# Phase Retrieval from 1D Fourier Measurements: Convexity, Uniqueness, and Algorithms

arXiv: https://arxiv.org/abs/1603.05215 · HTML: https://mlsanity.com/paper/1603.05215/

## Key facts

- Cited by 3 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 16 Mar 2016.

## Built on by (most cited first, 3 of 3)

- [Signal Reconstruction from Quantized Noisy Samples of the Discrete Fourier Transform](https://mlsanity.com/paper/2201.03114/) (2022)
- [Convex Augmentation for Total Variation Based Phase Retrieval](https://mlsanity.com/paper/2205.00834/) (2022)
- [Res-U2Net: Untrained Deep Learning for Phase Retrieval and Image Reconstruction](https://mlsanity.com/paper/2404.06657/) (2024)

## Abstract (arXiv)

This paper considers phase retrieval from the magnitude of 1D over-sampled Fourier measurements, a classical problem that has challenged researchers in various fields of science and engineering. We show that an optimal vector in a least-squares sense can be found by solving a convex problem, thus establishing a hidden convexity in Fourier phase retrieval. We also show that the standard semidefinite relaxation approach yields the optimal cost function value (albeit not necessarily an optimal solution) in this case. A method is then derived to retrieve an optimal minimum phase solution in polynomial time. Using these results, a new measuring technique is proposed which guarantees uniqueness of the solution, along with an efficient algorithm that can solve large-scale Fourier phase retrieval problems with uniqueness and optimality guarantees.