Surface mapping plays an important role in geometric processing. They induce both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasi-conformal mapping. Many surface mappings in our physical world are quasi-conformal. The angular distortion of a quasi-conformal map can be represented by the Beltrami differentials. According to quasi-conformal Teichmuller theory, there is an one to one correspondence between the set of Beltrami differentials and the set of quasiconformal surface mappings. Therefore, every quasiconformal surface map can be fully determined by the Beltrami differentials and can be reconstructed by solving the so-called Beltrami equation.
In this work, we propose an effective method to solve the Beltrami equation on general Riemann surfaces.
The solution is a quasi-conformal map associated with the prescribed Beltrami differential. The main strategy is to define an auxiliary metric of the source surface, such that the original quasi-conformal map becomes conformal under the auxiliary metric. The associated map can then be obtained by using the discrete curvature flow method. In our work, we use both Euclidean and hyperbolic Yamabe flow for the computation. We tested our algorithm on surfaces scanned from real life with different topologies. Experimental results demonstrate the generality and accuracy of our auxiliary metric method.