Choi Gary P T, Qiu Di, Lui Lok Ming
John A. Paulson School of Engineering and Applied Sciences, Harvard University, Cambridge, MA, USA.
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA.
Proc Math Phys Eng Sci. 2020 Oct;476(2242):20200147. doi: 10.1098/rspa.2020.0147. Epub 2020 Oct 7.
In this work, we develop a framework for shape analysis using inconsistent surface mapping. Traditional landmark-based geometric morphometr- ics methods suffer from the limited degrees of freedom, while most of the more advanced non-rigid surface mapping methods rely on a strong assumption of the global consistency of two surfaces. From a practical point of view, given two anatomical surfaces with prominent feature landmarks, it is more desirable to have a method that automatically detects the most relevant parts of the two surfaces and finds the optimal landmark-matching alignment between these parts, without assuming any global 1-1 correspondence between the two surfaces. Our method is capable of solving this problem using inconsistent surface registration based on quasi-conformal theory. It further enables us to quantify the dissimilarity of two shapes using quasi-conformal distortion and differences in mean and Gaussian curvatures, thereby providing a natural way for shape classification. Experiments on Platyrrhine molars demonstrate the effectiveness of our method and shed light on the interplay between function and shape in nature.
在这项工作中，我们开发了一个使用不一致表面映射进行形状分析的框架。传统的基于地标点的几何形态计量学方法存在自由度有限的问题，而大多数更先进的非刚性表面映射方法则强烈依赖于两个表面全局一致性的假设。从实际角度来看，给定两个具有显著特征地标的解剖表面，更希望有一种方法能够自动检测这两个表面的最相关部分，并在这些部分之间找到最优的地标匹配对齐方式，而无需假设两个表面之间存在任何全局的一一对应关系。我们的方法能够基于拟共形理论使用不一致表面配准来解决这个问题。它还使我们能够使用拟共形畸变以及平均曲率和高斯曲率的差异来量化两个形状的差异，从而为形状分类提供一种自然的方式。对阔鼻猴臼齿的实验证明了我们方法的有效性，并揭示了自然界中功能与形状之间的相互作用。
