Declipping and the recovery of vectors from saturated measurements
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arXiv
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| Autores principales: | , , , , |
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866911771123515392 |
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| author | Alharbi, Wedad Freeman, Daniel Ghoreishi, Dorsa Johnson, Brody Randrianarivony, N. Lovasoa |
| author_facet | Alharbi, Wedad Freeman, Daniel Ghoreishi, Dorsa Johnson, Brody Randrianarivony, N. Lovasoa |
| contents | A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ allows for a linear and stable reconstruction of any vector $x\in H$ from the linear measurements $(\langle x,x_j\rangle)_{j\in J}$. However, there are many situations where some information in the frame coefficients is lost. In applications where one is using sensors with a fixed dynamic range, any measurement above that range is registered as the maximum, and any measurement below that range is registered as the minimum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. We initiate a frame theoretic approach to saturation recovery in a similar way to what [BCE06] did for phase retrieval. We characterize when saturation recovery is possible, show optimal frames for use with saturation recovery correspond to minimal multi-fold packings in projective space, and prove that the classical frame algorithm may be adapted to this non-linear problem to provide a reconstruction algorithm. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_03237 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Declipping and the recovery of vectors from saturated measurements Alharbi, Wedad Freeman, Daniel Ghoreishi, Dorsa Johnson, Brody Randrianarivony, N. Lovasoa Functional Analysis Numerical Analysis Metric Geometry 42C15, 46T20, 51F30, 94A12 A frame $(x_j)_{j\in J}$ for a Hilbert space $H$ allows for a linear and stable reconstruction of any vector $x\in H$ from the linear measurements $(\langle x,x_j\rangle)_{j\in J}$. However, there are many situations where some information in the frame coefficients is lost. In applications where one is using sensors with a fixed dynamic range, any measurement above that range is registered as the maximum, and any measurement below that range is registered as the minimum. Depending on the context, recovering a vector from such measurements is called either declipping or saturation recovery. We initiate a frame theoretic approach to saturation recovery in a similar way to what [BCE06] did for phase retrieval. We characterize when saturation recovery is possible, show optimal frames for use with saturation recovery correspond to minimal multi-fold packings in projective space, and prove that the classical frame algorithm may be adapted to this non-linear problem to provide a reconstruction algorithm. |
| title | Declipping and the recovery of vectors from saturated measurements |
| topic | Functional Analysis Numerical Analysis Metric Geometry 42C15, 46T20, 51F30, 94A12 |
| url | https://arxiv.org/abs/2402.03237 |