Phase Retrieval From Quantized Measurements via Approximate Message Passing

In this letter, the problem of sparse signal reconstruction from quantized noisy magnitudes y = Q (|z + w| + n) is studied, where z = Ax, Q(·) denotes a quantizer, and x is the sparse signal. According to expectation propagation, the abovementioned problem can be solved by exchanging extrinsic information between the standard linear model (SLM) module and the minimum mean square error (MMSE) module. For the MMSE module, exploiting the fact that the likelihood is only a function of the absolute value of z, the magnitude and phase information of z are decoupled, and only the posterior means and variances of the magnitude of z are calculated. While for the SLM module, the approximate message passing (AMP) is used. To obtain the closed-form expression during the iteration, a novel approximation is adopted. We refer to the resulting algorithm as generalized AMP (Gr-AMP) based quantized phase retrieval algorithm. Finally, several numerical simulations are conducted to demonstrate the performances of the proposed approach.

I. Introduction

Phase Retrieval (PR) aims to recover a signal from the magnitudes of noisy linear measurements. PR has drawn a great deal of attention recently, since it has various applications such as array imaging [1], optics [2], X-ray diffraction imaging [3], X-ray crystallography [4], and microscopy [5].