无需带限约束的样本唯一小波符号恢复
Unique wavelet sign retrieval from samples without bandlimiting
AI总结：
本文证明实值信号可在无带限约束下，通过由Poisson小波及其Hilbert变换线性组合构成的三个小波框架系数幅值实现全局符号意义下的唯一恢复，而复值信号则无法做到。
摘要：
我们研究当分析小波为实值时，从其小波框架系数的幅值中恢复信号的问题。我们证明，对于任意 $α>1,β>0$ 且满足 $β\ln(α)\leq 4π/(1+4p)$，$p>0$，当三个小波 $ϕ_i$ 是 $p$ 阶 Poisson 小波 $P_p$ 及其 Hilbert 变换 $\mathscr{H}P_p$ 的适当线性组合时，每个实值信号都可以从其多小波框架系数 \\[ \{\lvert \mathcal{W}_{ϕ_i} f(α^{m}βn,α^{m}) \rvert: i\in\{1,2,3\}, m,n\in\mathbb{Z}\} \\] 中在全局符号意义下被唯一恢复。对于复值信号，我们发现对于任何参数选择 $α>1,β>0$ 以及任何窗函数，这都是不可能的。与现有关于小波符号恢复的文献相比，我们的唯一性结果不需要任何带限约束或关于实值信号的其他先验知识，即可保证其从小波系数绝对值中被唯一恢复。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
