Education is not the filling of a pail, it is the lighting of a fire. - W. B. Yeats

 

NEWS in the Department of Mathematics
Graduate Announcements

 

Robert Arns - PhD preliminary examination
Date: Friday, December 11, 2015
Place: Weber 201
Time: 11:00 a.m.


Title: On the Formulation and Uses of SVD-Based Generalized Curvatures

Advisor: Dr. Michael Kirby
Co-Advisor:  Dr. Chris Peterson


Committee: Dr. Dan Bates, Dr. Raoul Reiser


Abstract: In this dissertation proposal 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 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. Since the curve is not known, the mean-center produces a better approximation of the Frenet Frame as the effect of noise is generally reduced. The second modification is an adaptive method for selecting the size of the window to use for the singular value decomposition. This allows us to use a large window size when curvatures are small, thereby helping to reduce the effects of noise, and to use a small window size when curvatures are large, thereby best capturing the local information. 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 sensors 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 unique 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. This allows us to segment the time-series into individual motions.

 

Robert Arns - PhD preliminary examination
Date: Friday, December 11, 2015
Place: Weber 201
Time: 11:00 a.m.

Title: On the Formulation and Uses of SVD-Based Generalized Curvatures

Advisor: Dr. Michael Kirby
Co-Advisor:  Dr. Chris Peterson

Committee: Dr. Dan Bates, Dr. Raoul Reiser

Abstract: In this dissertation proposal 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 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. Since the curve is not known, the mean-center produces a better approximation of the Frenet Frame as the effect of noise is generally reduced. The second modification is an adaptive method for selecting the size of the window to use for the singular value decomposition. This allows us to use a large window size when curvatures are small, thereby helping to reduce the effects of noise, and to use a small window size when curvatures are large, thereby best capturing the local information. 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 sensors 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 unique 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. This allows us to segment the time-series into individual motions.


Tim Marrinan - PhD preliminary examination
Date: Friday, December 18, 2015
Place: Weber 201
Time: 11:00 a.m.

Title:  Grassmann, flag, and Schubert varieties in applications

Advisor: Dr. Michael Kirby
Co-Advisor:  Dr. Chris Peterson
Committee: Dr. Dan Bates, Dr. Bruce Draper, Dr. Ross Beveridge, Dr. Mahmood Azimi-Sadjadi


The proposed dissertation will develop mathematical tools for solving problems in signal processing and pattern recognition where data can be represented by points on Grassmann manifolds.  This work will build on the growing canon of techniques for analyzing and optimizing over data on Riemannian manifolds, and specifically the method referred to as the flag mean.  The flag mean finds a nested sequence of vector spaces that minimize a geometric optimization criterion based on the projection Frobenius norm to find an average representation for a collection of linear subspaces of possibly different dimensions.

The proposed dissertation will expand these ideas by showing implementable, real-life applications of the flag mean to chemical plume detection in long-wave infrared hyperspectral imagery.  It will build on the idea of preserving importance information in a flag structure by defining a data 'attribute' as an appropriate Schubert variety, and will utilize this definition via a novel algorithm for discovering attributes in data without human supervision that is motivated by the well known Random Sample Consensus paradigm (RanSaC).  The Schubert variety definition of attribute will also be employed in a novel algorithm for adaptive visual sort and summary.  That is, a method for creating an updatable 2-dimensional spatialization that visualizes relationships between image data.  Finally, the proposed dissertation will solve the problem of Schubert variety constrained optimization on the Grassmann manifold for a particular class of cost functions.