The generic crystallographic phase retrieval problem
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866918476510134272 |
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| author | Edidin, Dan Suresh, Arun |
| author_facet | Edidin, Dan Suresh, Arun |
| contents | In this paper we consider the problem of recovering a signal $x \in \mathbb{R}^N$ from its power spectrum assuming that the signal is sparse with respect to a generic basis for $\mathbb{R}^N$. Our main result is that if the sparsity level is at most $\sim\! N/2$ in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is $\sim\! N/4$ then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in $\mathbb{C}^N$ which extend earlier results of Wang and Xu \cite{arXiv:1310.0873}. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2307_06835 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The generic crystallographic phase retrieval problem Edidin, Dan Suresh, Arun Functional Analysis Information Theory Algebraic Geometry 42A10, 94A12, 94A15 In this paper we consider the problem of recovering a signal $x \in \mathbb{R}^N$ from its power spectrum assuming that the signal is sparse with respect to a generic basis for $\mathbb{R}^N$. Our main result is that if the sparsity level is at most $\sim\! N/2$ in this basis then the generic sparse vector is uniquely determined up to sign from its power spectrum. We also prove that if the sparsity level is $\sim\! N/4$ then every sparse vector is determined up to sign from its power spectrum. Analogous results are also obtained for the power spectrum of a vector in $\mathbb{C}^N$ which extend earlier results of Wang and Xu \cite{arXiv:1310.0873}. |
| title | The generic crystallographic phase retrieval problem |
| topic | Functional Analysis Information Theory Algebraic Geometry 42A10, 94A12, 94A15 |
| url | https://arxiv.org/abs/2307.06835 |