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.
This research project concerned the development of basic mathematical tools to be incorporated into algorithms for the detection and classification of anomalies in data cubes, e.g., video sequences or hyper-spectral imagery. The mathematical tools applied are motivated by geometry and permit large data sets to be encoded as points. This approach enables the processing and analyzing of large amounts of data generated by, e.g., wide area surveillance from airborne platforms including UAV-based and satellite-based imaging sensors. The scope of the research program is inherently interdisciplinary. The primary motivation of this work is to address the detection of chemical or biological threats. The award supported an eight month collaboration between Justin Marks, a Colorado State University Ph.D. candidate, and Group 97 at MIT Lincoln Laboratory. Marks relocated to Lexington, MA for the duration of the collaboration to facilitate direct daily interaction with project PI's, including Dimitris Manolakis. The primary interest of the investigation was in the quantification of information in data cubes. Using geometric representations on special manifolds, the similarity of two data cubes can be computed in a manner analogous to the similarity between two data points. Similarly, one can propose a meaning for the average of a set of data cubes. Specifically, new algorithms to compute average subspaces were developed including the Normal Mean, the Projection Mean, and the Flag Mean. These objects are proposed as integral components of threat detection algorithms. The award allowed for the development of the mathematical theory behind these data processing ideas that should facilitate their application. Investigations were also conducted to examine and exploit the (nonlinear) geometry of hyperspectral data. Manifold algorithms such as ISOMAP were applied to the data and compared to state-of-the-art statistical methods. The collaboration provided valuable training and experience for Marks both in the context of working on real world problems as well as delivering presentations on advanced mathematics to engineering audiences. Marks' research results were very well received by the DTRA/NSF/NGA community at the annual Algorithms Workshop held in San Diego in November 2012.