Super Resolution Phase Retrieval for Sparse Signals

by   Gilles Baechler, et al.

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.


page 8

page 10


Prony-Based Super-Resolution Phase Retrieval of Sparse, Multivariate Signals

Phase retrieval consists in the recovery of an unknown signal from phase...

MULAN: A Blind and Off-Grid Method for Multichannel Echo Retrieval

This paper addresses the general problem of blind echo retrieval, i.e., ...

Sampling Without Time: Recovering Echoes of Light via Temporal Phase Retrieval

This paper considers the problem of sampling and reconstruction of a con...

Toward a mathematical theory of the crystallographic phase retrieval problem

Motivated by the X-ray crystallography technology to determine the atomi...

Fourier Phase Retrieval with Extended Support Estimation via Deep Neural Network

We consider the problem of sparse phase retrieval from Fourier transform...

Finite alphabet phase retrieval

We consider the finite alphabet phase retrieval problem: recovering a si...

Subspace Phase Retrieval

In this paper, we propose a novel algorithm, termed Subspace Phase Retri...

Please sign up or login with your details

Forgot password? Click here to reset