基于形状控制的细分曲面的局部渐进插值方法

摘要

A new scheme for constructing a Catmull-Clark subdivision surface with shape control is proposed, which interpolates the local vertices of a quadrilateral mesh with arbitrary topology. Firstly, utilizing the local property of the progressive iterative approximation method, this subdivision scheme chooses a subset of the vertices on the control mesh to be adjusted and remains the others unchanged in the iterative process, it results the limit surface of the subdivision interoplates the corresponding subset of the vertices on the initial mesh. Secondly, the shape control for Catmull-Clark subdivision in the proposed scheme is based on a two-phase subdivision. The two-phase scheme works by applying a modified Catmull-Clark subdivision for the initial mesh to generate a new mesh firstly, and then applying the regular Catmull-Clark subdivision for the new mesh to resulting the limit surface. The modified subdivision scheme carries a parameter for each face of the initial mesh, and these parameters provide the degrees of freedom for the shape control of the interpolating subdivision surface. It is proven that the scheme based on Catmull-Clark subdivision with shape control is convergent. The experimental examples are given to show that the method is effective both in local progressive interpolation and shape control.%提出一种基于形状控制的 Catmull-Clark 细分曲面构造方法，实现局部插值任意拓扑的四边形网格顶点。首先该方法利用渐进迭代逼近方法的局部性质，在初始网格中选取若干控制顶点进行迭代调整，保持其他顶点不变，使得最终生成的极限细分曲面插值于初始网格中的被调整点；其次该方法的 Catmull-Clark 细分的形状控制建立在两步细分的基础上，第一步通过对初始网格应用改造的 Catmull-Clark 细分产生新的网格，第二步对新网格应用 Catmull-Clark 细分生成极限曲面，改造的 Catmull-Clark 细分为每个网格面加入参数值，这些参数值为控制局部插值曲面的形状提供了自由度。证明了基于形状控制的 Catmull-Clark 细分局部渐进插值方法的收敛性。实验结果验证了该方法可同时实现局部插值和形状控制。