A Smoothness Energy without Boundary Distortion for Curved Surfaces
A Smoothness Energy without Boundary Distortion for Curved Surfaces
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DOI：
10.1145/3377406
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发表时间：
2019-05
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影响因子：
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通讯作者：
Oded Stein;Alec Jacobson;M. Wardetzky;E. Grinspun
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文献类型：
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作者：
Oded Stein;Alec Jacobson;M. Wardetzky;E. Grinspun
当前曲面的二次光滑能量要么在边界附近由于零诺伊曼边界条件而表现出扭曲，要么不能正确地解释固有曲率，这导致了远离边界的不自然行为。这导致了一种不幸的权衡：一个人要么在内部有自然的行为，要么在边界有无扭曲的结果，但不能两者兼而有之。我们引入曲面的广义Hessian能量，用协变一形式Dirichlet能量、高斯曲率和外导数表示。能量最小化解决了拉普拉斯-贝尔特拉米双调和方程，正确地考虑了内在曲率，导致自然的等线。在边界上，最小化器尽可能是线性的，这减少了边界处等值线的畸变。我们利用Crouzeix-Raviart有限元对协变一形式Dirichlet能量进行离散化，得到了曲面上应用的Hessian能量的离散化表达式。我们在实验中观察到离散化的收敛性。
Current quadratic smoothness energies for curved surfaces either exhibit distortions near the boundary due to zero Neumann boundary conditions or they do not correctly account for intrinsic curvature, which leads to unnatural-looking behavior away from the boundary. This leads to an unfortunate trade-off: One can either have natural behavior in the interior or a distortion-free result at the boundary, but not both. We introduce a generalized Hessian energy for curved surfaces, expressed in terms of the covariant one-form Dirichlet energy, the Gaussian curvature, and the exterior derivative. Energy minimizers solve the Laplace-Beltrami biharmonic equation, correctly accounting for intrinsic curvature, leading to natural-looking isolines. On the boundary, minimizers are as-linear-as-possible, which reduces the distortion of isolines at the boundary. We discretize the covariant one-form Dirichlet energy using Crouzeix-Raviart finite elements, arriving at a discrete formulation of the Hessian energy for applications on curved surfaces. We observe convergence of the discretization in our experiments.