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
Statistical Shape Modeling is widely used to study the morphometrics of deformable objects in computer vision and biomedical studies. There are mainly two viewpoints to understand the shapes. On one hand, the outer surface of the shape can be taken as a two-dimensional embedding in space. On the other hand, the outer surface along with its enclosed internal volume can be taken as a three-dimensional embedding of interests. Most studies focus on the surface-based perspective by leveraging the intrinsic features on the tangent plane. But a two-dimensional model may fail to fully represent the realistic properties of shapes with both intrinsic and extrinsic properties. In this thesis, severalStochastic Partial Differential Equations (SPDEs) are thoroughly investigated and several methods are originated from these SPDEs to try to solve the problem of both two-dimensional and three-dimensional shape analyses. The unique physical meanings of these SPDEs inspired the findings of features, shape descriptors, metrics, and kernels in this series of works. Initially, the data generation of high-dimensional shapes, here, the tetrahedral meshes, is introduced. The cerebral cortex is taken as the study target and an automatic pipeline of generating the gray matter tetrahedral mesh is introduced. Then, a discretized Laplace-Beltrami operator (LBO) and a Hamiltonian operator (HO) in tetrahedral domain with Finite Element Method (FEM) are derived. Two high-dimensional shape descriptors are defined based on the solution of the heat equation and Schrödinger’s equation. Considering the fact that high-dimensional shape models usually contain massive redundancies, and the demands on effective landmarks in many applications, a Gaussian process landmarking on tetrahedral meshes is further studied. A SIWKS-based metric space is used to define a geometry-aware Gaussian process. The study of the periodic potential diffusion process further inspired the idea of a new kernel call the geometry-aware convolutional kernel. A series of Bayesian learning methods are then introduced to tackle the problem of shape retrieval and classification. Experiments of every single item are demonstrated. From the popular SPDE such as the heat equation and Schrödinger’s equation to the general potential diffusion equation and the specific periodic potential diffusion equation, it clearly shows that classical SPDEs play an important role in discovering new features, metrics, shape descriptors and kernels. I hope this thesis could be an example of using interdisciplinary knowledge to solve problems.
ContributorsFan, Yonghui (Author) / Wang, Yalin (Thesis advisor) / Lepore, Natasha (Committee member) / Turaga, Pavan (Committee member) / Yang, Yezhou (Committee member) / Arizona State University (Publisher)
Created2021