Algebraic Representations for Volumetric Frame Fields

Palmer, David; Bommes, David; Solomon, Justin (15 August 2019). Algebraic Representations for Volumetric Frame Fields (arXiv). Cornell University

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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.

Item Type:

Working Paper

Division/Institute:

08 Faculty of Science > Institute of Computer Science (INF)

UniBE Contributor:

Bommes, David

Subjects:

000 Computer science, knowledge & systems
500 Science > 510 Mathematics

Series:

arXiv

Publisher:

Cornell University

Language:

English

Submitter:

Nicolas Gallego Ortiz

Date Deposited:

08 Apr 2020 16:37

Last Modified:

08 Apr 2020 16:37

ArXiv ID:

1908.05411v1

BORIS DOI:

10.7892/boris.142372

URI:

https://boris.unibe.ch/id/eprint/142372

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