Surface Approximation via Asymptotic Optimal Geometric Partition

In this paper, we present a surface remeshing method with high approximation quality based on Principal Component Analysis. Given a triangular mesh and a user assigned polygon/vertex budget, traditional methods usually require the extra curvature metric field for the desired anisotropy to best approximate the surface, even though the estimated curvature metric is known to be imperfect and already self-contained in the surface. In our approach, this anisotropic control is achieved through the optimal geometry partition without this explicit metric field. The minimization of our proposed partition energy has the following properties: Firstly, on a C2 surface, it is theoretically guaranteed to have the optimal aspect ratio and cluster size as specified in approximation theory for L1 piecewise linear approximation. Secondly, it captures sharp features on practical models without any pre-tagging. We develop an effective merging-swapping framework to seek the optimal partition and construct polygonal/triangular mesh afterwards. The effectiveness and efficiency of our method are demonstrated through the comparison with other state-of-the-art remeshing methods.