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

TL;DR

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 and proves that the resulting mesh is closed, has bounded angles, and is compatible with cotangent-based discretizations and standard finite element methods.

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.

Figures (25)

• Figure 1: Overview of the SBMT execution pipeline. The process begins with a regular triangular grid, followed by boundary embedding, threshold-based preprocessing, discrete classification of boundary–triangle intersections, and local retriangulation via symbolic lookup templates. Each stage is deterministic, parallelizable, and free of global mesh coupling.
• Figure 2: Structured equilateral mesh over a bitmap-derived domain. Green squares indicate unit pixels; blue lines denote the triangular grid; red polylines represent boundary segments intersecting the mesh. This configuration satisfies the geometric protocol of Section \ref{['sec:geometric-protocol']} and supports lookup-based local retriangulation while preserving topological and numerical consistency. The grid shown uses parameters $a = 0.2$, $b = 0.08$, $c = 0$ (as defined in Section \ref{['subsubsec:theoretical-properties']}), with triangle edge length $\sqrt{0.7}$.
• Figure 3: Exhaustive enumeration of canonical intersection patterns induced by a single boundary segment intersecting a triangular cell. Each configuration is treated as an atomic unit, abstracted independently of any additional segments that may also intersect the same cell. Subfigures are labeled by the number of intersection points (e.g., 1 intersection, 2 intersections, etc.), omitting edge identities. These localized, template-based configurations form the basis of our lookup-driven remeshing framework, enabling robust and scalable subdivision of structured triangular domains with embedded polygonal boundaries.
• Figure 4: Geometric classification of all local configurations in which a single boundary segment intersects a triangle cell. The four cases include vertex intersection (a), edge piercing (b), single-edge contact (c), and edge-aligned overlap (d). This classification forms the base of the local retriangulation lookup table.
• Figure 5: Geometric configurations of type (1,1), where two boundary segments intersect a triangle at one point each. These cases are grouped by intersection count rather than edge location, forming a unified retriangulation pattern applicable to all edge combinations.

Theorems & Definitions (50)

• Lemma 3.1: Bound on segment--triangle intersections
• proof : Proof sketch
• Theorem 3.2: Completeness of retriangulation lookup table under SBMT constraints
• proof : Sketch
• Theorem 3.3: Edge Conformity under Finite Precision
• proof
• Remark 1
• Lemma 3.4: Uniqueness of Retriangulation per Atomic Type
• proof
• Remark 2