**Project Description:**

In this work, we propose an effective way to compute the Teichm¨uller
extremal maps between Riemann

surfaces. The goal is to develop a consistent and effective algorithm for
surface registration, which matches

corresponding landmark features. Surfaces are cut open along landmark curves
and become multiply-connected. According to Teichmuller Quasiconformal geometry,
there is an unique Teichm¨uller extremal map between two multiply-connected
surfaces that minimizes the conformality distortion as much as possible.
Given two multiplyconnected domains, we consider their quadratic differentials
which can be expressed explicitly in closed forms. By comparing the horizontal
and vertical trajectories, we can find the corresponding quadratic differentials
associated uniquely to the Teichmuller extremal map. The resulting extremal
map can then be computed by solving a Laplace equation on a unit disk with
a given boundary condition. This method can be easily applied for surface
registration between open surfaces with fixed landmark points on the boundary.
We have implemented our algorithm and experimented it on both synthetic
and real data. Experimental results show that our algorithm can effectively
compute the Teichmuller extremal map and is useful in surface registration
for shape analysis.

**Publication:
**

- Lok Ming Lui, Jingsong Liu, Xianfeng Gu, Tony
Chan and Shing-Tung Yau, Computation of Teichmuller extremal
map using Quadratic differentials, in preparation.