Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality

Abstract

We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^{\infty }(\ddot{I}†)$ from spectrogram measurements $|\mathcal{G}f(X)|$ where $\mathcal{G}$ is the Gabor transform and $X\subseteq {\mathbb{R}}^2$. An explicit reconstruction formula will demonstrate that such signals can be recovered from measurements located on parallel lines in the time-frequency plane by means of a Riesz basis expansion. Moreover, connectedness assumptions on $|f|$ result in stability estimates in the situation where one aims to reconstruct $f$ on compact intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements [Grohs, P., Liehr, L., Injectivity of Gabor phase retrieval from lattice measurements, Appl. Comput. Harmon. Anal. 62 (2023), pp. 173-193] we prove a sampling result on the stable approximation from finitely many spectrogram samples. The resulting algorithm provides a provably stable and convergent approximation technique. In addition, it constitutes a method of approximating signals in function spaces beyond $V^{\infty }(\ddot{I}†)$, such as Paley-Wiener spaces.

44 pages, 9 figures