Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms

TL;DR

This paper reveals hidden convexity in 1D Fourier phase retrieval, enabling the development of efficient algorithms that guarantee unique and optimal solutions for large-scale problems.

Contribution

It introduces a convex formulation for Fourier phase retrieval, guarantees solution uniqueness, and proposes an efficient polynomial-time algorithm for optimal recovery.

Findings

Convex formulation finds optimal vector in least-squares sense.

Semidefinite relaxation yields the optimal cost value.

New measurement technique guarantees solution uniqueness.

Abstract

This paper considers phase retrieval from the magnitude of 1D over-sampled Fourier measurements, a classical problem that has challenged researchers in various fields of science and engineering. We show that an optimal vector in a least-squares sense can be found by solving a convex problem, thus establishing a hidden convexity in Fourier phase retrieval. We also show that the standard semidefinite relaxation approach yields the optimal cost function value (albeit not necessarily an optimal solution) in this case. A method is then derived to retrieve an optimal minimum phase solution in polynomial time. Using these results, a new measuring technique is proposed which guarantees uniqueness of the solution, along with an efficient algorithm that can solve large-scale Fourier phase retrieval problems with uniqueness and optimality guarantees.