We present a novel topology-preserving 3D medial axis computation framework based on volumetric restricted power diagram (RPD), while preserving the medial features and geometric convergence simultaneously, for both 3D CAD and organic shapes. The volumetric RPD discretizes the input 3D volume into sub-regions given a set of medial spheres. With this intermediate structure, we convert the homotopy equivalency between the generated medial mesh and the input 3D shape into a localized contractibility checking for each restricted element (power cell, power face, power edge), by checking their connected components and Euler characteristics. We further propose a fractional Euler characteristic algorithm for efficient GPU-based computation of Euler characteristic for each restricted element on the fly while computing the volumetric RPD. Compared with existing voxel-based or point-cloud-based methods, our approach is the first to adaptively and directly revise the medial mesh without globally modifying the dependent structure, such as voxel size or sampling density, while preserving its topology and medial features. In comparison with the feature preservation method MATFP, our method provides geometrically comparable results with fewer spheres and more robustly captures the topology of the input 3D shape.

翻译：本文提出了一种新颖的基于体素化受限幂图的拓扑保持三维中轴计算框架，该框架在保持中轴特征和几何收敛性的同时，适用于三维CAD模型与有机形状。体素化受限幂图通过给定一组中轴球体将输入三维体离散化为多个子区域。借助这一中间结构，我们将生成的中轴网格与输入三维形状之间的同伦等价性，转化为对每个受限元素（幂胞体、幂面、幂边）的局部可收缩性检验，具体通过检查其连通分支与欧拉示性数实现。我们进一步提出了一种分数欧拉示性数算法，可在计算体素化受限幂图的同时，高效地基于GPU实时计算每个受限元素的欧拉示性数。与现有的基于体素或点云的方法相比，本方法首次实现了在不全局修改依赖结构（如体素尺寸或采样密度）的情况下自适应地直接修正中轴网格，同时保持其拓扑结构与中轴特征。与特征保持方法MATFP相比，本方法以更少的中轴球体获得了几何精度相当的结果，并能更鲁棒地捕捉输入三维形状的拓扑结构。