Injectivity of Gabor phase retrieval from lattice measurements

Abstract

We establish novel uniqueness results for the Gabor phase retrieval problem: if $\mathcal{G}:L^2(\mathbb{R})\to L^2({\mathbb{R}}^2)$ denotes the Gabor transform then every $f\in L^4[-\frac{c}{2},\frac{c}{2}]$ is determined up to a global phase by the values $|\mathcal{G}f(x,\ddot{I}‰)|$ where $(x,\ddot{I}‰)$ are points on the lattice $b^{-1}\mathbb{Z}\times (2c{)}^{-1}\mathbb{Z}$ and $b>0$ is an arbitrary positive constant. This for the first time shows that compactly-supported, complex-valued functions can be uniquely reconstructed from lattice samples of their spectrogram. Moreover, by making use of recent developments related to sampling in shift-invariant spaces by Gröchenig, Romero and Stöckler, we prove analogous uniqueness results for functions in shift-invariant spaces with Gaussian generator. Generalizations to nonuniform sampling are also presented. Finally, we compare our results to the situation where the considered signals are assumed to be real-valued.

26 pages, 1 figure, Section 3.2 and Section 3.4 added