Synchronizing Fields with Singularities

A variety of problems in geometry processing boil down to finding the most parallel field relative to a connection. Instances of this prototypical problem show up in computing direction fields and stripe patterns, quadrilateral meshing, and visualization of fluid flows. When the class of allowed fields includes those with topological defects, a relaxation is required to make the problem well-posed. We observe that these problems can be viewed as synchronization problems, which admit a natural semidefinite relaxation. We propose a unified method of solving all these problems via the efficient Burer-Monteiro factorization method. Geometrically, this amounts to lifting the field values to a higher-dimensional manifold, naturally resolving the singular nature of defects. Practically, we show that our convex relaxation method achieves better and more reliable optima than previous work employing alternative relaxations.