Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$
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| Format: | Preprint |
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2023
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| _version_ | 1866916720646553600 |
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| author | Alaifari, Rima Bartolucci, Francesca Wellershoff, Matthias |
| author_facet | Alaifari, Rima Bartolucci, Francesca Wellershoff, Matthias |
| contents | Sampled Gabor phase retrieval - the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice - is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in $L^2(\mathbb{R})$, but is not equal to the whole of $L^2(\mathbb{R})$ in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_03940 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$ Alaifari, Rima Bartolucci, Francesca Wellershoff, Matthias Functional Analysis Complex Variables 42C15, 94A12 Sampled Gabor phase retrieval - the problem of recovering a square-integrable signal from the magnitude of its Gabor transform sampled on a lattice - is a fundamental problem in signal processing, with important applications in areas such as imaging and audio processing. Recently, a classification of square-integrable signals which are not phase retrievable from Gabor measurements on parallel lines has been presented. This classification was used to exhibit a family of counterexamples to uniqueness in sampled Gabor phase retrieval. Here, we show that the set of counterexamples to uniqueness in sampled Gabor phase retrieval is dense in $L^2(\mathbb{R})$, but is not equal to the whole of $L^2(\mathbb{R})$ in general. Overall, our work contributes to a better understanding of the fundamental limits of sampled Gabor phase retrieval. |
| title | Uncovering the limits of uniqueness in sampled Gabor phase retrieval: A dense set of counterexamples in $L^2(\mathbb{R})$ |
| topic | Functional Analysis Complex Variables 42C15, 94A12 |
| url | https://arxiv.org/abs/2307.03940 |