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

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。这个项目的目标是开发一个信号处理理论，用于检测和分类经历了几何变换的图像。实际和专题的例子是在可变放大率和方向，以及在拥挤的场景中的任意方向的人的图像中查看的医疗功能。在这样的成像条件下分类问题的解决方案将推进医疗实践和国防。作为一个更广泛的影响，该项目为学生在数学和电气工程的职业生涯做好准备，并具有信号处理和成像科学的专业知识。该项目开发了匹配流形检测器的理论，基于通用流形嵌入，从图像中提取子空间基。基本身对图像的坐标变换进行编码，但其跨度对变换是不变的。因此，所提取的子空间是用于检测的不变统计量，并且基础是用于变换的参数的协变统计量。分类则是格拉斯曼流形上的子空间匹配问题，而坐标变换的识别则是分析斯蒂费尔流形上的子空间基的问题。我们的目标是适应这一理论的其他问题，一个变换组变成一个轨道的图像，所有这些都被归类为等价的。我们的目标是开发一个理论的信号处理流形上，是广泛的范围和精确的方法，作为现代子空间信号处理。这一理论将使统计推理与几何推理相结合，并带来新的数学方法。

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.