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

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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.