Surface Construction and Comparison using Manifolds

这项研究的重点是比较表面相似，但不相同。例如，几乎所有的腿骨都是相似的形状（一个圆柱体上的两端旋钮），但可能会有很大的不同，在长度，直径，大小和形状的旋钮等，本研究的目标是提供强大的方法进行比较，分类和分析这种类型的表面。生物医学、体质人类学和法医学领域的研究人员目前依靠人类专家来执行这些测量。它们可以进行的测量类型也仅限于简单的测量，例如长度和周长。这些测量用于回答以下问题：该骨骼来自哪个种族/性别？那人有多高？表面是否异常？在某个时候有受伤吗？为什么关节不能正常工作？分析测量不仅可以消除（有时是主观的）人为因素，而且还可以扩大表面比较的范围，而不仅仅是简单的长度测量。本研究的第一部分是一个建设性的方法来创建任意拓扑流形。这些流形提供了一种自然的机制，用于通过创建覆盖表面任何所需部分的局部映射来分析表面的特定子集。流形还提供了一种参数化机制，用于识别曲面上的对应点。该研究解决了与流形相关的理论问题，例如改进的局部贴图放置，更可控的混合策略，带边界的流形，以及使用流形来参数化现有网格或点云。他的研究的第二部分，详细的表面比较，是一个相对新颖的任务，并包括一个良好的陈述的问题，除了发展的算法解决方案。例如：什么是特征？我们如何以有意义的方式定量测量特征差异？我们如何衡量两个曲面之间对应的质量？

This research focuses on the comparison of surfaces which are similar, but not identical. For example, almost all leg bones are similar in shape (a cylinder with knobs on the ends) but may differ greatly in length, diameter, size and shape of the knobs, etc. The goal of this research is to provide robust methods for comparing, categorizing, and analyzing surfaces of this type. Researchers in the bio-medical, physical anthropology, and forensics fields, to name a few, currently rely on human experts to perform these measurements. The types of measurements they can make are also limited to simple ones, such as length and circumference. These measurements are used to answer questions such as: What race/sex is this bone from? How tall was the person? Is the surface is abnormal? Was there an injury at some time? Why does the joint not function normally? Analytical measurements will not only remove the (sometimes subjective) human component, but will also broaden the scope of surface comparison beyond simple length measurements. The first part of this research is a constructive approach for creating manifolds of arbitrary topology. These manifolds provide a natural mechanism for analyzing specific subsets of the surface via the creation of local maps that cover any desired portion of the surface. Manifolds also provide a parameterization mechanism for identifying corresponding points on surfaces. The research addresses theoretical issues related to manifolds such as improved local map placement, more controllable blend strategies, manifolds with boundary, and using manifolds to parameterize existing meshes or point clouds. The second part of his research, detailed surface comparison, is a relatively novel task, and includes a well-posed statement of the problem in addition to the development of algorithmic solutions. For example: What is a feature? How do we quantitatively measure feature differences in a meaningful way? How do we measure the quality of the correspondence between two surfaces?