基于小波和其他框架的无符号系数重构实值函数
Reconstructing real-valued functions from unsigned coefficients with respect to wavelet and other frames
摘要：
在本文中，我们考虑以下问题的相位恢复：给定一个集合的实值带限函数$\{ψ_λ\}_{λ\in Λ}\subset L^2(\mathbb{R}^d)$，构成一个半离散框架，我们问是否任何实值函数$f \in L^2(\mathbb{R}^d)$可以唯一地恢复从其无符号卷积${\{|f \ast ψ_λ|\}_{λ\in Λ}}$。我们发现，在一些温和的假设下的半离散框架，如果$f$在$\infty$指数衰减，它足以知道$|f \ast ψ_λ|$在适当的精细格唯一地确定$f$（到一个全球性的符号因子）。我们进一步建立了我们的重建问题的局部稳定性。最后，对于两个具体的例子（离散）帧的$L^2(\mathbb{R}^d)$，$d=1,2$，我们表明，通过足够的过采样得到一个帧，使任何实值函数与指数衰减可以唯一地恢复其无符号帧系数。
英文摘要：
In this paper we consider the following problem of phase retrieval: Given a collection of real-valued band-limited functions $\{ψ_λ\}_{λ\in Λ}\subset L^2(\mathbb{R}^d)$ that constitutes a semi-discrete frame, we ask whether any real-valued function $f \in L^2(\mathbb{R}^d)$ can be uniquely recovered from its unsigned convolutions ${\{|f \ast ψ_λ|\}_{λ\in Λ}}$. We find that under some mild assumptions on the semi-discrete frame and if $f$ has exponential decay at $\infty$, it suffices to know $|f \ast ψ_λ|$ on suitably fine lattices to uniquely determine $f$ (up to a global sign factor). We further establish a local stability property of our reconstruction problem. Finally, for two concrete examples of a (discrete) frame of $L^2(\mathbb{R}^d)$, $d=1,2$, we show that through sufficient oversampling one obtains a frame such that any real-valued function with exponential decay can be uniquely recovered from its unsigned frame coefficients.
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