Fourier phase retrieval: Uniqueness and algorithms

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
Subtitle of host publicationSecond International MATHEON Conference 2015
EditorsHolger Boche, Giuseppe Caire, Robert Calderbank, Maximilian März, Gitta Kutyniok, Rudolf Mathar
Place of PublicationSwitzerland
Number of pages37
ISBN (Electronic)9783319698021
StatePublished - 2017

Publication series

NameApplied and Numerical Harmonic Analysis


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

All Science Journal Classification (ASJC) codes

  • Applied Mathematics


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