Dissertation/Thesis Abstract

Computational Conformal Geometry and Its Applications in Medical Imaging
by Peng, Hao, Ph.D., State University of New York at Stony Brook, 2018, 90; 13425128
Abstract (Summary)

Conformal geometry has deep roots in pure mathematics, such as Complex analysis, Riemann surface theory, Differential geometry and Algebraic topology. Computational conformal geometry also plays an important role in digital geometry processing. In recent years, theory of discrete conformal geometry and algorithms of computational conformal geometry have advanced. A series of practical algorithms have been presented to compute conformal mapping, which has been broadly applied in numerous practical fields, including computer vision and graphics, visualization, medical imaging, and so on. In this dissertation, we present three applications of computational conformal geometry in medical imaging and geometry processing.

In medical imaging, conformal geometry has been applied to surface parameterization and extract intrinsic features for natural objects like brain, colon, spleen and other human organs. We apply conformal geometric methods to find the underlying physical connection between morphologically comparable structures and parameterize surface for further registration and recognition.

We propose a set of feature extraction algorithms for colonic modeling, in order to perform auto-detection of polyps and present accurate endoscopy view for physician to locate polyps, the precursors of colorectal cancer. Based on conformal flattening and modified surface geodesics, the algorithm automatically locates the Teniae Coli, Haustral Folds and extracts centerline, which gives the global geometric structure of a colon wall surface anatomy. More specifically, \emph{auxiliary Riemannian metric} algorithm for Teniae Coli tracking; \emph{geodesic clustering} method for Haustral folding location; \emph{harmonic mass center} method for centerline reconstruction.

In addition to the previous model, we present an adaptive approach for fully automated small intestine cleansing and an alternative solution for colon centerline extraction based on maximum a posteriori expectation-maximization (MAP-EM) partial volume segmentation, level set-based adaptive convolution (LSAC) and distance field based topology analysis. We combined distance field based centerline graph pruning and constrained label optimization for analyzing colon topology. We apply this approach to locate branches, remove noises and extract centerline.

Furthermore, we introduce conformal welding signature for measuring and analyzing the geometric relations among different functional areas on surfaces. Conformal welding shape signature is intrinsic to the Riemannian metric, invariant under conformal deformation, and is capable of encoding complete information of the functional area boundaries. The computational algorithm is based on discrete surface Ricci flow, which has theoretic guarantees for the existence and uniqueness of the solutions. In practice, discrete Ricci flow is equivalent to a convex optimization problem, therefore has high numerical stability.

We propose a novel approach to apply Teichm\"uller space theory and conformal welding method to study brain morphometry in Congenital Hand Deformities(CAD) patients. We compute the conformal welding signatures of contours on general 3D surfaces with surface Ricci flow method, which encodes both global and local surface contour information. Then we evaluated the signatures of pre-central and post-central gyrus on healthy control and CHD subjects to analyze brain cortical morphometry. Preliminary experimental results from 3D MRI data of CHD/control groups demonstrate the effectiveness of our method. The statistical comparison between left and right brain provides an augmented perception on brain morphometry of subjects with Congenital Hand Deformities.

Indexing (document details)
Advisor: Gu, Xianfeng
Commitee: Gao, Jie, Duan, Ye, Bi, Xiaojun
School: State University of New York at Stony Brook
Department: Computer Science
School Location: United States -- New York
Source: DAI-B 81/9(E), Dissertation Abstracts International
Source Type: DISSERTATION
Subjects: Computer science
Keywords: Computational conformal geometry
Publication Number: 13425128
ISBN: 9781658470032
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