Computation of Quasi-Conformal Surface Maps Using Discrete Beltrami Flow

The manipulation of surface homeomorphisms is an important aspect in 3D modeling and surface processing. Every homeomorphic surface map can be considered as a quasiconformal map, with its local non-conformal distortion given by its Beltrami differential. As a generalization of conformal maps, quasiconformal maps are of great interest in mathematical study and real applications. Efficient and accurate computational construction of desirable quasiconformal maps between general surfaces is crucial. However, in the literature we have reviewed, all existing computational works on construction of quasiconformal maps to or from a compact domain require global parametrization onto the plane, and have difficulty to be directly applied to maps between arbitrary surfaces. This work fills up the gap by proposing to compute quasiconformal homeomorphisms between arbitrary Riemann surfaces using discrete Beltrami flow, which is a vector field corresponding to the adjustment to the intrinsic Beltrami differential of the map. The vector field is defined by a partial differential equation (PDE) in a local conformal coordinate. Based on this formulation and a composition formula, we can compute the Beltrami flow of any homeomorphism adjustment as a vector field on the target domain defined from the source domain, with appropriate boundary conditions and correspondences. Numerical tests show that our method provides a robust and efficient way of adjusting surface homeomorphisms. It is also insensitive to surface representation and has no limitation to the classes of surfaces that can be processed. Extensive numerical examples will be shown.