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.

翻译：方向场，包括单位向量场、线场和叉场，是几何处理工具库中的核心工具。方向场的拓扑特性由其奇异点刻画。尽管奇异点在下游应用（如网格生成）中扮演重要角色，现有计算方向场的方法要么要求预先指定它们，要么完全忽略它们，要么将其视为松弛场中的零点。虽然场在其奇异点处定义不良，但含奇异点的方向场的图在圆丛中构成定义明确的曲面。通过将场的优化提升至其图上的优化，我们可以利用丛中电流空间上的最小截面问题所对应的自然凸松弛。这种松弛将奇异点作为一等公民对待，将场与奇异点的关系表达为显式边界条件。由于曲率阻碍了丛的有限元离散化，我们设计了一种混合谱方法来表征和优化最小截面。我们的方法支持平坦域和弯曲域上的场优化，并能更精确地控制奇异点布局。