Computing quasiconformal maps using an auxiliary metric and discrete curvature flow

Abstract

Surface mapping plays an important role in geometric processing, which induces both area and angular distortions. If the angular distortion is bounded, the mapping is called a quasiconformal mapping (QC-Mapping). Many surface mappings in our physical world are quasiconformal. The angular distortion of a QC mapping can be represented by the Beltrami differentials. According to QC Teichmuller theory, there is a one-to-one correspondence between the set of Beltrami differentials and the set of QC surface mappings under normalization conditions. Therefore, every QC surface mapping can be fully determined by the Beltrami differential and 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 QC mapping associated with the prescribed Beltrami differential. The main strategy is to define an auxiliary metric (AM) on the domain surface, such that the original QC mapping becomes conformal under the auxiliary metric. The desired QC-mapping can then be obtained by using the conventional conformal mapping method. In this paper, we first formulate a discrete analogue of QC mappings on triangular meshes. Then, we propose an algorithm to compute discrete QC mappings using the discrete Yamabe flow method. To the best of our knowledge, it is the first work to compute the discrete QC mappings for general Riemann surfaces, especially with different topologies. Numerically, the discrete QC mapping converges to the continuous solution as the mesh grid size approaches to 0. 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.