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.
Publication: