拓扑结构正确的三线性插值曲面的三角片逼近

A Topology Complexity Based Method to Approximate Isosurface with Trilinear Interpolated Triangular Patch

在等值面的三角片逼近问题中,采样点的选择对于逼近等值面拓扑结构的正确性和逼近的精确性都非常关键.现有的Marching Cubes以及对其进行改进的方法缺乏对原始曲面拓扑结构的考虑,通常选择同类采样点,无法保证逼近等值面具有正确的拓扑结构.为解决上述问题,将Morse理论的基本思想引入到等值面逼近问题中,提出基于拓扑复杂度的等值面逼近的新方法.该方法根据体元内部曲面拓扑复杂度不同,自适应地提取两类等值点作为采样点临界点和边界等值点.由于临界点是反映曲面拓扑结构的关键点,因此,无论原始曲面的拓扑结构复杂与否,新方法都能保证逼近等值面具有正确的拓扑结构、较高的逼近精度且基本不增加计算量和数据量.用实例对新方法和已有方法的逼近结果做了比较.

To approximate isosurface with triangular patch, the selection of sample points is pivotal to the topology correctness and approximation accuracy. In the marching cubes method and its variations, the topology of original surface is not taken into account, and only the same kind of isopoints is selected, and thus these methods can＇t guarantee correct topology of approximated isosurface. In this paper, Morse theory is incorporated into the study of triangular approximation, and a new method based on topology complexity is presented to approximate the isosurface patch inside a cell. According to the topology complexity of the original isosurface, the approximated isosurfaces can be adaptively constructed by triangulating two kinds of isopoints： critical points and the isopoints on cell edges. Because critical points are the key isopoints defining the surface topology, the new method can guarantee correct topology and high accuracy of the approximated isosurface without adding much computation and data. Examples are given for comparing the approximated isosurface generated from the new method with those from other methods.