Triangular mesh parameterization with trimmed surfaces



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Springer-Verlag France


Given a 2-manifold triangular mesh M subset of 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 subset of 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, redesign, 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 of 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 phi : M -> U subset of R-2, (2) an inverse mapping S : W subset of R-2 -> 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.


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