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Computing Optimal Alignments of Surfaces

计算表面的最佳对齐方式

基本信息

批准号：
1719582
负责人：
Joel Hass
金额：
$ 25万
依托单位：
University of California-Davis
依托单位国家：
美国
项目类别：
Standard Grant
财政年份：
2017
资助国家：
美国
起止时间：
2017-07-15 至 2021-06-30
项目状态：
已结题

来源：
https://www.nsf.gov/awardsearch/showAward?AWD_ID=1719582&HistoricalAwards=false
关键词：
Computing Optimal Alignments Surfaces

项目摘要

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.

我们周围的世界几乎都是二维的。无论是用眼睛、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，了解如何分析这些几何数据的对齐和比较表面对的目的。这个项目将开发和实施新的和严格的数学方法来比较形状，基于低维拓扑技术。它将发展新的数学成果，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。形状比较的基本数学理论将得到扩展，并将开发新的算法来实现所产生的理论，并提供软件。此外，该软件还将应用于生物形状的集合，如蛋白质表面、面部数据库、骨骼和牙齿集合以及大脑皮层。该项目将开发新的数学失真测量方法，该方法基于各种测量方法，测量一个表面在另一个表面上拉伸所需的能量。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被捕获在新的距离函数中，该距离函数捕获几何相似性的方面。该项目建立在成功的现有方法，用于比较表面的拓扑结构的一个领域。这些方法将得到改进并扩展到更一般的曲面，允许比较具有一个、两个或多个手柄的曲面。过去的方法的基础上保角映射将被扩展到允许使用更一般的表面仿射。该项目还将在部分曲面或具有边界的曲面之间生成明确的对齐。这种部分表面匹配允许在仅测量了每个表面的一部分而其他部分被遮挡时匹配表面。它还允许具有不同拓扑结构的表面对之间的比较，克服了当前方法所面临的问题。最近引入的对称畸变能量将被扩展到这些更一般的情况下。将开发并提供执行这些几何算法的软件。该软件将被设计为可供各种学科的科学家使用，并可能导致生物和医学理解的突破。