Theses and Dissertations
This collection includes both ASU Theses and Dissertations, submitted by graduate students, and the Barrett, Honors College theses submitted by undergraduate students.
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- Creators: Turaga, Pavan
Recursive Bayesian Estimation on Projective Spaces: Theoretical Foundations and Practical Algorithms
Description
This thesis develops geometrically and statistically rigorous foundations for multivariate analysis and bayesian inference posed on grassmannian manifolds. Requisite to the development of key elements of statistical theory in a geometric realm are closed-form, analytic expressions for many differential geometric objects, e.g., tangent vectors, metrics, geodesics, volume forms. The first part of this thesis is devoted to a mathematical exposition of these. In particular, it leverages the classical work of Alan James to derive the exterior calculus of differential forms on special grassmannians for invariant measures with respect to which integration is permissible. Motivated by various multi-sensor remote sensing applications, the second part of this thesis describes the problem of recursively estimating the state of a dynamical system propagating on the Grassmann manifold. Fundamental to the bayesian treatment of this problem is the choice of a suitable probability distribution to a priori model the state. Using the Method of Maximum Entropy, a derivation of maximum-entropy probability distributions on the state space that uses the developed geometric theory is characterized. Statistical analyses of these distributions, including parameter estimation, are also presented. These probability distributions and the statistical analysis thereof are original contributions. Using the bayesian framework, two recursive estimation algorithms, both of which rely on noisy measurements on (special cases of) the Grassmann manifold, are the devised and implemented numerically. The first is applied to an idealized scenario, the second to a more practically motivated scenario. The novelty of both of these algorithms lies in the use of thederived maximumentropy probability measures as models for the priors. Numerical simulations demonstrate that, under mild assumptions, both estimation algorithms produce accurate and statistically meaningful outputs. This thesis aims to chart the interface between differential geometry and statistical signal processing. It is my deepest hope that the geometric-statistical approach underlying this work facilitates and encourages the development of new theories and new computational methods in geometry. Application of these, in turn, will bring new insights and bettersolutions to a number of extant and emerging problems in signal processing.
ContributorsCrider, Lauren N (Author) / Cochran, Douglas (Thesis advisor) / Kotschwar, Brett (Committee member) / Scharf, Louis (Committee member) / Taylor, Thomas (Committee member) / Turaga, Pavan (Committee member) / Arizona State University (Publisher)
Created2021
Description
Today's world is seeing a rapid technological advancement in various fields, having access to faster computers and better sensing devices. With such advancements, the task of recognizing human activities has been acknowledged as an important problem, with a wide range of applications such as surveillance, health monitoring and animation. Traditional approaches to dynamical modeling have included linear and nonlinear methods with their respective drawbacks. An alternative idea I propose is the use of descriptors of the shape of the dynamical attractor as a feature representation for quantification of nature of dynamics. The framework has two main advantages over traditional approaches: a) representation of the dynamical system is derived directly from the observational data, without any inherent assumptions, and b) the proposed features show stability under different time-series lengths where traditional dynamical invariants fail.
Approximately 1\% of the total world population are stroke survivors, making it the most common neurological disorder. This increasing demand for rehabilitation facilities has been seen as a significant healthcare problem worldwide. The laborious and expensive process of visual monitoring by physical therapists has motivated my research to invent novel strategies to supplement therapy received in hospital in a home-setting. In this direction, I propose a general framework for tuning component-level kinematic features using therapists’ overall impressions of movement quality, in the context of a Home-based Adaptive Mixed Reality Rehabilitation (HAMRR) system.
The rapid technological advancements in computing and sensing has resulted in large amounts of data which requires powerful tools to analyze. In the recent past, topological data analysis methods have been investigated in various communities, and the work by Carlsson establishes that persistent homology can be used as a powerful topological data analysis approach for effectively analyzing large datasets. I have explored suitable topological data analysis methods and propose a framework for human activity analysis utilizing the same for applications such as action recognition.
Approximately 1\% of the total world population are stroke survivors, making it the most common neurological disorder. This increasing demand for rehabilitation facilities has been seen as a significant healthcare problem worldwide. The laborious and expensive process of visual monitoring by physical therapists has motivated my research to invent novel strategies to supplement therapy received in hospital in a home-setting. In this direction, I propose a general framework for tuning component-level kinematic features using therapists’ overall impressions of movement quality, in the context of a Home-based Adaptive Mixed Reality Rehabilitation (HAMRR) system.
The rapid technological advancements in computing and sensing has resulted in large amounts of data which requires powerful tools to analyze. In the recent past, topological data analysis methods have been investigated in various communities, and the work by Carlsson establishes that persistent homology can be used as a powerful topological data analysis approach for effectively analyzing large datasets. I have explored suitable topological data analysis methods and propose a framework for human activity analysis utilizing the same for applications such as action recognition.
ContributorsVenkataraman, Vinay (Author) / Turaga, Pavan (Thesis advisor) / Papandreou-Suppappol, Antonia (Committee member) / Krishnamurthi, Narayanan (Committee member) / Li, Baoxin (Committee member) / Arizona State University (Publisher)
Created2016