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Description
This dissertation constructs a new computational processing framework to robustly and precisely quantify retinotopic maps based on their angle distortion properties. More generally, this framework solves the problem of how to robustly and precisely quantify (angle) distortions of noisy or incomplete (boundary enclosed) 2-dimensional surface to surface mappings. This framework builds upon the Beltrami Coefficient (BC) description of quasiconformal mappings that directly quantifies local mapping (circles to ellipses) distortions between diffeomorphisms of boundary enclosed plane domains homeomorphic to the unit disk. A new map called the Beltrami Coefficient Map (BCM) was constructed to describe distortions in retinotopic maps. The BCM can be used to fully reconstruct the original target surface (retinal visual field) of retinotopic maps. This dissertation also compared retinotopic maps in the visual processing cascade, which is a series of connected retinotopic maps responsible for visual data processing of physical images captured by the eyes. By comparing the BCM results from a large Human Connectome project (HCP) retinotopic dataset (N=181), a new computational quasiconformal mapping description of the transformed retinal image as it passes through the cascade is proposed, which is not present in any current literature. The description applied on HCP data provided direct visible and quantifiable geometric properties of the cascade in a way that has not been observed before. Because retinotopic maps are generated from in vivo noisy functional magnetic resonance imaging (fMRI), quantifying them comes with a certain degree of uncertainty. To quantify the uncertainties in the quantification results, it is necessary to generate statistical models of retinotopic maps from their BCMs and raw fMRI signals. Considering that estimating retinotopic maps from real noisy fMRI time series data using the population receptive field (pRF) model is a time consuming process, a convolutional neural network (CNN) was constructed and trained to predict pRF model parameters from real noisy fMRI data
ContributorsTa, Duyan Nguyen (Author) / Wang, Yalin (Thesis advisor) / Lu, Zhong-Lin (Committee member) / Hansford, Dianne (Committee member) / Liu, Huan (Committee member) / Li, Baoxin (Committee member) / Arizona State University (Publisher)
Created2022
Description
Functional magnetic resonance imaging (fMRI) has been widely used to measure the retinotopic organization of early visual cortex in the human brain. Previous studies have identified multiple visual field maps (VFMs) based on statistical analysis of fMRI signals, but the resulting geometry has not been fully characterized with mathematical models. This thesis explores using concepts from computational conformal geometry to create a custom software framework for examining and generating quantitative mathematical models for characterizing the geometry of early visual areas in the human brain. The software framework includes a graphical user interface built on top of a selected core conformal flattening algorithm and various software tools compiled specifically for processing and examining retinotopic data. Three conformal flattening algorithms were implemented and evaluated for speed and how well they preserve the conformal metric. All three algorithms performed well in preserving the conformal metric but the speed and stability of the algorithms varied. The software framework performed correctly on actual retinotopic data collected using the standard travelling-wave experiment. Preliminary analysis of the Beltrami coefficient for the early data set shows that selected regions of V1 that contain reasonably smooth eccentricity and polar angle gradients do show significant local conformality, warranting further investigation of this approach for analysis of early and higher visual cortex.
ContributorsTa, Duyan (Author) / Wang, Yalin (Thesis advisor) / Maciejewski, Ross (Committee member) / Wonka, Peter (Committee member) / Arizona State University (Publisher)
Created2013
Description
This thesis introduces new techniques for clustering distributional data according to their geometric similarities. This work builds upon the optimal transportation (OT) problem that seeks global minimum cost for matching distributional data and leverages the connection between OT and power diagrams to solve different clustering problems. The OT formulation is based on the variational principle to differentiate hard cluster assignments, which was missing in the literature. This thesis shows multiple techniques to regularize and generalize OT to cope with various tasks including clustering, aligning, and interpolating distributional data. It also discusses the connections of the new formulation to other OT and clustering formulations to better understand their gaps and the means to close them. Finally, this thesis demonstrates the advantages of the proposed OT techniques in solving machine learning problems and their downstream applications in computer graphics, computer vision, and image processing.
ContributorsMi, Liang (Author) / Wang, Yalin (Thesis advisor) / Chen, Kewei (Committee member) / Karam, Lina (Committee member) / Li, Baoxin (Committee member) / Turaga, Pavan (Committee member) / Arizona State University (Publisher)
Created2020
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