  Conformal surface parameterization is useful in graphics, imaging and
visualization, with applications to texture mapping, atlas construction,
registration, remeshing and so on. With the increasing capability in scanning
and storing data, dense 3D surface meshes are common nowadays. While meshes
with higher resolution better resemble smooth surfaces, they pose computational
difficulties for the existing parameterization algorithms. In this work, we
propose a novel parallelizable algorithm for computing the global conformal
parameterization of simply-connected surfaces via partial welding maps. A given
simply-connected surface is first partitioned into smaller subdomains. The
local conformal parameterizations of all subdomains are then computed in
parallel. The boundaries of the parameterized subdomains are subsequently
integrated consistently using a novel technique called partial welding, which
is developed based on conformal welding theory. Finally, by solving the Laplace
equation for each subdomain using the updated boundary conditions, we obtain a
global conformal parameterization of the given surface, with bijectivity
guaranteed by quasi-conformal theory. By including additional shape
constraints, our method can be easily extended to achieve disk conformal
parameterization for simply-connected open surfaces and spherical conformal
parameterization for genus-0 closed surfaces. Experimental results are
presented to demonstrate the effectiveness of our proposed algorithm. When
compared to the state-of-the-art conformal parameterization methods, our method
achieves a significant improvement in both computational time and accuracy.
