Phase Retrieval From Binary Measurements

TL;DR

This paper introduces a novel binary phase retrieval algorithm (BPR) for reconstructing signals from binary quadratic measurements, demonstrating effective performance close to theoretical bounds even with noise.

Contribution

The paper proposes a convex optimization-based iterative algorithm for binary phase retrieval, incorporating momentum for faster convergence and deriving the Cramer-Rao Bound for noisy measurements.

Findings

BPR achieves ~25 dB SRER without noise.

Performance within 2-3 dB of CRB under noise.

Faster convergence with momentum in PGD.

Abstract

We consider the problem of signal reconstruction from quadratic measurements that are encoded as +1 or -1 depending on whether they exceed a predetermined positive threshold or not. Binary measurements are fast to acquire and inexpensive in terms of hardware. We formulate the problem of signal reconstruction using a consistency criterion, wherein one seeks to find a signal that is in agreement with the measurements. To enforce consistency, we construct a convex cost using a one-sided quadratic penalty and minimize it using an iterative accelerated projected gradient-descent (APGD) technique. The PGD scheme reduces the cost function in each iteration, whereas incorporating momentum into PGD, notwithstanding the lack of such a descent property, exhibits faster convergence than PGD empirically. We refer to the resulting algorithm as binary phase retrieval (BPR). Considering additive white noise contamination prior to quantization, we also derive the Cramer-Rao Bound (CRB) for the binary encoding model. Experimental results demonstrate that the BPR algorithm yields a signal-to- reconstruction error ratio (SRER) of approximately 25 dB in the absence of noise. In the presence of noise prior to quantization, the SRER is within 2 to 3 dB of the CRB.