Computational differential geometry and intrinsic surface processing
Computational differential geometry and intrinsic surface processing
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发表时间：
2010
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通讯作者：
T. Chan;Rongjie Lai
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作者：
T. Chan;Rongjie Lai
在这项工作中，我们专注于使用内在几何方法来研究变分问题和Laplace-Beltrami特征几何的3D三角化表面和它们的应用计算大脑解剖。本文将讨论两类问题。在第一部分中，我们研究了如何用变分方法处理曲面上的图像处理问题。从证明全变分方法适用于曲面上的图像处理问题出发，利用微分几何技术，将与全变分相关的成像模型推广到曲面上的成像问题.作为这种内在方法的一个优点，用于解决全变分相关问题的流行算法可以适用于解决曲面上的广义模型。我们还表明，这种内在的方法为我们提供了一个强大的和有效的方法来解决表面上的成像问题。在第二部分中，我们着重研究了曲面自身的几何性质。具体来说，我们将研究如何检测局部和全局表面几何形状及其在计算大脑解剖学中的应用。我们使用的主要工具是Laplace-Beltrami（LB）算子及其特征系统，这为我们提供了一个内在的和强大的工具来研究曲面几何。我们首先提出使用LB节点计数序列作为表面特征来表征表面，并展示其在等谱表面分辨和表面分类中的应用。然后，我们提供了一种新的方法来计算骨架的单连通曲面构造Reeb图的各向异性Laplace-Beltrami算子的特征函数。在关于LB特征几何的最后一个主题中，我们提出了一个通用的框架，通过使用LB算子的特征系统来定义表面之间的数学上严格的距离，然后我们展示了它的应用之一，以解决计算大脑解剖学中具有挑战性的脑沟区域识别问题。
In this work, we focus on using the intrinsic geometric method to study variational problems and Laplace-Beltrami eigen-geometry on 3D triangulated surfaces and their applications to computational brain anatomy. Two classes of problems will be discussed in this dissertation. In the first part, we study how to tackle image processing problems on surfaces by using variational approaches. Starting from the proof for the suitability of total variation for image processing problems on surfaces, we generalize the well-known total variation related imaging models to study imaging problems on surfaces by using differential geometry techniques. As an advantage of this intrinsic method, popular algorithms for solving the total variation related problems can be adapted to solve the generalized models on surfaces. We also demonstrate that this intrinsic method provides us a robust and efficient way to solve imaging problems on surfaces. In the second part, we focus on studying surfaces' own geometry. Specifically, we will study how to detect local and global surface geometry and its applications to computational brain anatomy. The main tool we use is the Laplace-Beltrami (LB) operator and its eigen-systems, which provide us an intrinsic and robust tool to study surface geometry. We first propose to use LB nodal count sequences as a surface signature to characterize surface and demonstrate its applications to isospectral surfaces resolving and surface classification. Then, we provide a novel approach of computing skeletons of simply connected surfaces by constructing Reeb graphs from the eigenftmctions of an anisotropie Laplace-Beltrami operator. In the last topic about the LB eigen-geometry, we propose a general framework to define a mathematically rigorous distance between surfaces by using the eigen-system of the LB operator, and then we demonstrate one of its applications to tackle the challenging sulci region identification problem in computational brain anatomy.