[论文解读] Construction of Injective Mappings Of Meshes.

Yaron Lipman|arXiv (Cornell University)|Oct 3, 2013

Computational Geometry and Mesh Generation参考文献 15被引用 3

一句话总结

本文提出了三组充分条件，用于构建流形网格的单射单纯映射，利用方向一致性与边界约束。该工作将经典全局反演定理推广至分段线性映射，通过线性约束实现灵活边界条件下的单射映射，并在保持方向性时确保包括边界在内的完全单射性，为几何处理提供了一套实用且适合优化的框架。

ABSTRACT

This paper introduces three sets of sufficient conditions, for generating injective simplicial mappings of manifold meshes. A necessary condition for a simplicial mapping of a mesh to be injective is that it consistently preserves or inverts the orientations of all elements. However, these conditions are insufficient to guarantee injectivity. 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 the map is a bijection. 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 that guarantees that the resulting map is injective over the interior of the mesh. These linear conditions provide a practical tool for optimizing an injective map of the mesh while allowing the boundary map to adjust freely. Allowing more freedom in the boundary conditions is useful for two reasons: a) it circumvents the hard task of providing a bijective boundary map, and b) it allows optimizing the boundary map together with the simplicial map to achieve lower energy levels. The third set of conditions adds to the second set the requirement that the boundary maps are orientation preserving as-well. 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 injectively. A secondary goal of this paper is to advocate and develop the tool of degree in the context of geometry processing and modeling of meshes.

研究动机与目标

• 为解决构建流形网格单射单纯映射的挑战，此类映射在几何处理与建模任务中至关重要。
• 通过识别额外的充分条件，克服仅靠方向性保持不足以保证单射性的局限。
• 通过放宽双射边界映射的严格要求，为优化单射映射提供实用工具。
• 倡导在基于网格的几何处理与建模中使用拓扑度概念。

提出的方法

• 将经典全局反演定理推广至分段线性（网格）设定，证明双射边界映射结合一致的方向性保持可确保全局单射性。
• 引入一组边界映射的线性约束，即使边界映射非双射，也能保证网格内部的单射性。
• 通过要求边界映射保持方向性，扩展第二组条件，确保映射在网格内部及边界上均保持单射性。
• 运用代数拓扑中的度概念，分析并确保单纯映射在分段线性网格上的单射性。
• 采用优化框架，允许同时调整边界映射与内部映射，以最小化能量并保持单射性。
• 通过多项实验将条件应用于三角网格，验证了理论保证在实际映射中的有效性。

实验结果

研究问题

• RQ1 如何利用拓扑与几何约束保证网格的单射单纯映射？
• RQ2 当边界映射无需为双射时，何种条件可确保单射性？
• RQ3 拓扑度概念在分段线性网格映射中如何有效用于验证单射性？
• RQ4 如何在不牺牲单射性的前提下，利用边界自由度进行优化？
• RQ5 何种附加约束可确保单射性不仅存在于网格内部，也存在于边界上？

主要发现

• 本文证明，双射边界映射结合一致的方向性保持可保证全局单射的单纯映射，将经典反演定理推广至网格设定。
• 边界映射的线性约束可替代双射性要求，同时仍确保网格内部的单射性，从而实现灵活优化。
• 当边界映射同时满足单射性与方向性保持时，所得单纯映射在整个网格（包括边界）上均为单射。
• 所提出的条件允许在边界非双射的前提下，对映射进行能量最小化优化，同时保持单射性。
• 实验表明，这些条件在生成三角网格单射映射方面具有实际有效性。
• 拓扑度的使用为验证网格映射中的单射性提供了稳健的理论基础，推动了其在几何处理中的应用。