Surface Quasi-Conformal Mapping by Solving Beltrami Equations

We consider the problem of constructing quasi-conformal mappings between surfaces by solving Beltrami equations. This is of great importance for shape registration. In the physical world, most surface deformations can be rigorously modeled as quasi-conformal maps. The local deformation is characterized by a complex-value function, Beltrami coefficient, which describes the deviation from conformality of the deformation at each point. We propose an effective algorithm to solve the quasi-conformal map from the Beltrami coefficient. The major strategy is to deform the conformal structure of the original surface to a new conformal structure by the Beltrami coefficient, such that the quasi-conformal map becomes a conformal map. By using holomorphic differential forms, conformal maps under the new conformal structure are calculated, which are the desired quasi-conformal maps. The efficiency and efficacy of the algorithms are demonstrated by experimental results. Furthermore, the algorithms are robust for surfaces scanned from real life, and general for surfaces with different topologies.