**Project Description:**

We propose a simple representation of bijective surface maps,
which helps the process of surface registration. In shape analysis, finding
an optimal 1-1 correspondence between surfaces within a large class of admissible
bijective mappings is of great importance. Such a process is called surface
registration. The difficulty lies in the fact that the space of all surface
homeomorphisms is a complicated functional space. Hence, the optimization
process over the search space of bijective surface maps becomes challenging.
To tackle this problem, we propose a novel representation of surface diffeomorphisms
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 Beltrami coefficients on the source domain.
Hence, every bijective surface map can be represented by a unique Beltrami
coefficient. Conversely, given a Beltrami coefficient, we can reconstruct
the unique surface map associated to it using the Beltrami Holomorphic
flow (BHF) method introduced in this paper. Using BCs to represent surface
maps is advantageous because it captures most essential features of a surface
map. By adjusting BCs, we can adjust the surface homeomorphism accordingly
to obtain desired properties of the map. Also, the Beltrami holomorphic
flow

gives us the variation of the associated map under the variation of
BC. Using this, variational problems over the space of surface diffeomorphisms
can be easily reformulated into variational problems over the space of
BCs. The space of BCs is a simple functional space, which makes the minimization
procedure much easier. More importantly, a bijective surface map is guaranteed
to be obtained during the optimization process. We applied our method
to synthetic and real examples, which shows the effectiveness of our proposed
method.

**Publication:
**

- Lok Ming Lui, Tsz Wai Wong, Xianfeng Gu, Tony Chan and Shing-Tung Yau, Representation and Property Adjustments of Surface Diffeomorphism using Beltrami Holomorphic Flow, UCLA CAM report 09-19, to be submitted