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Teichmuller mapping (T-Map) and its applications

Project Description:
Registration, which aims to find an optimal 1-1 correspondence between shapes (or images), is important in different areas such as in computer vision and medical imaging. Conformal mappings have been widely used to obtain a diffeomorphism between shapes that minimizes angular distortion. Conformal registrations are beneficial since it preserves the local geometry well. However, when extra constraints (such as landmark constraints) are enforced, conformal mappings generally do not exist. This motivates us to look for an optimal quasi-conformal registration, which satisfies the required constraints while minimizing the conformality distortion. Under suitable condition on the constraints, a unique diffeomporphism, called the Teichmuller  mapping between two surfaces can be obtained, which minimizes the maximal conformality distortion. In this work, we propose an efficient iterative algorithm, called the Quasi-conformal (QC) iterations, to compute the Teichmuller mapping. The basic idea is to represent the set of diffeomorphisms using Beltrami coefficients (BCs), and look for an optimal BC associated to the desired Teichmuller mapping. The associated diffeomorphism can be efficiently reconstructed from the optimal BC using the Linear Beltrami Solver(LBS). Using BCs to represent diffeomorphisms guarantees the diffeomorphic property of the registration. Using the proposed method, the Teichmuller mapping can be accurately and efficiently computed within 10 seconds. The obtained registration is guaranteed to be bijective. This proposed algorithm can also be practically applied to real applications. In our experiments, we have examined  how Teichmuller  mapping can be used for brain landmark matching registration, constrained texture mapping and face recognition.


  • K.C. Lam, X.F. Gu, S.T. Yau, L.M. Lui, Teichmuller mapping (T-Map) and its applications to landmark matching registrations, accepted, SIAM Journal on Imaging Sciences (2013)