# Phase Retrieval: Uniqueness and Stability

## TL;DR

This paper reviews mathematical results on phase retrieval, focusing on the conditions for unique and stable recovery of functions from Fourier magnitude data across various scientific fields.

## Contribution

It summarizes recent advances in understanding the uniqueness and stability of phase retrieval problems, integrating results from harmonic analysis, complex analysis, and geometry.

## Key findings

- Conditions for uniqueness in phase retrieval
- Stability properties of phase retrieval solutions
- Connections to applications in physics and imaging

## Abstract

The problem of phase retrieval, i.e., the problem of recovering a function from the magnitudes of its Fourier transform, naturally arises in various fields of physics, such as astronomy, radar, speech recognition, quantum mechanics and, perhaps most prominently, diffraction imaging. The mathematical study of phase retrieval problems possesses a long history with a number of beautiful and deep results drawing from different mathematical fields, such as harmonic analyis, complex analysis, or Riemannian geometry. The present paper aims to present a summary of some of these results with an emphasis on recent activities. In particular we aim to summarize our current understanding of uniqueness and stability properties of phase retrieval problems.