Subdivision Directional Fields

TL;DR

This paper introduces a structure-preserving subdivision scheme for face-based tangent directional fields on triangle meshes, enabling accurate and robust processing of directional data in geometric applications.

Contribution

A novel coordinate-free subdivision scheme that preserves curl-free and divergence-free properties of directional fields on meshes, extending to multiple vectors per face.

Findings

Reproduces curl-free fields exactly.

Reproduces divergence-free fields in the weak sense.

Enables applications in directional-field design and robust computation.

Abstract

We present a novel linear subdivision scheme for face-based tangent directional fields on triangle meshes. Our subdivision scheme is based on a novel coordinate-free representation of directional fields as halfedge-based scalar quantities, bridging the finite-element representation with discrete exterior calculus. By commuting with differential operators, our subdivision is structure-preserving: it reproduces curl-free fields precisely, and reproduces divergence-free fields in the weak sense. Moreover, our subdivision scheme directly extends to directional fields with several vectors per face by working on the branched covering space. Finally, we demonstrate how our scheme can be applied to directional-field design, advection, and robust earth mover's distance computation, for efficient and robust computation.