Phaseless Reconstruction and Geometric Analysis of Frames

框架的无相重建和几何分析

• 批准号: 1413249
• 负责人: Radu Balan
• 金额: $ 35.65万
• 依托单位: University of Maryland, College Park
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-15 至 2018-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1413249&HistoricalAwards=false
• 关键词: Phaseless; Reconstruction; Geometric; Analysis; Frames

The investigator studies two problems, each exploiting redundancy of representations in mathematics and engineering, and develops new methods to recover a signal from a nonlinear processing scheme. The first problem is related to signal reconstruction from magnitudes of a redundant linear representation (the so-called phase retrieval problem). The second problem involves a geometric analysis of frames and connections to deep problems in mathematics (such as the Kadison-Singer problem). This analysis leads to faster methods that offer better quality and resolution of the reconstructed signals in applications from X-ray crystallography, data communication on fiber optics, and speech processing. Undergraduate and graduate students involved in this project are trained for a globally competitive STEM workforce by learning to develop new mathematical tools to solve real-world problems. The investigator studies two problems, each exploiting redundancy of representations in mathematics and engineering. The first leads to new methods to recover a signal from a nonlinear processing scheme. Recently two far-reaching discoveries have been made that connect the nonlinear information (magnitudes of frame coefficients) to certain scalar products in larger embedding spaces. This way the original problem of recovering a signal, which is fundamentally nonlinear, is recast into a linear reconstruction problem coupled with a rank-one approximation problem. When the linear redundant representation is associated with a group representation (such as Weyl-Heisenberg, or windowed Fourier transform), then the relevant tensor operators inherit this invariance property. Thus a fast (nonlinear) reconstruction algorithm is possible. This approach suggests a new signal representation model, where signals are not represented simply by vectors in a Hilbert space, but rather by operators in a larger dimensional Hilbert-Schmidt-like space, similar to the quantum state theory. The methods developed here borrow from a wide range of mathematical areas such as harmonic analysis, operator theory, and algebraic geometry. In turn this approach allows for stable and efficient solutions relevant to areas of electrical engineering as diverse as array signal processing, speech processing, quantum computing, and X-ray crystallography. The second problem expands the solution of the Kadison-Singer problem in a different direction in frame theory. Specifically the issue is to "thin out" frames to subsets that remain frames and have density arbitrary close to one, the critical density associated to a Riesz basis. Such a result belongs to a larger body of results describing the geometry of frame sets. The unifying concept in all these problems is redundancy of representations and atomic decompositions.

研究人员研究了两个问题，每个问题都利用了数学和工程中表示的冗余，并开发了从非线性处理方案中恢复信号的新方法。 第一个问题与从冗余线性表示的幅度进行信号重建有关（所谓的相位恢复问题）。 第二个问题涉及框架的几何分析以及与数学深层问题（例如卡迪森-辛格问题）的联系。 这种分析带来了更快的方法，可以在 X 射线晶体学、光纤数据通信和语音处理等应用中提供更好的重建信号质量和分辨率。 参与该项目的本科生和研究生通过学习开发新的数学工具来解决现实世界问题，接受培训，成为具有全球竞争力的 STEM 劳动力。研究人员研究两个问题，每个问题都利用数学和工程学中的表示冗余。 第一个导致了从非线性处理方案中恢复信号的新方法。 最近取得了两个影响深远的发现，将非线性信息（帧系数的大小）与较大嵌入空间中的某些标量积联系起来。 这样，恢复信号的原始问题（基本上是非线性的）被重新转换为与一阶近似问题相结合的线性重建问题。 当线性冗余表示与群表示（例如 Weyl-Heisenberg 或加窗傅立叶变换）相关联时，相关张量算子继承此不变性。 因此，快速（非线性）重建算法是可能的。 这种方法提出了一种新的信号表示模型，其中信号不是简单地由希尔伯特空间中的向量表示，而是由更大维的类希尔伯特-施密特空间中的算子表示，类似于量子态理论。 这里开发的方法借鉴了广泛的数学领域，例如调和分析、算子理论和代数几何。 反过来，这种方法可以为阵列信号处理、语音处理、量子计算和 X 射线晶体学等不同的电气工程领域提供稳定、高效的解决方案。 第二个问题在框架理论的不同方向上扩展了Kadison-Singer问题的解决方案。 具体来说，问题是将帧“稀疏”为保留帧的子集，并且密度任意接近于一，即与里斯基相关的临界密度。 这样的结果属于描述框架集几何形状的较大结果体。 所有这些问题的统一概念是表示和原子分解的冗余。