边界对齐的平滑三维对称标架场

为了把广泛应用于网格四边形化和纹理合成的二维表面标架场拓展到三维,提出一种生成三维对称标架场的方法.不同于表面对称标架场(四对称方向场),二维标架场的对称性能用一个切平面的旋转角度来表示,而三维对称标架场的对称性却不能这样简单地表示.为了解决这个问题,利用球面函数来获得一个对称性表述,该表述对于绕任意一个轴的π/2旋转以及它们的复合是不变的.基于球面函数的表示可以获得一个有效的标架场光顺程度的度量,并以球面调和分析进行加速计算；基于一组边界约束,可以通过极小化这个度量函数来获得一个光顺的标架场,该标架场在表面上能很好地对齐法线.最后通过表面投影、流线追踪和奇异点来可视化这个标架场,并将这个光顺的标架场用于六面体网格生成,且讨论了它在生成高质量纯六面体网格方面的潜力,其与表面标架场在生成四边形网格方面的潜力是一致的.

Boundary Aligned Smooth 3D Cross-Frame Field

In this paper,we present a method for constructing a 3D cross-frame field,a 3D extension of the 2D cross-frame field as applied to surfaces in applications such as quadrangulation and texture synthesis.In contrast to the surface cross-frame field(equivalent to a 4-way rotational-symmetry vector field),symmetry for 3D cross-frame fields cannot be formulated by simple one-parameter 2D rotations in the tangent planes.To address this critical issue,we represent the 3D frames by spherical harmonics,in a manner invariant to combinations of rotations around any axis by multiples of π/2.With such a representation,we can formulate an efficient smoothness measure of the cross-frame field.Through minimization of this measure under certain boundary conditions,we can construct a smooth 3D cross-frame field that is aligned with the surface normal at the boundary.We visualize the resulting cross-frame field through restrictions to the boundary surface,streamline tracing in the volume,and singularities.We also demonstrate the application of the 3D cross-frame field to producing hexahedron-dominant meshes for given volumes,and discuss its potential in high-quality hexahedralization,much as its 2D counterpart has shown in quadrangulation.