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School of Electrical, Computer and Energy Engineering
PhD Final Oral Defense
Statistical and Dynamical Modeling of Riemannian Trajectories with Application to
Human Movement Analysis
by
Rushil Anirudh
Feb 23rd (Tuesday), 2016.
1.45pm, GWC 487.
Committee:
Dr. Pavan Turaga (chair)
Dr. George Runger
Dr. Douglas Cochran
Dr. Thomas Taylor
Abstract
The data explosion in the past decade is in part due to the widespread use of rich
sensors that measure different physical phenomenon -- such as gyroscopes that measure
orientation in phones, the Microsoft Kinect measures depth information, etc. A typical
application requires inferring the underlying physical phenomenon from data, which is
done using machine learning. A fundamental assumption in training models is that the
data is Euclidean, i.e. the metric is the standard Euclidean distance governed by the l2
norm. However in many cases this assumption is violated, such as when data lies on
Riemannian manifolds which can be non-Euclidean. For example, gyroscopes measure
rotation information, where different orientations result in moving along the 2-sphere, S2.
The distance between two points cannot be measured by the Euclidean distance, but
rather the length of the shortest arc along the sphere. While the underlying geometry
accounts for the non-linearity, accurate analysis of human activity also requires temporal
information to be taken into account. Human movement has a natural interpretation as a
trajectory on the underlying feature manifold, as it evolves smoothly in time. A
commonly occurring theme in many emerging problems is the need to represent,
compare, and manipulate such trajectories in a manner that respects the geometric
constraints. This dissertation is a comprehensive treatise on modeling Riemannian
trajectories to understand and exploit their statistical and dynamical properties. Such
properties allow us to formulate novel representations for Riemannian trajectories. For
example, the physical constraints on human movement are rarely considered, which
results in an unnecessarily large space of features, making search, classification and other
applications more complicated. Exploiting statistical properties can help us understand
the true space of such trajectories. In applications such as stroke rehabilitation where
there is a need to differentiate between very similar kinds of movement, dynamical
properties can be much more effective. In this regard, we propose a generalization to the
Lyapunov exponent to Riemannian manifolds and show its effectiveness for human
activity analysis. The theory developed in this thesis naturally leads to several benefits in
areas such as data mining, compression, dimensionality reduction, classification, and
regression.