Triangular mesh parameterization with trimmed surfaces

Abstract

Given a 2-manifold triangular mesh \(M \subset {\mathbb {R}}^3\), with border, a parameterization of \(M\) is a FACE or trimmed surface \(F=\{S,L_0,\ldots ,L_m\}\). \(F\) is a connected subset or region of a parametric surface \(S\), bounded by a set of LOOPs \(L_0,\ldots ,L_m\) 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 \(S \subset {\mathbb {R}}^3\) which approximates a superset of the mesh \(M\). Then, we compute a set of LOOPs trimming \(S\), and therefore completing the FACE \(F=\{S,L_0,\ldots ,L_m\}\). 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 \(\phi : M \rightarrow U \subset {\mathbb {R}}^2\), (2) an inverse mapping \(S: W \subset {\mathbb {R}}^2 \rightarrow {\mathbb {R}}^3\), with \(W\) being a rectangular grid containing and surpassing \(U\). To compute \(\phi \) 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 \(\phi (M)\). 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.