Integrable Frame Fields using Odeco Tensors

TL;DR

An energy formulation is constructed that computes smooth and integrable frame fields, in both isotropic and anisotropic settings, and expresses the frame field's Lie bracket solely in terms of the tensor representation.

Abstract

We propose a method for computing integrable orthogonal frame fields on planar surfaces. Frames and their symmetries are implicitly represented using orthogonally decomposable (odeco) tensors. To formulate an integrability criterion, we express the frame field's Lie bracket solely in terms of the tensor representation; this is made possible by studying the sensitivity of the frame with respect to perturbations in the tensor. We construct an energy formulation that computes smooth and integrable frame fields, in both isotropic and anisotropic settings. The user can prescribe any size and orientation constraints in input, and the solver creates and places the singularities required to fit the constraints with the correct topology. The computed frame field can be integrated to a seamless parametrization that is aligned with the frame field.

Figures (9)

• Figure 1: Overview of the approach on a machine composed of a rotor and stator. (Left) An integrable frame field is computed respecting user-prescribed size constraints: size of 1 on the inner and outer arcs, and size of 0.3 on the other boundaries. The frames indicate the local size and orientation of the quadrilateral mesh elements. (Right) A seamless parametrization is computed by integrating the frame field. Black triangles represent singularities and thick lines form the cut graph.
• Figure 2: A seamless parametrization across a cut. The isolines of the coordinate maps $u(\vb{p})$ and $v(\vb{p})$ are depicted in red and blue, respectively. Notice how the gradients $\grad{u},\grad{v}$ and the frame vectors $\vb{u},\vb{v}$ undergo a 90° rotation as they cross the cut.
• Figure 3: Equivalence class of the 90° rotations of an orthogonal frame $(\vb{u},\vb{v})$.
• Figure 4: Polynomial representation $p_{\vb{T}}(\theta)$ of (left) an arbitrary tensor, and (right) an odeco tensor, with the corresponding frame vectors. Green represents positive values for $p_{\vb{T}}(\theta)$ and magenta represents negative values.
• Figure 5: Orthonormal basis of circular harmonics used to represent odeco tensor polynomials. Green represents positive values for $p_{\vb{T}}(\theta)$ and magenta represents negative values.