将图像边界嵌入三角网格，实现稳定且高效的微分方程求解。
Structured Bitmap-to-Mesh Triangulation for Geometry-Aware Discretization of Image-Derived Domains
- 仅重剖分与边界相交的三角形，保持基底网格不变。
- 减少细长三角形，提升边界附近几何保真度。
- 支持并行计算，适合实时物理模拟与图像域分析。
我们提出一种模板驱动的重剖分框架，将栅格或分割得到的边界嵌入规则三角网格中，以实现图像衍生域上稳定的偏微分方程离散化。与可能引发全局连通性更新的约束德劳内三角剖分（CDT）不同，本方法仅重剖分与边界相交的三角形，保留原始网格结构，并支持无同步并行执行。为保证确定性和可扩展性，我们对所有局部边界交集配置按离散等价和三角形对称性分类，生成有限符号查找表，映射每种情形到无冲突的重剖分模板。证明了所得网格封闭、角度有界，兼容余切离散化与标准有限元方法。在椭圆与抛物型PDE、信号插值及结构度量实验中，结果表明更少的细长元素、更规整的三角形以及复杂边界处更高的几何保真度。该框架适用于图像域上的实时几何分析与物理仿真。
原文摘要 · Abstract (English)
We propose a template-driven triangulation framework that embeds raster- or segmentation-derived boundaries into a regular triangular grid for stable PDE discretization on image-derived domains. Unlike constrained Delaunay triangulation (CDT), which may trigger global connectivity updates, our method retriangulates only triangles intersected by the boundary, preserves the base mesh, and supports synchronization-free parallel execution. To ensure determinism and scalability, we classify all local boundary-intersection configurations up to discrete equivalence and triangle symmetries, yielding a finite symbolic lookup table that maps each case to a conflict-free retriangulation template. We prove that the resulting mesh is closed, has bounded angles, and is compatible with cotangent-based discretizations and standard finite element methods. Experiments on elliptic and parabolic PDEs, signal interpolation, and structural metrics show fewer sliver elements, more regular triangles, and improved geometric fidelity near complex boundaries. The framework is well suited for real-time geometric analysis and physically based simulation over image-derived domains.
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