Statistical and Dynamical Modeling of Riemannian Trajectories with Application to Human Movement Analysis

abstract: The data explosion in the past decade is in part due to the widespread use of rich sensors that measure various physical phenomenon -- gyroscopes that measure orientation in phones and fitness devices, the Microsoft Kinect which 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 L-2 norm. However in many cases this assumption is violated, when the data lies on non Euclidean spaces such as Riemannian manifolds. 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 \emph{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 \emph{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.Dissertation/ThesisDoctoral Dissertation Electrical Engineering 201

Visually Augmented Navigation for Autonomous Underwater Vehicles

As autonomous underwater vehicles (AUVs) are becoming routinely used in an exploratory context for ocean science, the goal of visually augmented navigation (VAN) is to improve the near-seafloor navigation precision of such vehicles without imposing the burden of having to deploy additional infrastructure. This is in contrast to traditional acoustic long baseline navigation techniques, which require the deployment, calibration, and eventual recovery of a transponder network. To achieve this goal, VAN is formulated within a vision-based simultaneous localization and mapping (SLAM) framework that exploits the systems-level complementary aspects of a camera and strap-down sensor suite. The result is an environmentally based navigation technique robust to the peculiarities of low-overlap underwater imagery. The method employs a view-based representation where camera-derived relative-pose measurements provide spatial constraints, which enforce trajectory consistency and also serve as a mechanism for loop closure, allowing for error growth to be independent of time for revisited imagery. This article outlines the multisensor VAN framework and demonstrates it to have compelling advantages over a purely vision-only approach by: 1) improving the robustness of low-overlap underwater image registration; 2) setting the free gauge scale; and 3) allowing for a disconnected camera-constraint topology.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/86054/1/reustice-16.pd