Stable Gabor Phase Retrieval in Gaussian Shift-Invariant Spaces via Biorthogonality

Abstract

AbstractWe study the phase reconstruction of signals f belonging to complex Gaussian shift-invariant spaces $$V^\infty (\varphi )$$ V ∞ ( φ ) from spectrogram measurements $$|{\mathcal {G}} f(X)|$$ | G f ( X ) | where $${\mathcal {G}}$$ G is the Gabor transform and $$X \subseteq {{\mathbb {R}}}^2$$ X ⊆ 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 compacts intervals. Driven by a recent observation that signals in Gaussian shift-invariant spaces are determined by lattice measurements (Grohs and Liehr in Injectivity of Gabor phase retrieval from lattice measurements. Appl. Comput. Harmon. Anal. 62, 173–193 (2023)) 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 $$V^\infty (\varphi )$$ V ∞ ( φ ) , such as Paley–Wiener spaces.