# Cost Minimizing Local Anisotropic Quad Mesh Refinement

Quad meshes as a surface representation have many conceptual advantages over triangle meshes. Their edges can naturally be aligned to principal curvatures of the underlying surface and they have the flexibility to create strongly anisotropic cells without causing excessively small inner angles. While in recent years a lot of progress has been made towards generating high quality uniform quad meshes for arbitrary shapes, their adaptive and anisotropic refinement remains difficult since a single edge split might propagate across the entire surface in order to maintain consistency. In this paper we present a novel refinement technique which finds the optimal trade-off between number of resulting elements and inserted singularities according to a user prescribed weighting. Our algorithm takes as input a quad mesh with those edges tagged that are prescribed to be refined. It then formulates a binary optimization problem that minimizes the number of additional edges which need to be split in order to maintain consistency. Valence 3 and 5 singularities have to be introduced in the transition region between refined and unrefined regions of the mesh. The optimization hence computes the optimal trade-off and places singularities strategically in order to minimize the number of consistency splits — or avoids singularities where this causes only a small number of additional splits. When applying the refinement scheme iteratively, we extend our binary optimization formulation such that previous splits can be undone if this prevents degenerate cells with small inner angles that otherwise might occur in anisotropic regions or in the vicinity of singularities. We demonstrate on a number of challenging examples that the algorithm performs well in practice.