On the connection between uniqueness from samples and stability in Gabor phase retrieval

Rima Alaifari, Francesca Bartolucci*, Stefan Steinerberger, Matthias Wellershoff

*Corresponding author for this work

Research output: Contribution to journalArticleScientificpeer-review

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Abstract

Gabor phase retrieval is the problem of reconstructing a signal from only the magnitudes of its Gabor transform. Previous findings suggest a possible link between unique solvability of the discrete problem (recovery from measurements on a lattice) and stability of the continuous problem (recovery from measurements on an open subset of R2). In this paper, we close this gap by proving that such a link cannot be made. More precisely, we establish the existence of functions which break uniqueness from samples without affecting stability of the continuous problem. Furthermore, we prove the novel result that counterexamples to unique recovery from samples are dense in L2(R) . Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

Original languageEnglish
Article number6
Number of pages36
JournalSampling Theory, Signal Processing, and Data Analysis
Volume22
Issue number1
DOIs
Publication statusPublished - 2024

Keywords

  • Bargmann transform
  • Cheeger constant
  • Counterexamples
  • Gabor transform
  • Laplace eigenvalues
  • Phase retrieval
  • Poincaré inequality
  • Sampled Gabor phase retrieval

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