Fourier phase retrieval: Uniqueness and algorithms

Tamir Bendory*, Robert Beinert, Yonina C. Eldar

*Corresponding author for this work

Research output: Chapter in Book/Report/Conference proceedingChapterpeer-review


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.

Original languageEnglish
Title of host publicationApplied and Numerical Harmonic Analysis
PublisherSpringer International Publishing
Number of pages37
StatePublished - 2017
Externally publishedYes

Publication series

NameApplied and Numerical Harmonic Analysis
ISSN (Print)2296-5009
ISSN (Electronic)2296-5017


  • Alternating projections
  • Finitelysupported and sparse signals
  • Masked fourier phase retrieval
  • Non-convex optimization
  • Phase retrieval
  • Ptychography
  • Semidefiniteprogramming
  • Ultra-short pulse characterization
  • Uniqueness guarantees


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