NEWS in the Department of Mathematics
Robert Arn - PhD Defense
Date: Tuesday, May 31, 2016
Place: Weber 201
Time: 2:15 p.m.
Title: On the Formulation and Uses of SVD-based Generalized Curvatures
Advisor: Dr. Michael Kirby Co-Advisor: Chris Peterson
Committee: Dr. Dan Bates, Dr. Raoul Reiser
Abstract: In this dissertation we consider the problem of computing generalized curvature values from noisy, discrete data and applications of the provided algorithms. We first establish a connection between the Frenet-Serret Frame, typically defined on an analytical curve, and the vectors from the local Singular Value Decomposition (SVD) of a discretized time-series. Next, we expand upon this connection to relate generalized curvature values, or curvatures, to a scaled ratio of singular values. Initially, the local singular value decomposition is centered on a point of the discretized time-series. This provides for an efficient computation of curvatures when the underlying curve is known. However, when the structure of the curve is not known, for example, when noise is present in the tabulated data, we propose two modifications. The first modification computes the local singular value decomposition on the mean-centered data of a windowed selection of the time-series. We observe that the mean-center version increases the stability of the curvature estimations in the presence of signal noise. The second modification is an adaptive method for selecting the size of the window, or local ball, to use for the singular value decomposition. This allows us to use a large window size when curvatures are small, which reduces the effects of noise thanks to the use of a large number of points in the SVD, and to use a small window size when curvatures are large, thereby best capturing the local curvature. Overall we observe that adapting the window size to the data, enhances the estimates of generalized curvatures. The combination of these two modifications produces a tool for computing generalized curvatures with reasonable precision and accuracy. Finally, we compare our algorithm, with and without modifications, to existing numerical curvature techniques on different types of data such as that from the Microsoft Kinect 2 sensor. To address the topic of action segmentation and recognition, a popular topic within the field of computer vision, we created a new dataset from this sensor showcasing a pose space skeletonized representation of individuals performing continuous human actions as defined by the MSRC-12 challenge. When this data is optimally projected onto a low-dimensional space, we observed each human motion lies on a distinguished line, plane, hyperplane, etc. During transitions between motions, either the dimension of the optimal subspace significantly, or the trajectory of the curve through pose space nearly reverses. We use our methods of computing generalized curvature values to identify these locations, categorized as either high curvatures or changing curvatures. The geometric characterization of the time-series allows us to segment individual,or geometrically distinct, motions. Finally, using these segments, we construct a methodology for selecting motions to conjoin for the task of action classification.