带2^(1/2)细分结构的四边形化

Quadriangulation with 2^(1/2)-subdivision structure

在Morse函数理论的基础上,提出一种新的从三角网格中建立四边形网格多分辨率表示的方法.先由人工指定或从拉普拉斯矩阵的特征函数中提取临界点,计算带约束的拉普拉斯方程得到光滑的Morse函数.函数的临界点(极大、极小和鞍点)有规律地分布在模型表面,在三角网格表面梯度场的引导下,生成临界点间流线,得到临界点间的拓扑关系.通过临界点交换规则,同样是采用流线的方法,得到更精细的四边形网格。最终可实现无需参数化而仅用流线方法来建立不同多分辨率表示的四边形网格。

Based on the Morse theory, a novel approach to create quad-mesh multiresolution from triangular mesh models is proposed. A smooth Morse function on a given triangular mesh is firstly defined as the solution of a Laplacian equation with constraints in which critical points are either specified by user or exacted from an eigenfunction of the Laplaeian matrix of the mesh. As critical point layout has been carefully treated, maximum, minimum and saddle points of the function will automatically possess of structure of a quad mesh whose connectivity is then produced by tracing the stream lines of the function guided under the gradient field of the function. A critical point exchange rule is employed to generate the structure of the next finer quad mesh whose connectivity is finally also generated through stream line tracing. Parameterization is not required in the process as all resolution levels are created using the stream lines method.