[论文解读] Decomposition and Modeling in the Non-Manifold Domain

Franco Morando|arXiv (Cornell University)|Jan 1, 2003

Digital Image Processing Techniques参考文献 81被引用 1

一句话总结

本文提出了一种组合框架，通过单纯复形和商格（quotient lattices）对非流形几何对象进行建模与分解。该框架基于单纯形粘合指令与顶点标识，从简单组件构建复杂胞复形，实现非流形领域中稳健的拓扑推理与分解。

ABSTRACT

The problem of decomposing non-manifold object has already been studied in solid modeling. However, the few proposed solutions are limited to the problem of decomposing solids described through their boundaries. In this thesis we study the problem of decomposing an arbitrary non-manifold simplicial complex into more regular components. A formal notion of decomposition is developed using combinatorial topology. The proposed decomposition is unique, for a given complex, and is computable for complexes of any dimension. A decomposition algorithm is proposed that is linear w.r.t. the size of the input. In three or higher dimensions a decomposition into manifold parts is not always possible. Thus, in higher dimensions, we decompose a non-manifold into a decidable super class of manifolds, that we call, Initial-Quasi-Manifolds. We also defined a two-layered data structure, the Extended Winged data structure. This data structure is a dimension independent data structure conceived to model non-manifolds through their decomposition into initial-quasi-manifold parts. Our two layered data structure describes the structure of the decomposition and each component separately. In the second layer we encode the connectivity structure of the decomposition. We analyze the space requirements of the Extended Winged data structure and give algorithms to build and navigate it. Finally, we discuss time requirements for the computation of topological relations and show that, for surfaces and tetrahedralizations, embedded in real 3D space, all topological relations can be extracted in optimal time. This approach offers a compact, dimension independent, representation for non-manifolds that can be useful whenever the modeled object has few non-manifold singularities.

研究动机与目标

• 开发一种系统化方法，将非流形几何对象分解为更简单的拓扑分量。
• 实现对CAD和几何设计中常见的复杂非流形形状的稳健建模。
• 形式化基于格基代数结构的单纯形粘合与顶点标识过程。
• 为检测和处理几何模型中非流形奇点提供计算基础。

提出的方法

• 以完全展开分解（Ω⊤）作为起点，其中所有最高维单纯形均被隔离。
• 基于从展开复形导出的顶点副本等价关系，定义分解格。
• 应用缝合方程与半模格性质，以建模有效的粘合操作。
• 采用基于形式逻辑的系统（类似Prolog的谓词）来编码与操作粘合指令。
• 引入“顶点副本”概念，以追踪同一顶点在不同单纯形之间的多重标识。
• 利用商格表示通过粘合指令标识顶点所形成的拓扑空间。

实验结果

研究问题

• RQ1 如何系统化地将非流形几何对象分解为更简单的拓扑分量？
• RQ2 何种代数结构可支持非流形复形中单纯形之间有效粘合操作？
• RQ3 在构建复杂胞复形过程中，如何追踪与验证顶点标识？
• RQ4 何种条件可确保一组粘合指令生成一个良定义的胞复形？
• RQ5 如何算法化地检测与处理非流形奇点（例如，一个d-1维面被三个或更多d-单纯形共享）？

主要发现

• 分解格为给定复形的所有可能分解提供了完整且代数上严谨的表示。
• 使用顶点副本可精确追踪多重标识，支持正确的拓扑重构。
• 通过谓词如`nonPseudoManifold`与`nonPseudoManifoldPair`检测非伪流形邻接关系。
• 系统支持通用粘合（`doGluingInstruction`）与伪流形感知粘合（`doPseudoManifoldGluingInstruction`），并具备有效性检查。
• 格结构确保所有有效粘合序列被完整捕获，且生成的复形保持拓扑一致性。
• 通过Prolog谓词实现，支持对分解与粘合操作的有效符号化处理与验证。