Phase Retrieval Using Feasible Point Pursuit: Algorithms and Cram\'er-Rao Bound

TL;DR

This paper introduces two novel phase retrieval algorithms based on feasible point pursuit, demonstrating near-optimal performance close to the Cramér-Rao bound and advancing the state-of-the-art in signal reconstruction from quadratic measurements.

Contribution

It proposes two new non-convex QCQP-based phase retrieval algorithms, B-FPP and LS-FPP, and derives explicit CRB expressions for performance benchmarking.

Findings

LS-FPP outperforms existing algorithms

Algorithms operate close to the CRB

CRB expressions are explicitly derived for various cases

Abstract

Reconstructing a signal from squared linear (rank-one quadratic) measurements is a challenging problem with important applications in optics and imaging, where it is known as phase retrieval. This paper proposes two new phase retrieval algorithms based on non-convex quadratically constrained quadratic programming (QCQP) formulations, and a recently proposed approximation technique dubbed feasible point pursuit (FPP). The first is designed for uniformly distributed bounded measurement errors, such as those arising from high-rate quantization (B-FPP). The second is designed for Gaussian measurement errors, using a least squares criterion (LS-FPP). Their performance is measured against state-of-the-art algorithms and the Cram\'er-Rao bound (CRB), which is also derived here. Simulations show that LS-FPP outperforms the state-of-art and operates close to the CRB. Compact CRB expressions, properties, and insights are obtained by explicitly computing the CRB in various special cases -- including when the signal of interest admits a sparse parametrization, using harmonic retrieval as an example.