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

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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.