Geometric Properties of Gabor Frames and Their Applications to the Phase Retrieval Problem
Geometric Properties of Gabor Frames and Their Applications to the Phase Retrieval Problem
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Gabor框架的几何性质及其在相位恢复问题中的应用
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发表时间：
2017
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通讯作者：
Palina Salanevich
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作者：
Palina Salanevich
在这篇论文中，我们解决了两个不同的数学研究领域中出现的问题，即，在有限维的Gabor框架领域，属于应用谐波分析领域，相位恢复问题，属于信号处理领域。我们的工作是受相位恢复问题的启发，这是由真实的世界的应用，如光学，语音识别，天文成像，量子力学和无线通信的动机。与此同时，对这个问题的研究导致了不同领域交叉点上美丽而富有洞察力的数学。本文的重点是研究时频结构框架下的Gabor框架的几何性质及其在相位恢复问题中的作用。尽管相位恢复问题已经被研究了很长一段时间，直到最近很少有人知道如何实现稳定和有效的重建。如今，当测量框架是具有独立框架向量的高斯框架时的情况被充分地研究。与此同时，人们对结构化的、与应用相关的框架的情况知之甚少。其主要原因是结构框架的一些几何性质尚未完全理解。在我们的工作中，我们研究了具有随机窗口的Gabor帧的最佳帧边界和帧顺序统计等帧属性。所得到的结果使我们可以得出结论，随机窗口的Gabor框架的属性往往是非常相似的属性的高斯框架与独立向量，这是最佳的许多应用。我们还设计了一个有效的相位恢复算法从几乎最佳数量的时频结构的测量，并显示其鲁棒性的情况下，测量时被破坏的加性噪声。所构造的算法的鲁棒性分析原来也密切相关的几何性质的Gabor框架。
In this thesis we address questions arising in two different research areas of mathematics, namely, in the area of Gabor frames in finite dimensions, belonging to the field of applied harmonic analysis, and phase retrieval problem, belonging to the field of signal processing. Our work is inspired by phase retrieval problem, which is motivated by real world applications, such as optics, speech recognition, astronomical imaging, quantum mechanics, and wireless communication. At the same time, study of this problem leads to beautiful and insightful mathematics on the intersection of different fields. The focus of this thesis is the investigation of geometric properties of Gabor frames and their role in the phase retrieval problem in the case of time-frequency structured frames. Even though phase retrieval problem has been studied for a long time, until recently very little was known about how to achieve stable and efficient reconstruction. Nowadays, the case when the measurement frame is a Gaussian frame with independent frame vectors is sufficiently well studied. At the same time very little is known about the case of structured, application relevant frames. The main reason for this is that some geometric properties of structured frames are not yet fully understood. In our work, we investigate such frame properties as optimal frame bounds and frame order statistics in the case of Gabor frames with random windows. The obtained results allow us to conclude that the properties of Gabor frames with random windows are often quite similar to the properties of Gaussian frames with independent vectors, which are optimal for many applications. We also design an efficient phase retrieval algorithm from nearly optimal number of time-frequency structured measurements and show its robustness in the case when measurements are corrupted by additive noise. Robustness analysis of the constructed algorithm turns out to be also closely linked to the geometric properties of Gabor frames.