Hilbert Space Frames and their applications

该奖项支持首席研究员在“框架理论”领域的研究计划，这是一个与信号、图像和信息的数学表示有关的课题。这个项目涉及希尔伯特空间框架理论的应用-当今数学中最活跃和最实用的学科之一，应用于纯数学、应用数学、工程学、医学、计算机科学等。框架理论是信息数字化处理和存储的基础。该项目的一部分将开发“相位恢复”的新方法，该方法在X射线结晶学、电子显微镜、天文成像、光学、量子物理等方面都有应用。在计划于2018年发射的新詹姆斯·韦伯太空望远镜中，甚至需要恢复相位来对准镜子。这项研究项目的另一个部分涉及为现代应用开发“融合框架”。最近硬件技术的进步使大量低成本部件的经济生产和部署成为可能，这些部件通过协作能够可靠和高效地运行。无线传感器网络作为一种新技术已经出现，具有实现经济有效和可靠的监视的潜力。雷达成像正在从单一平台向多平台协同发展，以获得更好的性能。传感器硬件的最新进展为检测化学和生物武器的生产、部署和使用提供了前所未有的能力。所有这些应用都涉及大量数据流，这些数据流需要在中央处理器上集成-通常是在多个级别的信息融合中。融合框架就是为了解决这些问题而设计的。相位恢复是一个百年前的问题，它是根据一个函数(如信号或图像)的傅里叶变换的大小来重建它。它的起源来自于这样一个事实，在许多情况下，探测器只能记录强度测量，而丢失了相位。该项目正在利用框架理论将相位恢复推向几个相对较新的方向。确定确保信号或图像能够从其强度测量(在没有任何先验知识的情况下)唯一地重建所需的最小测量数量将为探测器的设计提供至关重要的洞察。该项目还将使用时频分析、量子物理和离散几何等工具的创新组合，提供以严格的数学理论和高效的数值算法为后盾的新颖测量设计。研究的另一个方向将是将现有的用于相位恢复的方法扩展到更高等级的投影，该方法使用基于矢量的测量。这将对晶体孪生有重要的应用。融合框架为围绕信息融合的许多问题提供了巨大的希望，但需要更好的设计、算法和更多的通用性来适应当前的需求。该项目将解决所有这些问题，使融合框架成为主流。

This award supports the research program of the Principal Investigator in the area of "frame theory," a subject concerned with the mathematical representation of signals, images, and information in general. This project concerns applications of Hilbert space frame theory---one of the most active and applied subjects in mathematics today, with applications to pure mathematics, applied mathematics, engineering, medicine, computer science, and more. Frame theory is fundamental to the digitalizing of information for processing and storing. One part of this project will develop new methods for "phase retrieval," which has applications to X-ray crystallography, electron microscopy, astronomical imaging, optics, quantum physics, and more. Phase retrieval will even be needed to align the mirrors in the new James Webb Space Telescope scheduled for launch in 2018. Another part of this research project involves developing "fusion frames" for modern applications. Recent advances in hardware technology have enabled the economic production and deployment of a large number of low-cost components, which through collaboration enable reliable and efficient operation. Wireless sensor networks have emerged as a new technology with the potential to enable cost-effective and reliable surveillance. Radar imaging is moving away from single platform to multiple platforms that cooperate to achieve better performance. Recent advances in sensor hardware provide an unprecedented capability to detect the production, deployment, and use of chemical and biological weapons. All of these applications involve a large number of data streams, which need to be integrated at a central processor---often at multiple levels of information fusion. Fusion frames are being designed to deal with these problems.Phase retrieval is the century-old problem of reconstructing a function (such as a signal or an image) from the magnitude of its Fourier transform. Its origin comes from the fact that in many scenarios, detectors can only record intensity measurements and lose the phases. This project is pushing phase retrieval in several relatively new directions using frame theory. Determining the minimal number of measurements needed to ensure that a signal or image can be uniquely reconstructed from its intensity measurements (without any prior knowledge) will provide crucial insight into designing detectors. This project will also use an innovative combination of tools from time-frequency analysis, quantum physics, and discrete geometry to provide novel measurement designs that are backed up by a rigorous mathematical theory and efficient numerical algorithms. Another direction of the research will be to extend the existing approach used for phase retrieval, which uses vector-based measurements, to higher-rank projections. This will have important applications to crystal twinning. Fusion frames offer great promise to a host of problems surrounding information fusion but need much better designs, algorithms, and much more generality to fit current needs. This project will address all of these problems to bring fusion frames into the mainstream.