Phase retrieval with background information

TL;DR

This paper introduces a new phase retrieval model using background information, proving uniqueness and developing an efficient projected gradient descent algorithm that is robust to noise for recovering signals from Fourier intensity data.

Contribution

It establishes the theoretical guarantee of uniqueness with background info and proposes a practical, robust algorithm for phase retrieval.

Findings

Uniqueness of phase retrieval is guaranteed with sufficient background info.

Projected gradient descent converges to the global optimum with high probability.

The method performs well for both 1-D and 2-D signals and is noise-robust.

Abstract

Phase retrieval problem has been studied in various applications. It is an inverse problem without the standard uniqueness guarantee. To make complete theoretical analyses and devise efficient algorithms to recover the signal is sophisticated. In this paper, we come up with a model called \textit{phase retrieval with background information} which recovers the signal with the known background information from the intensity of their combinational Fourier transform spectrum. We prove that the uniqueness of phase retrieval can be guaranteed even considering those trivial solutions when the background information is sufficient. Under this condition, we construct a loss function and utilize the projected gradient descent method to search for the ground truth. We prove that the stationary point is the global optimum with probability 1. Numerical simulations demonstrate the projected gradient descent method performs well both for 1-D and 2-D signals. Furthermore, this method is quite robust to the Gaussian noise and the bias of the background information.