Genus-One Surface Registration via Teichmüller Extremal Mapping

  • Ka Chun Lam
  • Xianfeng Gu
  • Lok Ming Lui
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8675)


This paper presents a novel algorithm to obtain landmark-based genus-1 surface registration via a special class of quasi-conformal maps called the Teichmüller maps. Registering shapes with important features is an important process in medical imaging. However, it is challenging to obtain a unique and bijective genus-1 surface matching that satisfies the prescribed landmark constraints. In addition, as suggested by [11], conformal transformation provides the most natural way to describe the deformation or growth of anatomical structures. This motivates us to look for the unique mapping between genus-1 surfaces that matches the features while minimizing the maximal conformality distortion. Existence and uniqueness of such optimal diffeomorphism is theoretically guaranteed and is called the Teichmüller extremal mapping. In this work, we propose an iterative algorithm, called the Quasi-conformal (QC) iteration, to find the Teichmüller extremal mapping between the covering spaces of genus-1 surfaces. By representing the set of diffeomorphisms using Beltrami coefficients (BCs), we look for an optimal BC which corresponds to our desired diffeomorphism that matches prescribed features and satisfies the periodic boundary condition on the covering space. Numerical experiments show that our proposed algorithm is efficient and stable for registering genus-1 surfaces even with large amount of landmarks. We have also applied the algorithm on registering vertebral bones with prescribed feature curves, which demonstrates the usefulness of the proposed algorithm.


Fundamental Domain Extremal Mapping Vertebral Bone Beltrami Equation Universal Covering Space 
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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • Ka Chun Lam
    • 1
  • Xianfeng Gu
    • 2
  • Lok Ming Lui
    • 1
  1. 1.Department of MathematicsThe Chinese University of Hong KongHong Kong
  2. 2.Department of Computer ScienceState University of New York at Stony BrookUSA

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