Optimizing 3D Triangulations Using Discrete Curvature Analysis

A tool for constructing a “good” 3D triangulation of a given set of vertices in 3D is developed and studied. The constructed triangulation is “optimal” in the sense that it locally minimizes a cost function which measures a certain discrete curvature over the resulting triangle mesh. The algorithm for obtaining the optimal triangulation is that of swapping edges sequentially, such that the cost function is reduced maximally by each swap. In this paper three easy-to-compute cost functions are derived using a simple model for defining discrete curvatures of triangle meshes. The results obtained by the different cost functions are compared. Operating on data sampled from simple 3D models, we compare the approximation error of the resulting optimal triangle meshes to the sampled model in various norms. The conclusion is that all three cost functions lead to similar results, and none of them can be said to be superior to the others. The triangle meshes generated by our algorithm, when serving as initial triangle meshes for the butterfly subdivision scheme, are found to improve significantly the limit butterfly-surfaces compared to arbitrary initial triangulations of the given sets of vertices. Based upon this observation, we believe that any algorithm operating on triangle meshes such as subdivision, finite element solution of PDE, or mesh simplification, can obtain better results if applied to a “good” triangle mesh with small discrete curvatures. Thus our algorithm can serve for modelling surfaces from sampled data as well as for initialization of other triangle mesh based algorithms.