Algebraic Representations for Volumetric Frame Fields
Algebraic Representations for Volumetric Frame Fields
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DOI：
10.1145/3366786
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发表时间：
2019-08
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影响因子：
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通讯作者：
David R Palmer;D. Bommes;J. Solomon
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文献类型：
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作者：
David R Palmer;D. Bommes;J. Solomon
场引导的参数化方法已被证明是有效的四网格的表面，这些方法计算光滑的交叉领域，以指导网格化过程，然后整合的领域，以构建一个离散的网格。然而，将这些方法扩展到三维的一个关键挑战是字段值的表示。交叉场可以用线性空间的切向量场来表示，而三维模拟-八面体框架场-则在非线性流形中取值。在这项工作中，我们描述的八面体框架的微分和代数几何的语言空间。有了这样的理解，我们开发了用于优化八面体场的几何感知工具，即测地线步进和通过半定松弛的精确投影。我们的代数方法不仅提供了一个优雅的和数学上健全的描述空间的八面体框架，但也建议一个泛化的框架，其三个轴的规模独立，更好地捕捉奇异的行为，我们希望看到在体积框架领域。这些新的odeco框架，所谓的，因为它们是由正交可分解的张量，也承认一个半定的程序为基础的投影算子。我们的描述的八面体和odeco框架的空间建议计算框架领域通过基于流形的优化算法，我们表明，这些算法有效地产生高质量的领域，同时保持稳定性和光滑性。
Field-guided parameterization methods have proven effective for quad meshing of surfaces; these methods compute smooth cross fields to guide the meshing process and then integrate the fields to construct a discrete mesh. A key challenge in extending these methods to three dimensions, however, is representation of field values. Whereas cross fields can be represented by tangent vector fields that form a linear space, the 3D analog—an octahedral frame field—takes values in a nonlinear manifold. In this work, we describe the space of octahedral frames in the language of differential and algebraic geometry. With this understanding, we develop geometry-aware tools for optimization of octahedral fields, namely geodesic stepping and exact projection via semidefinite relaxation. Our algebraic approach not only provides an elegant and mathematically sound description of the space of octahedral frames but also suggests a generalization to frames whose three axes scale independently, better capturing the singular behavior we expect to see in volumetric frame fields. These new odeco frames, so called as they are represented by orthogonally decomposable tensors, also admit a semidefinite program–based projection operator. Our description of the spaces of octahedral and odeco frames suggests computing frame fields via manifold-based optimization algorithms; we show that these algorithms efficiently produce high-quality fields while maintaining stability and smoothness.