TY - JOUR
T1 - On the connection between uniqueness from samples and stability in Gabor phase retrieval
AU - Alaifari, Rima
AU - Bartolucci, Francesca
AU - Steinerberger, Stefan
AU - Wellershoff, Matthias
PY - 2024
Y1 - 2024
N2 - 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.
AB - 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.
KW - Bargmann transform
KW - Cheeger constant
KW - Counterexamples
KW - Gabor transform
KW - Laplace eigenvalues
KW - Phase retrieval
KW - Poincaré inequality
KW - Sampled Gabor phase retrieval
UR - http://www.scopus.com/inward/record.url?scp=85182436071&partnerID=8YFLogxK
U2 - 10.1007/s43670-023-00079-1
DO - 10.1007/s43670-023-00079-1
M3 - Article
AN - SCOPUS:85182436071
SN - 2730-5716
VL - 22
JO - Sampling Theory, Signal Processing, and Data Analysis
JF - Sampling Theory, Signal Processing, and Data Analysis
IS - 1
M1 - 6
ER -