  In shape analysis, finding an optimal 1-1 correspondence between surfaces
within a large class of admissible bijective mappings is of great importance.
Such process is called surface registration. The difficulty lies in the fact
that the space of all surface diffeomorphisms is a complicated functional
space, making exhaustive search for the best mapping challenging. To tackle
this problem, we propose a simple representation of bijective surface maps
using Beltrami coefficients (BCs), which are complex-valued functions defined
on surfaces with supreme norm less than 1. Fixing any 3 points on a pair of
surfaces, there is a 1-1 correspondence between the set of surface
diffeomorphisms between them and the set of BCs. Hence, every bijective surface
map can be represented by a unique BC. Conversely, given a BC, we can
reconstruct the unique surface map associated to it using the Beltrami
Holomorphic flow (BHF) method. Using BCs to represent surface maps is
advantageous because it is a much simpler functional space, which captures many
essential features of a surface map. By adjusting BCs, we equivalently adjust
surface diffeomorphisms to obtain the optimal map with desired properties. More
specifically, BHF gives us the variation of the associated map under the
variation of BC. Using this, a variational problem over the space of surface
diffeomorphisms can be easily reformulated into a variational problem over the
space of BCs. This makes the minimization procedure much easier. More
importantly, the diffeomorphic property is always preserved. We test our method
on synthetic examples and real medical applications. Experimental results
demonstrate the effectiveness of our proposed algorithm for surface
registration.
