and the efficiency of the proposed FASPR algorithm. Most importantly, it does not need many finely-tuned parameters, and only the iteration times need to be specified.