Computing Optimal Alignments of Surfaces

我们在世界上看到的几乎所有东西都是一个二维表面。无论是用我们的眼睛、雷达还是激光扫描仪，我们都能接收到形成物体外边界的表面几何形状的数据。有一个基本的需要是了解如何分析这些几何数据，以便对准和比较成对的曲面。该项目将基于低维拓扑技术，开发和实现新的严格的数学方法来比较形状。它将发展新的数学结果并创造新的计算工具。应用范围从面部识别到大脑映射到蛋白质分类。将扩展形状比较的基础数学理论，并开发新的算法来实现所产生的理论，并提供软件。此外，该软件将应用于生物形状的集合，如蛋白质表面、面部数据库、骨骼和牙齿的集合以及大脑皮层。该项目将基于将一个表面拉伸到另一个表面所需能量的各种测量，开发新的扭曲数学测量方法。给定两个表面或形状，将开发算法来产生它们之间的最佳对应或对齐。两个形状的差异或相似性将在新的距离函数中捕获，该距离函数捕获几何相似性的各个方面。该项目建立在比较具有球面拓扑结构的表面的成功现有方法的基础上。这些方法将得到改进并扩展到更一般的表面，允许比较具有一个，两个或多个手柄的表面。过去基于共形映射的方法将得到扩展，以允许使用更一般的表面微分同态。该项目还将在部分表面或具有边界的表面之间产生显式对齐。这种局部表面匹配允许在每个表面只有部分被测量时匹配表面，而其他部分被遮挡。它还允许对具有不同拓扑结构的表面进行比较，克服了当前方法面临的一个问题。最近介绍的对称畸变能将扩展到这些更一般的情况。将开发并提供实现这些几何算法的软件。该软件将被设计成供各种学科的科学家使用，并可能在生物学和医学理解方面取得突破。

Almost everything we see in the world around us is a two-dimensional surface. Whether with our eyes, with radar or with a laser scanner, we receive data about the geometry of the surface that forms the outer boundary of an object. There is a fundamental need to understand how to analyze this geometric data for purposes of aligning and comparing pairs of surfaces. This project will develop and implements new and rigorous mathematical methods to compare shapes, based on techniques of low-dimensional topology. It will develop new mathematical results and create new computational tools. Applications range from facial recognition to brain mapping to protein classification. The underlying mathematical theory of shape comparison will be extended, and new algorithms to implement the resulting theory will be developed and software made available. Moreover this software will be applied and tested on collections of biological shapes, such as protein surfaces, databases of faces, collections of bones and teeth, and brain cortices.This project will develop new mathematical measures of the distortion based on a variety of measures of the energy needed to stretch one surface over another. Given two surfaces, or shapes, algorithms will be developed to produce an optimal correspondence or alignment between them. The difference or similarity of two shapes will be captured in new distance functions that capture aspects of geometric similarity. The project builds on successful existing approaches used in comparing surfaces that have the topology of a sphere. These methods will be improved and extended to more general surfaces, allowing for comparing surfaces with one, two, or more handles. Past methods based on conformal maps will be extended to allow use of more general surface diffeomorphisms. The project will also produce explicit alignments between partial surfaces, or surfaces with boundary. This partial surface matching allows for matching surfaces when only portions of each have been measured, with other parts obscured. It also allows comparison between pairs of surfaces having different topologies, overcoming a problem faced by current methods. The symmetric distortion energy recently introduced will be extended to these more general contexts. Software implementing these geometric algorithms will be developed and made available. This software will be designed to be usable by scientists in a wide variety disciplines and could lead to breakthroughs in biological and medical understanding.