A multigrid approach for generating harmonic measured foliations

Abstract

Harmonic measured foliation has demonstrated its usefulness for many geometric problems, including conformal parameterization and mesh quadrangulation. Due to the non-linearity and hard constraints for its computation, existing iterative solvers converge very slowly, and so impractical for large meshes. Though the multigrid approach is well-known for speeding up iterative solvers, a general multigrid solver cannot be applied here in a plug-and-play fashion, because the constraints for computing the harmonic measured foliations would be broken. In this article, we design a novel multigrid solver for this problem, where we propose specific multi-resolution mesh hierarchies and interpolation schemes to fulfill the requirements of the harmonic measured foliation. Experimental results show that our multigrid solver converges much faster than the original algorithm on meshes ranging from a few thousands to over one million edges, even by over a hundred times. This would benefit scalable geometry processing using harmonic measured foliations.