Abstract
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.
Chapter PDF
Similar content being viewed by others
Keywords
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
References
Bookstein, F.L.: Principal Warps: Thin-Plate splines and the decomposition of deformations. IEEE Trans. Pattern Anal. Machine Intell. 11(6), 567–585 (1989)
Gardiner, F., Lakic, N.: Quasiconformal Teichmuller Theory. American Mathematics Society (2000)
Glaunès, J., Vaillant, M., Miller, M.I.: Landmark Matching via Large Deformation Diffeomorphisms on the Sphere. JMIV 20(1-2), 179–200 (2004)
Gu, X.F., Wang, Y., Chan, T.F., Thompson, P.M., Yau, S.T.: Genus zero surface conformal mapping and its application to brain surface mapping. IEEE Trans. Med. Imag. 23(8), 949–958 (2004)
Hurdal, M.K., Stephenson, K.: Discrete conformal methods for cortical brain flattening. Neuroimage 45(1), 86–98 (2009)
Jin, M., Kim, J., Luo, F., Gu, X.F.: Discrete surface Ricci flow. IEEE Trans. Visual. Comput. Graphics 14(5), 1030–1043 (2008)
Joshi, S., Miller, M.I.: Landmark matching via large deformation diffeomorphisms. IEEE Trans. Image Processing 9(8), 1357–1370 (2000)
Lui, L.M., Thiruvenkadam, S., Wang, Y., Chan, T.F., Thompson, P.M.: Optimized conformal parameterization of cortical surfaces using shape based matching of landmark curves. In: Metaxas, D., Axel, L., Fichtinger, G., Székely, G. (eds.) MICCAI 2008, Part I. LNCS, vol. 5241, pp. 494–501. Springer, Heidelberg (2008)
Lui, L.M., Wang, Y., Chan, T.F., Thompson, P.M.: Landmark constrained genus zero surface conformal mapping and its application to brain mapping research. Appl. Numer. Math. 57(5), 847–858 (2007)
Lui, L.M., Wen, C.F.: Geometric Registration of High-Genus Surfaces. SIIMS 7(1), 337–365 (2014)
Thompson, D.W.: On growth and form (1942)
Lui, L.M., Gu, X.F., Yau, S.T.: Convergence analysis of an iterative algorithm for Teichmüller maps via harmonic energy optimization. Math. Comp. (2014)
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2014 Springer International Publishing Switzerland
About this paper
Cite this paper
Lam, K.C., Gu, X., Lui, L.M. (2014). Genus-One Surface Registration via Teichmüller Extremal Mapping. In: Golland, P., Hata, N., Barillot, C., Hornegger, J., Howe, R. (eds) Medical Image Computing and Computer-Assisted Intervention – MICCAI 2014. MICCAI 2014. Lecture Notes in Computer Science, vol 8675. Springer, Cham. https://doi.org/10.1007/978-3-319-10443-0_4
Download citation
DOI: https://doi.org/10.1007/978-3-319-10443-0_4
Publisher Name: Springer, Cham
Print ISBN: 978-3-319-10442-3
Online ISBN: 978-3-319-10443-0
eBook Packages: Computer ScienceComputer Science (R0)