Quasi-isometric and Quasi-conformal Development of Triangulated Surfaces for Computerized Tomography

  • Eli Appleboim
  • Emil Saucan
  • Yehoshua Y. Zeevi
  • Ofir Zeitoun
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4040)


In this paper we present a simple method for minimal distortion development of triangulated surfaces for mapping and imaging. The method is based on classical results of F. Gehring and Y. Väisälä regarding the existence of quasi-conformal and quasi-isometric mappings between Riemannian manifolds. A random starting triangle version of the algorithm is presented. A curvature based version is also applicable. In addition the algorithm enables the user to compute the maximal distortion errors. Moreover, the algorithm makes no use to derivatives, hence it is suitable for analysis of noisy data. The algorithm is tested on data obtained from real CT images of the human brain cortex.


Quasiconformal Mapping Circle Packing Minimal Distortion Maximal Dilatation Triangulate Surface 
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  1. 1.
    Appleboim, E., Saucan, E., Zeevi, Y.: Minimal-Distortion Mappings of Surfaces for Medical Imaging. In: Proceedings of VISAPP 2006 (to appear)Google Scholar
  2. 2.
    Appleboim, E., Saucan, E., Zeevi, Y.Y.: On Sampling and Reconstruction of Surfaces, Technion CCIT Report (2006)Google Scholar
  3. 3.
    Caraman, P.: n-Dimensional Quasiconformal (QCf) Mappings, Editura Academiei Române, Bucharest. Abacus Press, Tunbridge Wells Haessner Publishing, Inc., Newfoundland, New Jersey (1974)Google Scholar
  4. 4.
    Gehring, W.F., Väisälä, J.: The coefficients of quasiconformality. Acta Math. 114, 1–70 (1965)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Gu, X., Wang, Y., Yau, S.T.: Computing Conformal Invariants: Period Matrices. Communications In Information and Systems 2(2), 121–146 (2003)Google Scholar
  6. 6.
    Gu, X., Yau, S.T.: Computing Conformal Structure of Surfaces. Communications In Information and Systems 2(2), 121–146 (2002)MATHMathSciNetGoogle Scholar
  7. 7.
    Gu, X., Yau, S.T.: Global Conformal Surface Parameterization. In: Eurographics Symposium on Geometry Processing (2003)Google Scholar
  8. 8.
    Haker, S., Angenet, S., Tannenbaum, A., Kikinis, R.: Non Distorting Flattening Maps and the 3-D visualization of Colon CT Images. IEEE Transauctions on Medical Imaging 19(7) (July 2000)Google Scholar
  9. 9.
    Haker, S., Angenet, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal Surface Parametrization for Texture Mapping. IEEE Transauctions on Visualization and Computer Graphics 6(2) (June 2000)Google Scholar
  10. 10.
    Hurdal, M.K., Bowers, P.L., Stephenson, K., Sumners, D.W.L., Rehm, K., Schaper, K., Rottenberg, D.A.: Quasi Conformally Flat Mapping the Human Crebellum. In: Taylor, C., Colchester, A. (eds.) MICCAI 1999. LNCS, vol. 1679, pp. 279–286. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  11. 11.
    Stephenson, K.: Personal communicationGoogle Scholar
  12. 12.
    Sheffer, A., de Stuler, E.: Parametrization of Faceted Surfaces for Meshing Using Angle Based Flattening. Enginneering with Computers 17, 326–337 (2001)MATHCrossRefGoogle Scholar
  13. 13.
    Surazhsky, T., Magid, E., Soldea, O., Elber, G., Rivlin, E.: A Comparison of Gaussian and Mean Curvatures Estimation Methods on Triangular Meshes. In: Proceedings of the IEEE International Conference on Robotics and Automation, Taipei, Taiwan, September 2003, pp. 1021–1026 (2003)Google Scholar
  14. 14.
    Thurston, W.: Three-Dimensional Geometry and Topology. Levy, S. (ed.), vol. 1. Princeton University Press, Princeton (1997)Google Scholar
  15. 15.
    Väisalä, J.: Lectures on n-dimensional quasiconformal mappings. Lecture Notes in Mathematics, vol. 229. Springer, Heidelberg (1971)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • Eli Appleboim
    • 1
  • Emil Saucan
    • 1
  • Yehoshua Y. Zeevi
    • 1
  • Ofir Zeitoun
    • 2
  1. 1.Electrical Engineering Department, TechnionHaifaIsrael
  2. 2.FRS Ltd.Rosh HaAyinIsrael

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