Triangular mesh parameterization with trimmed surfaces

Abstract

Given a 2-manifold triangular mesh M ⊂ R 3 , with border, a parameterization of M is a FACE or trimmed surface F = {S, L 0 , . . . , L m }. F is a connected subset or region of a parametric surface S, bounded by a set of LOOPs L 0 , . . . , L m such that each L i ⊂ 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 ⊂ 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 , . . . , 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 → U ⊂ R 2 , (2) an inverse mapping S : W ⊂ R 2 → 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 calcu-B Oscar E. Ruiz

Quasi-Isometric Mesh Parameterization Using Heat-Based Geodesics and Poisson Surface Fills

“…However, such a parameterization is improved by applying the Poisson reconstruction, reducing the distortion close to mesh holes and boundary concavities (Figure 11b). A more extreme case is illustrated in Figure 12, with the S-trimmed-on-cone data set from [14]. Since the S shape is trimmed on a cone, M is a fully developable surface.…”

“…As a consequence, these algorithms have been applied successfully in mesh parameterization applications. Such algorithms include Laplacian Eigenmaps [14] and Hessian Locally Linear Embedding [15].…”

“…Ruiz et al [14] used a dimensionality-reduction geodesic-based algorithm (Isomap) for the computation of isometric parameterization of quasi-developable meshes. However, in addition to the classic nonconvex parameterization problems, such algorithms estimate geodesics using shortest-path graph algorithms which introduce additional distortions in the resulting parameterization.…”

“…Figure 14 plots the parameterization results for the segmented mesh. Each submesh bijective parameterization presents low distortion, enabling further reverse engineering operations such as NURBs reparameterization [14], finite element analysis, structural optimization, and/or dimensional inspection [31].…”

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“…Failures of IsoMap are not due to the coarse graph approximation of geodesic curves. Instead,holes or to the non-developable character of the mesh hinder IsoMap performance [15].…”