Surface Approximation via Asymptotic Optimal Geometric Partition

TL;DR

This paper introduces a surface remeshing technique that achieves high approximation quality by optimizing geometric partitions based on PCA, without relying on explicit curvature metrics, and guarantees optimal aspect ratios on smooth surfaces.

Contribution

It proposes a novel partition energy minimization approach that inherently captures surface features and achieves optimal aspect ratios without explicit curvature metrics.

Findings

Outperforms state-of-the-art remeshing methods in quality and efficiency.

Guarantees optimal aspect ratio and cluster size on smooth surfaces.

Effectively captures sharp features without pre-tagging.

Abstract

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.