Algebraic Representations for Volumetric Frame Fields

Abstract

Fig. 1. Octahedral fields generated with mMBO + RTR on various models.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.

Designing 2D and 3D Non-Orthogonal Frame Fields

“…As for the 2D case, generalization to n-symmetry 3D vector fields [22] and to anisotropic orthogonal frame field [23,24] have been proposed by studying the properties of the spherical harmonic basis. However, to the best of our knowledge, there is no existing work on non-orthogonal 3D frame field design.…”

“…Therefore, we keep the three unit vectors u, v, w ∈ S 2 as variables. Like previous works [13,21,24], performing computation in the space of direction fields requires to solve a non-convex optimization problem.…”

Frame Field Operators

“…The odeco tensors form an algebraic variety known as an odeco variety [Rob16; BDHR17]. An odeco field is a field of odeco tensors [PBS20].…”

Octahedral Frames for Feature-Aligned Cross Fields

“…While there is no canonical three-dimensional generalization of arbitrary n-RoSy fields, the spherical harmonic representation allows for frames that mimic the symmetries of all platonic solids [Shen et al 2016], including octahedral fields [Corman and Crane 2019;Liu et al 2018;Solomon et al 2017]. Algebraic characterization of the orbit of д(x, y, z) under the space of rotations as a subset of all possible SH coefficients was presented by Palmer et al [2019] and Chemin et al [2018].…”

“…In the case that octahedral frames do degenerate significantly, their norms can be too small to project robustly. We locate these by using the octahedral projection from [Palmer et al 2019] to measure the distance from f t to the octahedral variety…”