Triangular mesh parameterization with trimmed surfaces: Fid-Bau Portal

Triangular mesh parameterization with trimmed surfaces

Ruiz, Oscar E. / Mejia, Daniel / Cadavid, Carlos A.

Abstract Given a 2-manifold triangular mesh $M \subset R^{3}$ , with border, a parameterization of $M$ is a FACE or trimmed surface $F = {S, L_{0}, \dots, L_{m}}$ . $F$ is a connected subset or region of a parametric surface $S$ , bounded by a set of LOOPs $L_{0}, \dots, L_{m}$ such that each $L_{i} \subset S$ is a closed 1-manifold having no intersection with the other $L_{j}$ LOOPs. The parametric surface $S$ is a statistical fit of the mesh $M$ . $L_{0}$ is the outermost LOOP bounding $F$ and $L_{i}$ is the LOOP of the i-th hole in $F$ (if any). The problem of parameterizing triangular meshes is relevant for reverse engineering, tool path planning, feature detection, re-design, etc. State-of-art mesh procedures parameterize a rectangular mesh $M$ . To improve such procedures, we report here the implementation of an algorithm which parameterizes meshes $M$ presenting holes and concavities. We synthesize a parametric surface $S \subset R^{3}$ which approximates a superset of the mesh $M$ . Then, we compute a set of LOOPs trimming $S$ , and therefore completing the FACE $F = {S, L_{0}, \dots, L_{m}}$ . Our algorithm gives satisfactory results for $M$ having low Gaussian curvature (i.e., $M$ being quasi-developable or developable). This assumption is a reasonable one, since $M$ is the product of manifold segmentation pre-processing. Our algorithm computes: (1) a manifold learning mapping $ϕ : M \to U \subset R^{2}$ , (2) an inverse mapping $S : W \subset R^{2} \to R^{3}$ , with $W$ being a rectangular grid containing and surpassing $U$ . To compute $ϕ$ we test IsoMap, Laplacian Eigenmaps and Hessian local linear embedding (best results with HLLE). For the back mapping (NURBS) $S$ the crucial step is to find a control polyhedron $P$ , which is an extrapolation of $M$ . We calculate $P$ by extrapolating radial basis functions that interpolate points inside $ϕ (M)$ . We successfully test our implementation with several datasets presenting concavities, holes, and are extremely non-developable. Ongoing work is being devoted to manifold segmentation which facilitates mesh parameterization.