全局意义下的测地距离场

项目名称： 全局意义下的测地距离场

项目编号： No.61300168

项目类型： 青年科学基金项目

立项/批准年度： 2014

项目学科： 自动化技术、计算机技术

项目作者： 辛士庆

作者单位： 宁波大学

项目金额： 23万元

中文摘要： 测地距离是直线距离在弯曲空间中的推广，在以网格曲面为研究对象的数字几何领域有广泛的应用。传统的测地线算法都是基于"测地距离场的梯度是单位向量"这一基本性质而设计的，在程式上由近及远、层层展开，逐步求出从源点到其它点的测地距离。然而，随着三维扫描技术的发展，出现了大量的残缺模型。由于算法程式的限制，传统算法不能在这些残缺模型上计算出有意义的测地距离场，模型上的一个裂痕或者拓扑短路会使计算结果发生根本的改变。为了在残缺模型上计算出更有意义的测地距离场，国际国内的学者开始寻求新的解决思路，包括热核方法和桥接法等。然而，这些算法要么仅适用于部分缺陷，要么求得的结果与测地度量相去甚远。因此，我们提出从全局的观点刻画测地度量的性质，建立在残缺模型上的目标函数，通过优化方法找到对各种缺陷均不敏感的测地距离场。鲁棒的测地线算法将在已有的几何分析方法与残缺模型之间搭起一座桥梁，推动数字几何处理继续向前发展。

中文关键词： 测地线；几何优化；最优传输；围长；

英文摘要： Geodesic distance is an extension of straight-line distance into curved spaces and central to numerous applications in digital geometry processing field that deals with polygonal surfaces. Most of conventional geodesic algorithms are designed based on the fact that a geodesic distance field has a unit gradient vector. These algorithms run from near to far, in a sweeping fashion. More and more defect models emerge with the development of 3D scanning technology. Due to the limitation of the sweeping paradigm, conventional methods cannot compute a robust geodesic distance field. A small gap or topological shortcut of the original model will lead to a fundamental change of the distance field. In order to compute a defect-tolerant geodesic distance field, domestic and foreign researchers have been experimenting with new ideas in recent years, such as heat kernel based methods and bridge methods. However, these approaches either apply to models with part of defects, or produce a totally different metric from the geodesic distance. Therefore, we propose characterizing the geodesic metric in a global perspective, building an object function on defect models, and computing defect-tolerant geodesic distance field by optimization techniques. A robust geodesic algorithm can serve as a bridge between existing geometry analy

英文关键词： geodesic；geometric optimization；optimal transport；girth；