Surface Construction and Comparison using Manifolds

本研究的重点是比较相似但不相同的曲面。例如，几乎所有的腿部骨骼在形状上都相似(末端带有旋钮的圆柱体)，但在旋钮的长度、直径、大小和形状等方面可能有很大的不同。本研究的目标是为比较、分类和分析这类曲面提供可靠的方法。仅举几例，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进行这些测量。他们可以进行的测量类型也仅限于简单的测量，如长度和周长。这些测量被用来回答这样的问题：这块骨头是什么种族/性别的？那个人有多高？表面是不是不正常？有没有在某个时候受伤？为什么关节不能正常工作？分析性测量将不仅消除(有时是主观的)人为因素，而且还将拓宽表面比较的范围，而不仅仅是简单的长度测量。本研究的第一部分是建立任意拓扑流形的构造性方法。这些流形提供了一种自然机制，用于通过创建覆盖曲面任何所需部分的局部贴图来分析曲面的特定子集。流形还提供了用于识别曲面上的对应点的参数化机制。这项研究解决了与流形相关的理论问题，如改进的局部地图放置，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的网格或点云。他的研究的第二部分，详细的表面比较，是一项相对新颖的任务，除了算法解决方案的开发外，还包括对问题的恰当陈述。例如：什么是功能？我们如何以一种有意义的方式定量地衡量功能差异？我们如何衡量两个表面之间对应的质量？

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?