Lifting Directional Fields to Minimal Sections

TL;DR

This paper introduces a novel approach to compute directional fields with singularities by lifting the problem to a minimal section formulation, enabling better control and handling of singularities in geometry processing.

Contribution

It proposes a convex relaxation of directional field optimization as a minimal section problem, explicitly incorporating singularities as boundary conditions.

Findings

Supports field optimization on flat and curved domains.

Provides more precise control over singularity placement.

Uses a hybrid spectral method for optimization.

Abstract

Directional fields, including unit vector, line, and cross fields, are essential tools in the geometry processing toolkit. The topology of directional fields is characterized by their singularities. While singularities play an important role in downstream applications such as meshing, existing methods for computing directional fields either require them to be specified in advance, ignore them altogether, or treat them as zeros of a relaxed field. While fields are ill-defined at their singularities, the graphs of directional fields with singularities are well-defined surfaces in a circle bundle. By lifting optimization of fields to optimization over their graphs, we can exploit a natural convex relaxation to a minimal section problem over the space of currents in the bundle. This relaxation treats singularities as first-class citizens, expressing the relationship between fields and singularities as an explicit boundary condition. As curvature frustrates finite element discretization of the bundle, we devise a hybrid spectral method for representing and optimizing minimal sections. Our method supports field optimization on both flat and curved domains and enables more precise control over singularity placement.