ATD: Geometric and Statistical Data Analysis on Special Manifolds for Threat Detection

这项建议涉及开发新的几何算法，用于探测和分类来自空气传播的生物剂和化学剂的威胁。 研究人员提出了一个数学框架，其中心是将与流式高光谱图像相关的大量数据集编码为特殊流形上的点，例如，作为Grassmann，Stiefel以及旗流形的表示。 在这种情况下，将制定算法计算描述性统计。 这种算法的一个著名例子是由Karcher介绍的，用于计算格拉斯曼流形上一组点的平均值。 研究人员主要关注的是开发新的算法，这些算法在格拉斯曼人身上具有改进的计算特性，以及可以应用于，例如，Stiefel和旗流形。 这些算法将被设计用于真实的时间非常大的数据集，并将使用国防威胁减少局提供的时间演变的高光谱数据集进行评估。这些包括（但不限于）使用法布里-珀罗干涉仪和频率捷变激光雷达获得的数据。拟议的跨学科研究计划解决了与国家安全有关的主要挑战，即， 根据真实的观测数据查明和评估环境中的化学和生物风险。 提出了用于探索大量化学和生物高光谱数据的新数学工具，以协助进行威胁检测和表征。该研究计划的主要目标是应用这些工具，以超越目前用于生物和化学威胁分类的技术的性能能力。 预计这项研究计划的结果将有助于与国家安全有关的其他应用，如超光谱图像数据异常检测。

This proposal concerns the development of new geometric algorithms for detecting and classifying threats from airborne biological agents and chemical agents. The investigators propose a mathematical framework centered on encoding massive data sets associated with streaming hyperspectral imagery as points on special manifolds, e.g., as representations on Grassmann, Stiefel as well as flag manifolds. In this setting, algorithms will be developed for computing descriptive statistics. A well-known example of such an algorithm was introduced by Karcher to compute the mean of a set of points on a Grassmann manifold. The investigators are primarily concerned with developing new algorithms with improved computational properties on Grassmannians, as well algorithms that can be applied to, e.g., Stiefel and flag manifolds. These algorithms will be designed to be applied to very large data sets in real time and will be evaluated using temporally-evolving hyperspectral data sets made available by the Defense Threat Reduction Agency. These include (but are not limited to) data acquired using a Fabry-Perot Interferometer and Frequency Agile Lidar.The proposed interdisciplinary research program addresses a major challenge related to National Security, i.e., identifying and assessing chemical and biological risks in the environment from observational data in real time. New mathematical tools for exploring massive quantities of chemical and biological hyperspectral data are proposed to assist with threat detection and characterization. A primary goal of the research program is to apply these tools to exceed performance capabilities of current techniques used for classification of biological and chemical threats. It is anticipated that the results of this research program will be useful to other applications related to National Security such as detection of anomalies in data beyond hyperspectral imagery.