  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.
