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.

翻译：我们提出了一种模板驱动的三角剖分框架，将基于栅格或分割得到的边界嵌入规则三角形网格，以实现图像衍生域上稳定的偏微分方程离散化。与可能触发全局连通性更新的约束Delaunay三角剖分不同，本方法仅对边界穿过的三角形进行重新三角剖分，保留基础网格结构，并支持无同步并行计算。为确保确定性与可扩展性，我们基于离散等价类与三角形对称性对所有局部边界相交构型进行分类，构建有限符号查找表，将每种情况映射至无冲突的重新三角剖分模板。我们证明所得网格具有封闭性、有界角度特性，且兼容基于余切的离散化方案及标准有限元方法。在椭圆/抛物型偏微分方程、信号插值与结构度量方面的实验表明，该方法能减少狭长单元、生成更规则的三角形，并在复杂边界附近提升几何保真度。该框架非常适用于图像衍生域的实时几何分析与基于物理的仿真。