带边界网格的双射映射与网格处理中的度
Bijective Mappings Of Meshes With Boundary And The Degree In Mesh Processing
摘要：
本文给出了流形网格上双射单纯形映射生成的三组充分条件。一个网格的单纯映射是单射的必要条件是它要么保持所有元素的方向，要么翻转所有元素。然而，已知这些条件对于单纯映射的内射性是不够的。在本文中，我们提供了额外的简单条件，连同上述必要条件保证单形映射的内射性。第一组条件将经典的全局反演定理推广到网格（分段线性）的情况。也就是说，证明了如果边界单纯形映射是双射的，并且必要条件成立，则该映射是内射的并且到目标域上。第二组条件与网格到多面体的映射有关，并且用线性约束的集合替换双射边界映射的（通常是硬的）要求，并且保证所得到的映射在网格的内部上是单射的。这些线性条件提供了一个实用的工具，用于优化到一个给定的多面体的网格的地图，同时允许的边界图，以自由调整，并保持在网格的内部的注入性能。第三组条件向第二组条件添加了边界图也保持方向的要求（具有边界图方向的适当定义）。这组条件保证了映射在网格的边界上以及其内部都是单射的。几个实验使用的充分条件映射三角形网格。本文的第二个目标是在网格处理的背景下倡导和发展度的工具。
英文摘要：
This paper introduces three sets of sufficient conditions, for generating bijective simplicial mappings of manifold meshes. A necessary condition for a simplicial mapping of a mesh to be injective is that it either maintains the orientation of all elements or flips all the elements. However, these conditions are known to be insufficient for injectivity of a simplicial map. In this paper we provide additional simple conditions that, together with the above mentioned necessary conditions guarantee injectivity of the simplicial map. The first set of conditions generalizes classical global inversion theorems to the mesh (piecewise-linear) case. That is, proves that in case the boundary simplicial map is bijective and the necessary condition holds then the map is injective and onto the target domain. The second set of conditions is concerned with mapping of a mesh to a polytope and replaces the (often hard) requirement of a bijective boundary map with a collection of linear constraints and guarantees that the resulting map is injective over the interior of the mesh and onto. These linear conditions provide a practical tool for optimizing a map of the mesh onto a given polytope while allowing the boundary map to adjust freely and keeping the injectivity property in the interior of the mesh. The third set of conditions adds to the second set the requirement that the boundary maps are orientation preserving as-well (with a proper definition of boundary map orientation). This set of conditions guarantees that the map is injective on the boundary of the mesh as-well as its interior. Several experiments using the sufficient conditions are shown for mapping triangular meshes. A secondary goal of this paper is to advocate and develop the tool of degree in the context of mesh processing.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
