Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality
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arXiv
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| Format: | Preprint |
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2021
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| author | Grohs, Philipp Liehr, Lukas |
| author_facet | Grohs, Philipp Liehr, Lukas |
| contents | We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^\infty(φ)$ 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 $V^\infty(φ)$, such as Paley-Wiener spaces. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2109_02494 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality Grohs, Philipp Liehr, Lukas Functional Analysis Numerical Analysis 42C15, 46B15, 94A12, 94A20 We study the phase reconstruction of signals $f$ belonging to complex Gaussian shift-invariant spaces $V^\infty(φ)$ 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 $V^\infty(φ)$, such as Paley-Wiener spaces. |
| title | Stable Gabor phase retrieval in Gaussian shift-invariant spaces via biorthogonality |
| topic | Functional Analysis Numerical Analysis 42C15, 46B15, 94A12, 94A20 |
| url | https://arxiv.org/abs/2109.02494 |