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CCF-BSF:CIF:Small:Signal Processing and Machine Learning on Manifolds, with Applications to Invariant Detection and Covariant Estimation

CCF-BSF:CIF:Small：流形上的信号处理和机器学习，及其在不变检测和协变估计中的应用

基本信息

批准号：
1712788
负责人：
Louis Scharf
金额：
$ 45万
依托单位：
Colorado State University
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-01 至 2020-12-31
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1712788&HistoricalAwards=false
关键词：
CCF BSF CIF Small Signal

项目摘要

In many fields of engineering and applied science the problem is to extract relevant information from a signal or image. Certainly this describes the problem of identifying cyber attackers in data networks, spotting objects of interest in closed-circuit TV records, and classifying anomalies in medical images. The goal of this project is to develop a signal processing theory for detecting and classifying images that have undergone geometric transformations. Practical and topical examples are medical features viewed in variable magnification and orientation, and images of people in arbitrary orientations in crowded scenes. The solution to classification problems under such imaging conditions will advance medical practice and national defense. As a broader impact, the project prepares students for careers in mathematics and electrical engineering, with an expertise in signal processing and imaging science.This project develops a theory of matched manifold detectors, based on a universal manifold embedding that extracts a subspace basis from an image. The basis itself codes for the coordinate transformation of the image, but its span is invariant to the transformation. Consequently the extracted subspace is an invariant statistic for detection, and the basis is a covariant statistic for the parameters of the transformation. Classification is then a problem of subspace matching on a Grassmann manifold, and identification of coordinate transformation is a problem of analyzing a subspace basis in a Stiefel manifold. We aim to adapt this theory to other problems where a transformation group turns out an orbit of images, all of which are to be classified as equivalent. The objective is to develop a theory of signal processing on manifolds that is as broad in its scope and as precise in its methodologies as modern subspace signal processing. Such a theory will augment statistical reasoning with geometrical reasoning, and bring new mathematical methods into play.

在工程和应用科学的许多领域中，问题是从信号或图像中提取相关信息。当然，这描述了识别数据网络中的网络攻击者、在闭路电视记录中发现感兴趣的对象以及对医学图像中的异常进行分类的问题。该项目的目标是开发一种信号处理理论，用于检测和分类经过几何变换的图像。实际和热门示例包括以可变放大倍率和方向查看的医学特征，以及拥挤场景中任意方向的人物图像。这种成像条件下分类问题的解决将推动医疗实践和国防。作为更广泛的影响，该项目为学生在数学和电气工程领域的职业生涯做好准备，并具备信号处理和成像科学方面的专业知识。该项目基于从图像中提取子空间基础的通用流形嵌入，开发了匹配流形检测器的理论。基本身编码图像的坐标变换，但其跨度对于变换是不变的。因此，提取的子空间是用于检测的不变统计量，并且基础是变换参数的协变统计量。那么分类就是格拉斯曼流形上的子空间匹配问题，而坐标变换的识别就是分析斯蒂菲尔流形上的子空间基的问题。我们的目标是将该理论应用于其他问题，其中变换群产生图像轨道，所有这些都被归类为等效的。目标是发展一种流形信号处理理论，其范围与现代子空间信号处理一样广泛，方法论也一样精确。这样的理论将用几何推理增强统计推理，并带来新的数学方法。