Super Resolution Phase Retrieval for Sparse Signals

TL;DR

This paper introduces a novel super-resolution phase retrieval algorithm for sparse signals that estimates the original continuous-domain signal from limited measurements, with theoretical performance bounds and improved noise robustness.

Contribution

It presents the first continuous-domain sparse phase retrieval algorithm leveraging super-resolution and auto-correlation, with theoretical analysis and practical enhancements.

Findings

The algorithm successfully recovers sparse signals from limited auto-correlation samples.

It outperforms Charge Flipping in crystallography applications.

Theoretical bounds validate the algorithm's performance.

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 to recover the phase information of a signal from the magnitude of its Fourier transform to enable the reconstruction of the original signal. Solving the phase retrieval problem is equivalent to recovering a signal from its auto-correlation function. In this paper, we assume the original signal to be sparse; this is a natural assumption in many applications, such as X-ray crystallography, speckle imaging and blind channel estimation. We propose an algorithm that resolves the phase retrieval problem in three stages: i) we leverage the finite rate of innovation sampling theory to super-resolve the auto-correlation function from a limited number of samples, ii) we design a greedy algorithm that identifies the locations of a sparse solution given the super-resolved auto-correlation function, iii) we recover the amplitudes of the atoms given their locations and the measured auto-correlation function. Unlike traditional approaches that recover a discrete approximation of the underlying signal, our algorithm estimates the signal on a continuous domain, which makes it the first of its kind. Along with the algorithm, we derive its performance bound with a theoretical analysis and propose a set of enhancements to improve its computational complexity and noise resilience. Finally, we demonstrate the benefits of the proposed method via a comparison against Charge Flipping, a notable algorithm in crystallography.