Algebraic Representations for Volumetric Frame Fields
| Autor: | Palmer, David, Bommes, David, Solomon, Justin |
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| Rok vydání: | 2019 |
| Předmět: | |
| Zdroj: | ACM Transactions on Graphics (TOG), 39(2), 1-17 (2020) |
| Druh dokumentu: | Working Paper |
| DOI: | 10.1145/3366786 |
| Popis: | Field-guided parametrization 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. Comment: 17 pages, 20 figures |
| Databáze: | arXiv |
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