Quantization-Aware Phase Retrieval

TL;DR

This paper introduces a quantization-aware phase retrieval algorithm that improves signal reconstruction accuracy from quantized measurements, incorporating consistency enforcement, sparsity constraints, and theoretical bounds, outperforming existing methods especially with coarse quantization.

Contribution

It proposes a novel rank-1 projection algorithm for quantized phase retrieval that enforces measurement consistency and includes sparsity constraints, with theoretical analysis and superior empirical performance.

Findings

Higher reconstruction accuracy than state-of-the-art algorithms.

Performance close to the Cramér-Rao lower bound.

Significant improvement with coarse quantization.

Abstract

We address the problem of phase retrieval (PR) from quantized measurements. The goal is to reconstruct a signal from quadratic measurements encoded with a finite precision, which is indeed the case in many practical applications. We develop a rank-1 projection algorithm that recovers the signal subject to ensuring consistency with the measurement, that is, the recovered signal when encoded must yield the same set of measurements that one started with. The rank-1 projection stems from the idea of lifting, originally proposed in the context of PhaseLift. The consistency criterion is enforced using a one-sided quadratic cost. We also determine the probability with which different vectors lead to the same set of quantized measurements, which makes it impossible to resolve them. Naturally, this probability depends on how correlated such vectors are, and how coarsely/finely the measurements get quantized. The proposed algorithm is also capable of incorporating a sparsity constraint on the signal. An analysis of the cost function reveals that it is bounded, both above and below, by functions that are dependent on how well correlated the estimate is with the ground truth. We also derive the Cram\'er-Rao lower bound (CRB) on the achievable reconstruction accuracy. A comparison with the state-of-the- art algorithms shows that the proposed algorithm has a higher reconstruction accuracy and is about 2 to 3 dB away from the CRB. The edge, in terms of the reconstruction signal-to-noise ratio, over the competing algorithms is higher (about 5 to 6 dB) when the quantization is coarse.