The focus of this thesis is to provide an introduction to Lie groups—the study of continuous symmetries—and related concepts. Starting with topology and geometry, progressing through the construction of Cayley-Dickson algebras, the thesis ultimately explores a few of the most common Lie groups. Finally, geometric and visual interpretations of SO(3) and SU(2), as well as the relations between them, are investigated.

abstract: Continuous monitoring of sensor data from smart phones to identify human activities and gestures, puts a heavy load on the smart phone's power consumption. In this research study, the non-Euclidean geometry of the rich sensor data obtained from the user's smart phone is utilized to perform compressive analysis and efficient classification of human activities by employing machine learning techniques. We are interested in the generalization of classical tools for signal approximation to newer spaces, such as rotation data, which is best studied in a non-Euclidean setting, and its application to activity analysis. Attributing to the non-linear nature of the rotation data space, which involve a heavy overload on the smart phone's processor and memory as opposed to feature extraction on the Euclidean space, indexing and compaction of the acquired sensor data is performed prior to feature extraction, to reduce CPU overhead and thereby increase the lifetime of the battery with a little loss in recognition accuracy of the activities. The sensor data represented as unit quaternions, is a more intrinsic representation of the orientation of smart phone compared to Euler angles (which suffers from Gimbal lock problem) or the computationally intensive rotation matrices. Classification algorithms are employed to classify these manifold sequences in the non-Euclidean space. By performing customized indexing (using K-means algorithm) of the evolved manifold sequences before feature extraction, considerable energy savings is achieved in terms of smart phone's battery life. / Dissertation/Thesis / M.S. Electrical Engineering 2014