Natural Boundary Conditions for Smoothing in Geometry Processing
Natural Boundary Conditions for Smoothing in Geometry Processing
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DOI：
10.1145/3186564
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发表时间：
2017-07
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影响因子：
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通讯作者：
Oded Stein;E. Grinspun;M. Wardetzky;Alec Jacobson
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文献类型：
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作者：
Oded Stein;E. Grinspun;M. Wardetzky;Alec Jacobson
在几何处理中，平滑能量通常用于在形状优化期间对离散数据插值、密集数据去噪和正则化进行建模。平方拉普拉斯能量是一种流行的能量选择，并有相应的标准实现：对离散拉普拉斯矩阵进行平方。对于紧域，当沿边界的值沿着是事先未知的，这种结构烘烤在低阶边界条件。这导致边界的几何形状强烈地偏向解。对于许多应用，这是不期望的。相反，我们建议使用Hessian的平方Frobenius范数作为平滑能量。与平方拉普拉斯能量不同，这种能量的自然边界条件（最大限度地减少能量的边界条件）对应于有意义的高阶边界条件。这些边界条件对自由边界进行建模，其中边界的形状不应使解局部偏置。我们的分析开始于平滑设置，并在2D网格上使用有限差分或三角形网格的混合有限元进行离散化。我们展示了平方海森函数作为各种任务的平滑能量的核心行为。
In geometry processing, smoothness energies are commonly used to model scattered data interpolation, dense data denoising, and regularization during shape optimization. The squared Laplacian energy is a popular choice of energy and has a corresponding standard implementation: squaring the discrete Laplacian matrix. For compact domains, when values along the boundary are not known in advance, this construction bakes in low-order boundary conditions. This causes the geometric shape of the boundary to strongly bias the solution. For many applications, this is undesirable. Instead, we propose using the squared Frobenius norm of the Hessian as a smoothness energy. Unlike the squared Laplacian energy, this energy’s natural boundary conditions (those that best minimize the energy) correspond to meaningful high-order boundary conditions. These boundary conditions model free boundaries where the shape of the boundary should not bias the solution locally. Our analysis begins in the smooth setting and concludes with discretizations using finite-differences on 2D grids or mixed finite elements for triangle meshes. We demonstrate the core behavior of the squared Hessian as a smoothness energy for various tasks.