  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 {\it quasi-conformal} map. Many surface maps in our
physical world are quasi-conformal. The angular distortion of a quasi-conformal
map can be represented by Beltrami differentials. According to quasi-conformal
Teichm\"uller theory, there is an 1-1 correspondence between the set of
Beltrami differentials and the set of quasi-conformal surface maps. Therefore,
every quasi-conformal surface map can be fully determined by the Beltrami
differential 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. We firstly formulate a discrete
analog of quasi-conformal maps on triangular meshes. Then, we propose an
algorithm to compute discrete quasi-conformal maps. The main strategy is to
define a discrete auxiliary metric of the source surface, such that the
original quasi-conformal map becomes conformal under the newly defined discrete
metric. The associated map can then be obtained by using the discrete Yamabe
flow method. Numerically, the discrete quasi-conformal map converges to the
continuous real solution as the mesh 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.
