Structured Bitmap-to-Mesh Triangulation for Geometry-Aware Discretization of Image-Derived Domains

TL;DR

This paper introduces a template-driven triangulation method for discretizing image-derived domains that improves stability, regularity, and scalability over traditional methods, enabling real-time PDE solving and geometric analysis.

Contribution

It presents a novel retriangulation framework that preserves the base mesh, supports parallel execution, and guarantees mesh quality for PDE discretization on complex image-derived boundaries.

Findings

Fewer sliver elements and more regular triangles.

Improved geometric fidelity near complex boundaries.

Compatible with finite element methods and cotangent discretizations.

Abstract

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.