  Registration, which aims to find an optimal one-to-one correspondence between
different data, is an important problem in various fields. This problem is
especially challenging when large deformations occur. In this paper, we present
a novel algorithm to obtain diffeomorphic image or surface registrations with
large deformations via quasi-conformal maps. The basic idea is to minimize an
energy functional involving a Beltrami coefficient term, which measures the
distortion of the quasi-conformal map. The Beltrami coefficient effectively
controls the bijectivity and smoothness of the registration, even with very
large deformations. Using the proposed algorithm, landmark-based registration
between images or surfaces can be effectively computed. The obtained
registration is guaranteed to be diffeomorphic (1-1 and onto), even with a
large deformation or large number of landmark constraints. The proposed
algorithm can also be combined with matching intensity (such as image intensity
or surface curvature) to improve the accuracy of the registration. Experiments
have been carried out on both synthetic and real data. Results demonstrate the
efficacy of the proposed algorithm to obtain diffeomorphic registration between
images or surfaces.
