TY - CHAP
T1 - Fourier phase retrieval
T2 - Uniqueness and algorithms
AU - Bendory, Tamir
AU - Beinert, Robert
AU - Eldar, Yonina C.
N1 - Publisher Copyright:
© Springer International Publishing AG 2017.
PY - 2017
Y1 - 2017
N2 - 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.
AB - 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.
KW - Alternating projections
KW - Finitelysupported and sparse signals
KW - Masked fourier phase retrieval
KW - Non-convex optimization
KW - Phase retrieval
KW - Ptychography
KW - Semidefiniteprogramming
KW - Ultra-short pulse characterization
KW - Uniqueness guarantees
UR - http://www.scopus.com/inward/record.url?scp=85047258235&partnerID=8YFLogxK
U2 - 10.1007/978-3-319-69802-1_2
DO - 10.1007/978-3-319-69802-1_2
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AN - SCOPUS:85047258235
T3 - Applied and Numerical Harmonic Analysis
SP - 55
EP - 91
BT - Applied and Numerical Harmonic Analysis
PB - Springer International Publishing
ER -