ATD: Frame-Theoretic Information Fusion for Threat Detection

本项目解决了希尔伯特空间框架数学领域中几个新出现的问题。这项研究有两个重点。第一个要点涉及使用框架理论来提供数据的稀疏表示。在这里，该项目关注的是该领域的首要问题：提供可实现大小的受限等距性质（RIP）矩阵的可验证构造；寻找真实和复杂格拉斯曼框架的算法结构；找出以尽可能稀疏的方式表示一组给定数据的帧。第二个推力涉及使用框架理论来融合从不同传感器模式收集的信息。在这里，工作集中在多模态数据处理的几个基本开放问题上，例如如何从数学上确定给定的模态集合产生最佳检测精度，以及如何注册来自不同模态的数据。传感器硬件的最新进展为探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。与此同时，误判此类威胁的代价是极其高昂的。此外，人类情报分析师越来越被越来越大的数据集所淹没。该项目的目的是开发新的数学方法，以帮助尽可能多地自动化这种威胁检测过程，使人类分析师能够专注于最高级别的任务。特别是，对稀疏表示的良好理解使人们能够将非常复杂的现象分解为相对较少的更容易理解的部分。同时，对多模态数据的良好理解允许人们将来自许多不同传感器的数据结合起来，以便做出比单独使用任何一个传感器的数据更明智的决策。

This project addresses several emerging problems in the mathematical field of Hilbert space frames. This research has two thrusts. The first thrust involves using frame theory to provide sparse representations of data. Here, the project is concerned with the paramount problems of the field: to provide verifiable constructions of restricted isometry property (RIP) matrices of implementable size; to find algorithmic constructions of real and complex Grassmannian frames; to find frames which, in the sparsest possible manner, represent a given set of data. The second thrust involves using frame theory to fuse information gathered from disparate sensor modalities. Here, the work focuses on several fundamental open problems of multimodal data processing, such as how to mathematically determine what given collections of modalities yields optimal detection accuracies, and how data from different modes should be registered.Recent advances in sensor hardware provide an unprecedented capability to detect the production, deployment and use of chemical and biological weapons. At the same time, the cost of misdetecting such threats is extremely high. Moreover, human intelligence analysts are increasingly overwhelmed by ever-larger data sets. The purpose of this project is to develop new mathematics which will help automate as much of this threat detection process as is possible, freeing human analysts to focus on the most high-level tasks. In particular, a good understanding of sparse representations allows one to break very complicated phenomena into a relatively small number of much-easier-to-understand pieces. Meanwhile, a good understanding of multimodal data allows one to combine data from many different sensors, in order to make a decision which is wiser than could be made from the data of any single sensor alone.