基于边长的三维形状插值

形状插值在计算机图形学和几何处理中是一个极其重要而基础的问题，在计算机动画等领域有 着广泛应用。注意到在平面三角网格和三维四面体网格插值问题中，对边长平方插值等价于对回拉度量进行插 值，因此具有等距扭曲和共形扭曲同时有界的良好性质。通过将其推广至曲面三角网格，提出了一种完全基于 边长的曲面三角网格插值算法。给定边长，在重建网格阶段，使用牛顿法对边长误差能量进行优化。并且给出 了其海森矩阵的解析正定化形式，从而避免了高代价的特征值分解步骤。注意到四面体网格的边长平方插值结 果具有极低曲率，意味着只需少许修改即可将其压平从而嵌入三维空间。因此提出先将曲面三角网格四面体化， 再从四面体网格的插值结果提取表面。然后将这表面作为初始化用于边长误差能量的牛顿迭代，从而使得收敛 结果更加接近全局最优。在一系列三角网格上进行了实验，结果说明了本文方法比之前方法的边长误差更小， 且得到的结果还是有界扭曲的。

Edge length based 3D shape interpolation

Shape interpolation is of important and fundamental significance to computer graphics and geometry processing, which is widely employed in computer animation and other fields. It is noted that for planar triangular meshes and 3D tetrahedral meshes, interpolating squared edge lengths is equivalent to interpolating pullback metric. Therefore, it has the good property that isometric distortion and conformation distortion are bounded simultaneously. A triangular mesh interpolation algorithm based on edge lengths was proposed by extending that to triangular meshes. Given the edge lengths, the edge length error energy was optimized using Newtonʹs method in the stage of mesh reconstruction. In addition, the costly eigenvalue decomposition could be avoided by giving the analytic positive definite form of its Hessian matrix. It was noted that the interpolation of squared edge lengths of the tetrahedral meshes resulted in very low curvature, meaning that it could be flattened and embedded in 3D space with only a few modifications. Therefore, we proposed to first convert the triangular meshes into tetrahedral meshes, and then extract the surface from the interpolation result of the tetrahedral meshes. After that the surface served as an initialization on the Newton iteration of the edge length error energy, thus bringing the convergence result closer to the global optimum. Experiments performed on a series of triangular meshes show that the proposed method leads to smaller edge length error than that of previous edge length-based methods, and that the results obtained have bounded distortion.