Phase Retrieval for Sparse Signals: Uniqueness Conditions

TL;DR

This paper establishes conditions under which sparse signals can be uniquely reconstructed from magnitude-only Fourier measurements, linking phase retrieval to the turnpike problem and addressing both 1D and multi-dimensional cases.

Contribution

It provides a new sufficient condition for the uniqueness of sparse phase retrieval solutions, extending results to multi-dimensional signals and connecting with combinatorial turnpike problem.

Findings

Uniqueness is guaranteed when the autocorrelation has no collisions.

The result extends to multi-dimensional signals via 1D problem solutions.

Autocorrelation collision absence ensures almost sure uniqueness in 1D.

Abstract

In a variety of fields, in particular those involving imaging and optics, we often measure signals whose phase is missing or has been irremediably distorted. Phase retrieval attempts the recovery of the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. A fundamental question then is: "Under which conditions can we uniquely recover the signal of interest from its measured magnitudes?" In this paper, we assume the measured signal to be sparse. This is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. In this work, we derive a sufficient condition for the uniqueness of the solution of the phase retrieval (PR) problem for both discrete and continuous domains, and for one and multi-dimensional domains. More precisely, we show that there is a strong connection between PR and the turnpike problem, a classic combinatorial problem. We also prove that the existence of collisions in the autocorrelation function of the signal may preclude the uniqueness of the solution of PR. Then, assuming the absence of collisions, we prove that the solution is almost surely unique on 1-dimensional domains. Finally, we extend this result to multi-dimensional signals by solving a set of 1-dimensional problems. We show that the solution of the multi-dimensional problem is unique when the autocorrelation function has no collisions, significantly improving upon a previously known result.