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
Nearly one percent of the population over 65 years of age is living with Parkinson’s disease (PD) and this population worldwide is projected to be approximately nine million by 2030. PD is a progressive neurological disease characterized by both motor and cognitive impairments. One of the most serious challenges for an individual as the disease progresses is the increasing severity of gait and posture impairments since they result in debilitating conditions such as freezing of gait, increased likelihood of falls, and poor quality of life. Although dopaminergic therapy and deep brain stimulation are generally effective, they often fail to improve gait and posture deficits. Several recent studies have employed real-time feedback (RTF) of gait parameters to improve walking patterns in PD. In earlier work, results from the investigation of the effects of RTF of step length and back angle during treadmill walking demonstrated that people with PD could follow the feedback and utilize it to modulate movements favorably in a manner that transferred, at least acutely, to overground walking. In this work, recent advances in wearable technologies were leveraged to develop a wearable real-time feedback (WRTF) system that can monitor and evaluate movements and provide feedback during daily activities that involve overground walking. Specifically, this work addressed the challenges of obtaining accurate gait and posture measures from wearable sensors in real-time and providing auditory feedback on the calculated real-time measures for rehabilitation. An algorithm was developed to calculate gait and posture variables from wearable sensor measurements, which were then validated against gold-standard measurements. The WRTF system calculates these measures and provides auditory feedback in real-time. The WRTF system was evaluated as a potential rehabilitation tool for use by people with mild to moderate PD. Results from the study indicated that the system can accurately measure step length and back angle, and that subjects could respond to real-time auditory feedback in a manner that improved their step length and uprightness. These improvements were exhibited while using the system that provided feedback and were sustained in subsequent trials immediately thereafter in which subjects walked without receiving feedback from the system.
ContributorsMuthukrishnan, Niveditha (Author) / Abbas, James (Thesis advisor) / Krishnamurthi, Narayanan (Thesis advisor) / Shill, Holly A (Committee member) / Honeycutt, Claire (Committee member) / Turaga, Pavan (Committee member) / Ingalls, Todd (Committee member) / Arizona State University (Publisher)
Created2022