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

Kejun Huang, Yonina C. Eldar, Nicholas D. Sidiropoulos

Research output: Contribution to journalArticlepeer-review

38 Scopus citations

Abstract

This paper considers phase retrieval from the magnitude of one-dimensional 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 then show that the standard semidefinite relaxation approach yields the optimal cost function value (albeit not necessarily an optimal solution). 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.

Original languageEnglish (US)
Article number7547374
Pages (from-to)6105-6117
Number of pages13
JournalIEEE Transactions on Signal Processing
Volume64
Issue number23
DOIs
StatePublished - Dec 1 2016

Bibliographical note

Funding Information:
The work of K. Huang and N. D. Sidiropoulos was supported by the National Science Foundation under Grants CIF-1525194 and IIS-1247632. The work of Y. C. Eldar was supported in part by the European Unions Horizon 2020 Research and Innovation Program through the ERC-BNYQ Project, and in part by the Israel Science Foundation under Grant 335/14.

Keywords

  • Phase retrieval
  • alternating direction method of multipliers
  • auto-correlation retrieval
  • holography
  • minimum phase
  • over-sampled Fourier measurements
  • semi-definite programming

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