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

Rima Alaifari; Francesca Bartolucci; Stefan Steinerberger; Matthias Wellershoff

doi:10.1007/s43670-023-00079-1

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

Sampl Theory Signal Process Data Anal. 2024;22(1):6. doi: 10.1007/s43670-023-00079-1. Epub 2024 Jan 17.

Authors

Rima Alaifari¹, Francesca Bartolucci², Stefan Steinerberger³, Matthias Wellershoff⁴

Affiliations

¹ Department of Mathematics, Seminar for Applied Mathematics, ETH Zürich, Rämistrasse 101, 8092 Zürich, Switzerland.
² Institute of Applied Mathematics, TU Delft, Mekelweg 4, 2628 CD Delft, The Netherlands.
³ Department of Mathematics, University of Washington, C-138 Padelford, Seattle, WA 98195-4350 USA.
⁴ Department of Mathematics, University of Maryland, 4176 Campus Drive, College Park, MD 20742 USA.

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 $R^{2}$ ). 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 $L^{2} (R)$ . Finally, we develop an intuitive argument on the connection between directions of instability in phase retrieval and certain Laplacian eigenfunctions associated to small eigenvalues.

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