Extrinsic Vector Field Processing

TL;DR

This work introduces an extrinsic, continuous discretization of tangent vector fields on triangle meshes by transporting vertex tangents into triangle interiors via Rodrigues rotations aligned to a Phong-interpolated normal field, enabling pointwise evaluation of covariant derivatives. The approach yields a finite-element framework with mass and stiffness matrices for scalar and vector-field bases, and supports energies (Connection, Hodge, Killing) as well as the Lie bracket, all decomposed into symmetric, antisymmetric, and trace components. Through comprehensive experiments, including sphere spectrums and bracket evaluations, the method is shown to be competitive with or superior to prior discretizations in several regimes while maintaining gradient- and rotation-related invariants. The framework enables robust vector-field processing tasks such as sparse interpolation, vector heat diffusion, and parallel transport, with potential extensions to frame-field computation and higher-order elements. Overall, extrinsic vector-field processing provides a practical, evaluatable, and extensible discretization for geometry-processing pipelines that rely on tangent-field energies and differential operators on meshes.

Abstract

We propose a novel discretization of tangent vector fields for triangle meshes. Starting with a Phong map continuously assigning normals to all points on the mesh, we define an extrinsic bases for continuous tangent vector fields by using the Rodrigues rotation to transport tangent vectors assigned to vertices to tangent vectors in the interiors of the triangles. As our vector fields are continuous and weakly differentiable, we can use them to define a covariant derivative field that is evaluatable almost-everywhere on the mesh. Decomposing the covariant derivative in terms of diagonal multiple of the identity, anti-symmetric, and trace-less symmetric components, we can define the standard operators used for vector field processing including the Hodge Laplacian energy, Connection Laplacian energy, and Killing energy. Additionally, the ability to perform point-wise evaluation of the covariant derivative also makes it possible for us to define the Lie bracket.

Figures (10)

• Figure 1: Visualization of smooth interpolation of sparse vectors (draw in blue) using a Dirichlet energy defined by the Connection (center) and Hodge (right) Laplacians.
• Figure 2: Input point constraints, with associated Voronoi partition of the geometry (left) and the result of applying the Vector Heat method (right).
• Figure 3: Visualization of The four smallest eigenvectors of the Connection (top), Hodge (middle), and Killing (bottom) energies.
• Figure 4: The differences between the analytic eigenvalues and the estimated ones, averaged over ten random tessellations of the unit sphere, as a function of eigenvalue index are show on the right. The top chart plots results when the sphere was sampled isotropically. The bottom plots results for the more challenging case of anisotropic sampling. (The associated eigenvalues are $\{1,5,11,19,29,41,55,71,89,109\}$.) Representative tessellations of the sphere, using $10K$ vertices for ease of visibility, are shown on the right.
• Figure 5: The relative difference between the eigenvalue of the Hodge Laplacian and the co-tangent Laplacian, computed using the holmorphic part of the connection Laplacian and the discretization using the Whitney basis. The plot also gives the genus, $g$, of the model and the ratio, $\rho$, of eigenvalues $2g$ and $2g+1$.