Wirtinger Flow Algorithms for Phase Retrieval from Binary Measurements

Abstract

We consider the problem of Binary Phase Retrieval, wherein we attempt to recover signals from their quadratic measurements, which are further encoded as +1 or −1 depending on whether they exceed a threshold or not. Binary encoding is the extreme case of quantization in the phase retrieval setting and has been introduced only recently. We formulate a consistency based non-convex cost function, which requires the signal estimate to agree with the binary measurements. Since lifting the measurements does not scale well with respect to the signal dimension, we solve the problem using Wirtinger Flow and the recently proposed accelerated Wirtinger Flow algorithms. We also propose a spectral initialization for the binary measurement model. The proposed algorithms have low computational complexity and also scale well with respect to the signal dimension. Simulation results show that the Wirtinger flow solution to the Binary Phase Retrieval problem is nearly on par with that obtained using the principle of lifting — the loss in signal-to-reconstruction error ratio (SRER) is about 1 dB, but the consistency with the binary observations is 100% in the absence of noise. Further, simulation results show that the accelerated Binary Wirtinger Flow (BWF) gives a 2 dB improvement in SRER over BWF.