一种基于局部平均法向变形的网格参数化方法

针对有边界无边界的网格参数化问题, 提出一种局部平均法向变形的网格参数化方法, 以平均曲率流的方式为参考, 将顶点推向其邻居的平均位置, 使网格变形至平面或球面. 首先, 计算每个三角形邻居面的平均法向, 并以该法向为目标, 计算每个面法向变化的旋转矩阵; 其次, 基于Poisson方程将整个网格重新“缝合”, 通过优化拉伸能量, 计算顶点的新坐标. 交替迭代上述两个步骤, 将网格变形至常平均曲率曲面. 该算法与一般的基于能量优化的方法不同, 每次迭代只需求解稀疏线性方程, 因此可以快速处理大型数据集. 通过在形变过程中加入惩罚函数动态地调整全局平均法向量的权重, 避免了变形过程中三角形退化或翻转的问题. 实验结果表明, 与其他参数化方法相比, 该方法具有实用可靠、 计算效率高等优点, 并能在同一框架下计算低扭曲的平面参数化和球面参数化.

Mesh Parameterization Method Based on Local Average Normal Deformation

Aiming at the parameterization problems for bounded and closed meshes, we proposed a mesh parameterization method based on local average  normal deformation. Taking the mean curvature flow as a reference, the vertices were pushed to the average position of their neighbors, thus deforming a mesh to a plane or a sphere. Firstly, the average normal of neighbor faces of each triangle was calculated and the rotation matrix of the normal was calculated. Secondly, based on Poisson system, the whole mesh were reconstructed by optimizing a stretching energy, the new coordinates of vertices were calculated. By alternating the above two steps, the original mesh was deformed to a constant mean curvature surface. The algorithm was different from the general methods based on energy optimization, which noly needed to solve a sparse linear system in each iteration, therefore, it could process large data sets quickly. By adding a penalty function during the deformation to dynamically adjust the weights of global average normals, the problem of triangle degeneration or flipping in the deformation process was avoided. The experimental results show that compared with other parameterization methods, the proposed method has the advantages of practical reliability and high efficiency, and can compute low-distortion plane parameterization and spherical parameterization in the same frame.