# Fourier Phase Retrieval: Uniqueness and Algorithms

## TL;DR

This paper surveys the theoretical uniqueness conditions and various algorithms for Fourier phase retrieval, a fundamental problem in signal processing with applications across science and engineering.

## Contribution

It provides a comprehensive overview of conditions ensuring uniqueness and reviews multiple algorithmic strategies for practical signal recovery.

## Key findings

- Almost all multidimensional signals are uniquely recoverable from Fourier magnitude
- Various algorithms have been developed for practical signal retrieval
- Open questions remain in the theoretical and algorithmic aspects

## Abstract

The problem of recovering a signal from its phaseless Fourier transform measurements, called Fourier phase retrieval, arises in many applications in engineering and science. Fourier phase retrieval poses fundamental theoretical and algorithmic challenges. In general, there is no unique mapping between a one-dimensional signal and its Fourier magnitude and therefore the problem is ill-posed. Additionally, while almost all multidimensional signals are uniquely mapped to their Fourier magnitude, the performance of existing algorithms is generally not well-understood. In this chapter we survey methods to guarantee uniqueness in Fourier phase retrieval. We then present different algorithmic approaches to retrieve the signal in practice. We conclude by outlining some of the main open questions in this field.