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 $|\mathcal{G}f(x,\omega )|$ where $(x,\omega )$ 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

References in corpus (6)

• The Numerics of Phase Retrieval
• Uniqueness of STFT phase retrieval for bandlimited functions
• On foundational discretization barriers in STFT phase retrieval
• Phase retrieval from sampled Gabor transform magnitudes: Counterexamples
• Stable Phase Retrieval from Locally Stable and Conditionally Connected Measurements
• A Note on the Phase Retrieval of Holomorphic Functions

Cited by in corpus (13)

• On foundational discretization barriers in STFT phase retrieval
• Phase retrieval from sampled Gabor transform magnitudes: Counterexamples
• Non-uniqueness theory in sampled STFT phase retrieval
• Phase retrieval of bandlimited functions for the wavelet transform
• Injectivity of sampled Gabor phase retrieval in spaces with general integrability conditions
• Sampling at twice the Nyquist rate in two frequency bins guarantees uniqueness in Gabor phase retrieval
• Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality
• Phase retrieval of entire functions and its implications for Gabor phase retrieval
• Phaseless sampling on square-root lattices
• On the connection between uniqueness from samples and stability in Gabor phase retrieval
• Arithmetic progressions and holomorphic phase retrieval
• Translation-based completeness on compact intervals
• Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$