Foundations of Computational Mathematics ( IF 3 ) Pub Date : 2024-02-08 , DOI: 10.1007/s10208-024-09640-3 Philipp Grohs , Lukas Liehr
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Due to its appearance in a remarkably wide field of applications, such as audio processing and coherent diffraction imaging, the short-time Fourier transform (STFT) phase retrieval problem has seen a great deal of attention in recent years. A central problem in STFT phase retrieval concerns the question for which window functions \(g \in {L^2({\mathbb R}^d)}\) and which sampling sets \(\Lambda \subseteq {\mathbb R}^{2d}\) is every \(f \in {L^2({\mathbb R}^d)}\) uniquely determined (up to a global phase factor) by phaseless samples of the form
$$\begin{aligned} |V_gf(\Lambda )| = \left\{ |V_gf(\lambda )|: \lambda \in \Lambda \right\} , \end{aligned}$$where \(V_gf\) denotes the STFT of f with respect to g. The investigation of this question constitutes a key step towards making the problem computationally tractable. However, it deviates from ordinary sampling tasks in a fundamental and subtle manner: recent results demonstrate that uniqueness is unachievable if \(\Lambda \) is a lattice, i.e \(\Lambda = A{\mathbb Z}^{2d}, A \in \textrm{GL}(2d,{\mathbb R})\). Driven by this discretization barrier, the present article centers around the initiation of a novel sampling scheme which allows for unique recovery of any square-integrable function via phaseless STFT-sampling. Specifically, we show that square-root lattices, i.e., sets of the form
$$\begin{aligned} \Lambda = A \left( \sqrt{{\mathbb Z}} \right) ^{2d}, \ \sqrt{{\mathbb Z}} = \{ \pm \sqrt{n}: n \in {\mathbb N}_0 \}, \end{aligned}$$guarantee uniqueness of the STFT phase retrieval problem. The result holds for a large class of window functions, including Gaussians
中文翻译：
方根格子上的无相采样
由于其在音频处理和相干衍射成像等非常广泛的应用领域的出现，短时傅里叶变换（STFT）相位恢复问题近年来引起了广泛的关注。 STFT 相位检索的一个中心问题涉及哪些窗口函数\(g \in {L^2({\mathbb R}^d)}\)和哪些采样集\(\Lambda \subseteq {\mathbb R} ^{2d}\)是每个\(f \in {L^2({\mathbb R}^d)}\)由以下形式的无相样本唯一确定（直到全局相位因子）
$$\begin{对齐} |V_gf(\Lambda )| = \left\{ |V_gf(\lambda )|: \lambda \in \Lambda \right\} , \end{对齐}$$其中\(V_gf\)表示f相对于g的 STFT 。对这个问题的研究是使问题易于计算处理的关键一步。然而，它在根本上和微妙的方式上偏离了普通的采样任务：最近的结果表明，如果\(\Lambda \)是一个格子，即\(\Lambda = A{\mathbb Z}^{2d}，则唯一性是无法实现的， A \in \textrm{GL}(2d,{\mathbb R})\)。在这种离散化障碍的驱动下，本文围绕一种新颖的采样方案的启动展开，该方案允许通过无相 STFT 采样对任何平方可积函数进行独特的恢复。具体来说，我们证明了平方根格，即形式的集合
$$\begin{对齐} \Lambda = A \left( \sqrt{{\mathbb Z}} \right) ^{2d}, \ \sqrt{{\mathbb Z}} = \{ \pm \sqrt{n }: n \in {\mathbb N}_0 \}, \end{对齐}$$保证STFT相位检索问题的唯一性。结果适用于一大类窗函数，包括高斯函数
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