中文摘要：

一般情况下,以密度函数作为权重的带权测地距离并不满足严格的三角不等式,给诸多几何问题的解决带来了一定的困难.为此,提出一种基于密度函数重构非退化度量的鲁棒算法.该算法将给定密度函数与网格曲面的缺省密度场相结合重设网格曲面的边长,并保证每个三角形的新边长仍然满足三角不等式;然后使用精确的测地线算法计算任意两点之间的带权测地距离.实验结果表明,文中算法以平均曲率作为密度函数重构度量,在自适应采样与重新网格化问题上得到了高质量的结果,展示了该算法的有用性和有效性.

英文摘要：

As an important tool for controlling the local significance of a surface, density functions are widely used in many computer graphics occasions, ranging from sampling to remeshing. However, the weighted geodesic distance induced from an arbitrarily given density function generally doesn＇t satisfy the triangle inequality, and degenerate cases often occur. This leads to difficulties in solving geometry processing problems. In this paper, we propose a novel technique to transform an arbitrary input density function to a non-degenerate metric by re-configuring the edge lengths of the mesh, which facilitates further geometry processing tasks. In order to demonstrate the usefulness and effectiveness of our algorithm, we construct a new metric from mean curvatures, employ exact geodesic algorithm to compute distances, and finally obtain high-quality adaptive sampling and remeshing results with the help of the new metric.