On the 2D Phase Retrieval Problem

Dani Kogan, Yonina C. Eldar, Dan Oron

Research output: Contribution to journalArticle

6 Citations (Scopus)

Abstract

The recovery of a signal from the magnitude of its Fourier transform, also known as phase retrieval, is of fundamental importance in many scientific fields. It is well known that due to the loss of Fourier phase the problem in one-dimensional (1D) is ill-posed. Without further constraints, there is no unique solution to the problem. In contrast, uniqueness up to trivial ambiguities very often exists in higher dimensions, with mild constraints on the input. In this paper, we focus on the 2D phase retrieval problem and provide insight into this uniqueness property by exploring the connection between the 2D and 1D formulations. In particular, we show that 2D phase retrieval can be cast as a 1D problem with additional constraints, which limit the solution space. We then prove that only one additional constraint is sufficient to reduce the many feasible solutions in the 1D setting to a unique solution for almost all signals. These results allow to obtain an analytical approach (with combinatorial complexity) to solve the 2D phase retrieval problem when it is unique.

Original languageEnglish
Article number7752976
Pages (from-to)1058-1067
Number of pages10
JournalIEEE Transactions on Signal Processing
Volume65
Issue number4
DOIs
Publication statusPublished - Feb 15 2017

Keywords

  • 2D autocorrelation
  • Phase retrieval
  • Uniqueness

ASJC Scopus subject areas

  • Signal Processing
  • Electrical and Electronic Engineering

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