Optimising 3D Triangulations: Improving the Initial Triangulation for the Butterfly Subdivision Scheme

Abstract

This work is concerned with the construction of a “good”3D triangulation of a given set of points in 3D, to serve as an initialtriangulation for the generation of a well shaped surface by thebutterfly scheme. The optimisation method is applied to manifoldmeshes, and conserves the topology of the triangulations. The constructedtriangulation is “optimal” in the sense that it locallyminimises a cost function. The algorithm for obtaining a locally-optimaltriangulation is an extension of Lawson’s Local OptimisationProcedure (LOP) algorithm to 3D, combined with a priority queue.The first cost function designed in this work measures an approximationof the discrete curvature of the surface generated by the butterflyscheme, based on the normals to this surface at the given 3D vertices.These normals can be expressed explicitly in terms of the verticesand the connectivity between them in the initial mesh. The secondcost function measures the deviations of given normals at the givenvertices from averages of normals to the surface generated by thebutterfly scheme in neighbourhoods of the corresponding vertices.It is observed from numerical simulations that our optimisation procedureleads to good results for vertices sampled from analytic objects.The first cost function is appropriate for analytic surfaces witha large proportion of convex vertices. Furthermore, the optimisationwith this cost function improves convex regions in non-convex complexmodels. The results of optimisation with respect to the second costfunction are satisfactory even when all the vertices are non-convex,but this requires additional initial information which is obtainableeasily only from analytic surfaces.