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| Main Authors: | , , |
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| Format: | Preprint |
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2023
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| Online Access: | https://arxiv.org/abs/2302.08129 |
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| _version_ | 1866911939384311808 |
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| author | Alaifari, Rima Bartolucci, Francesca Wellershoff, Matthias |
| author_facet | Alaifari, Rima Bartolucci, Francesca Wellershoff, Matthias |
| contents | We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multi-wavelet frame coefficients \[ \{\lvert \mathcal{W}_{ϕ_i} f(α^{m}βn,α^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\} \] for every $α>1,β>0$ with $β\ln(α)\leq 4π/(1+4p)$, $p>0$, when the three wavelets $ϕ_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathscr{H}P_p$. For complex-valued signals we find that this is not possible for any choice of the parameters $α>1,β>0$, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2302_08129 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Unique wavelet sign retrieval from samples without bandlimiting Alaifari, Rima Bartolucci, Francesca Wellershoff, Matthias Functional Analysis We study the problem of recovering a signal from magnitudes of its wavelet frame coefficients when the analyzing wavelet is real-valued. We show that every real-valued signal can be uniquely recovered, up to global sign, from its multi-wavelet frame coefficients \[ \{\lvert \mathcal{W}_{ϕ_i} f(α^{m}βn,α^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\} \] for every $α>1,β>0$ with $β\ln(α)\leq 4π/(1+4p)$, $p>0$, when the three wavelets $ϕ_i$ are suitable linear combinations of the Poisson wavelet $P_p$ of order $p$ and its Hilbert transform $\mathscr{H}P_p$. For complex-valued signals we find that this is not possible for any choice of the parameters $α>1,β>0$, and for any window. In contrast to the existing literature on wavelet sign retrieval, our uniqueness results do not require any bandlimiting constraints or other a priori knowledge on the real-valued signals to guarantee their unique recovery from the absolute values of their wavelet coefficients. |
| title | Unique wavelet sign retrieval from samples without bandlimiting |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2302.08129 |