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

by R. Alaifari and F. Bartolucci and S. Steinerberger and M. Wellershoff

(Report number 2022-25)

Abstract
For every lattice \(\Lambda\), we construct pairs of functions which are arbitrarily close to the Gaussian, do not agree up to global phase but have Gabor transform magnitudes agreeing on \(\Lambda\). Additionally, we prove that the Gaussian can be uniquely recovered (up to global phase) in \(L^2(\mathbb{R})\) from Gabor magnitude measurements on a sufficiently fine lattice. These two facts give evidence for the existence of functions which break uniqueness from samples without affecting stability. We prove that a uniform bound on the local Lipschitz constant of the signals is not sufficient to restore uniqueness in sampled Gabor phase retrieval and more restrictive a priori knowledge of the functions is necessary. With this, we show that there is no direct connection between uniqueness from samples and stability in Gabor phase retrieval. Finally, we provide an intuitive argument about the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

Keywords: Gabor transform, Phase retrieval, Sampled Gabor phase retrieval, Poincaré inequality, Cheeger constant, Laplace eigenvalues, Bargmann transform, Counterexamples

@Techreport{ABSW22_1013,
  author = {R. Alaifari and F. Bartolucci and S. Steinerberger and M. Wellershoff},
  title = {On the connection between uniqueness from samples and stability in Gabor phase retrieval},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2022-25},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2022/2022-25.pdf },
  year = {2022}
}

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