Phase Retrieval via Wirtinger Flow Algorithm and Its Variants

Abstract

Almost three-quarters of the underling information in the light wave field is embodied in the phase. However, the early optical detectors can only record the intensity or amplitude of the light wave field and cannot directly extract the phase information of the light wave field. Therefore, it is necessary to use the measured amplitude or strength to reconstruct the phase information of the object, this problem is denoted phase retrieval. Phase retrieval is a matter of cardinal significance in signal processing and machine learning. The phase retrieval by convex optimization algorithm is ideal but the computational complexity is high. In 2015, Candès proposed a very effective non-convex optimization algorithm-Wirtinger flow algorithm which used spectral initialization to get a better initial value and then gradient iteration to get a promised recovery effect. Subsequently, in line with the idea, a large number of variants are devised, such as: Wirtinger flow(WF), Truncated Wirtinger Flow (TWF), Truncated Amplitude Flow (TAF), Reshaped Wirtinger Flow (RWF), Incremental Truncated Wirtinger Flow (ITWF), Incremental Reshaped Wirtinger Flow (IRWF), Robust Wirtinger Flow (Robust-WF), Sparse Wirtinger Flow (SWF), Median-TWF, Median-RWF, Generalized Wirtinger Flow (GWF), Accelerated Wirtinger Flow (AWF), Thresholded Wirtinger Flow Revisited (THWFR), Thresholded Wirtinger Flow (THWF), Reweighted Wirtinger Flow (REWF), Wirtinger Flow Method With Optimal Stepsize (WFOS), Stochastic Truncated Wirtinger Flow Algorithm (STWF), Stochastic Truncated Amplitude Flow (STAF), Reweighted Amplitude Flow (RAF), Compressive Reweighted Amplitude Flow (CRAF), SPARse Truncated Amplitude flow (SPARTA) and Sparse Wirtinger Flow Algorithm with Optimal Stepsize (SWFOS), etc. This paper analyzes and summarizes these algorithms according to their characteristics such as: initialization method, step size, iteration times, sample complexity, computational complexity, etc., so that readers can intuitively and clearly see the characteristics of each algorithm. Finally, we provide the website of the source code of some algorithms, facilitate to access and use it for readers.