Theory and Application of Hilbert Space Frames

本项目涉及相位检索和框架理论相关的一些重要问题。在一些工程应用中，信号的相位在处理过程中会丢失。在项目的第一部分，首席研究员（PI）将解决有关相位检索的几个问题。相位检索在x射线晶体学、电子显微镜、天文成像、光学、x射线断层扫描等领域都有应用，比如在对准新詹姆斯·韦伯太空望远镜的反射镜方面。结合融合框架理论，相位检索有助于设计用于检测化学、生物、放射和核武器的传感器网络。PI还将研究双角紧框架中的基本问题。双角紧框架（BTF）有无数的应用，包括量子态层析成像、量子密码学、通信理论、球面设计和强正则图。这项研究将解决该领域一些最古老的问题，比如150年前的Hadamard猜想，它阻碍了应用。该项目的这一部分应用于设计在短波波段困难（低信噪比加多径传播）条件下工作的数字无线电协议，平衡重复复制，编码孔径光谱，反馈延迟网络，Plackett-Burman实验设计等。这个项目有一个重要的学生培训组成部分，因为几个PI的博士生和一个本科生都直接参与了研究。在这个项目的第一部分，PI将研究相位检索。如果空间中的每个向量都是由其与每个单位向量的内积的模唯一确定的（直到一个通用相位），则一组单位向量进行相位检索。这里的主要目标是构建进行相位检索的向量族，并找出在每个维度中进行相位检索所需的最少向量数。对于二维和三维实空间和复杂空间，答案是已知的，方法是通过对维度的归纳法在所有维度上解决问题。这个项目的第二部分处理双角紧框架，它是单位向量族，具有任意两个不同向量的内积的模取两个值中的一个的性质。目标是构造这样的集合的无限族。PI为此开发了一种新工具。有许多已知的等角紧框架（ETF），新工具将把这些ETF变成btf。项目的最后一部分旨在使用PI开发的解决这个问题的新方法来解决著名的Hadamard猜想。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

This project is concerned with a number of important problems related to phase retrieval and frame theory. In several engineering applications the phase of a signal is lost during processing. In the first part of the project the principal investigator (PI) will resolve several problems concerning phase retrieval. Phase retrieval has applications to x-ray crystallography, electron microscopy, astronomical imaging, optics, x-ray tomography, and more, such as in aligning the mirrors of the new James Webb Space Telescope. Combined with Fusion Frame Theory, phase retrieval helps to design sensor networks for detecting chemical, biological, radiological and nuclear weapons. The PI will also study fundamental problems in biangular tight frames. Biangular tight frames (BTF) have a myriad of applications including quantum state tomography, quantum cryptography, communication theory, spherical designs, and strongly regular graphs. This research will address some of the oldest problems in this area, such as the 150 year old Hadamard Conjecture, which are holding up the applications. This part of the project has applications to digital radio protocols designed to work in difficult (low signal-to-noise ratio plus multipath propagation) conditions on shortwave bands, balanced repeated replication, coded aperture spectrometry, feedback delay networks, Plackett-Burman design of experiments, among others. This project has a significant student training component, as several of the PI's Ph.D. students and one undergraduate student are all directly involved in the research. In the first part of this project the PI will study phase retrieval. A family of unit vectors does phase retrieval if every vector in the space is uniquely determined (up to a universal phase) by the modulus of its inner product with each of the unit vectors. The main goal here is to construct families of vectors doing phase retrieval and to find out the least number of vectors needed to do phase retrieval in each dimension. The answer is known for two- and three-dimensional real and complex spaces and the approach is to solve the problem in all dimensions by induction with respect to the dimension. The second part of this project deals with biangular tight frames, which are families of unit vectors with the property that the modulus of the inner product of any two distinct vectors takes one of two values. The goal is to construct infinite families of such sets. The PI has developed a new tool for this construction. There are many known equiangular tight frames (ETF) and the new tool will turn these ETFs into BTFs. The last part of the project aims to produce a solution to the celebrated Hadamard Conjecture using a new approach for this problem developed by the PI.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.