Quasi-conformal Flat Representation of Triangulated Surfaces for Computerized Tomography

  • Eli Appleboim
  • Emil Saucan
  • Yehoshua Y. Zeevi
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4241)


In this paper we present a simple method for flattening 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 and dilatation errors. Moreover, the algorithm makes no use to derivatives, hence it is robust and suitable for analysis of noisy data. The algorithm is tested on data obtained from real CT images of the human brain cortex and colon, as well as on a synthetic model of the human skull.


Riemannian Manifold Quasiconformal Mapping Angle Dilatation Circle Packing 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, 2006)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.
    Appleboim, E., Saucan, E., Zeevi, Y.Y.: http://www.ee.technion.ac.il/people/eliap/Demos.html
  4. 4.
    Caraman, P.: n-Dimensional Quasiconformal (QCf) Mappings, Editura Academiei Române, Bucharest. Abacus Press, Tunbridge Wells Haessner Publishing, Inc., Newfoundland (1974)Google Scholar
  5. 5.
    Gehring, W.F., Väisälä, J.: The coefficients of quasiconformality. Acta Math. 114, 1–70 (1965)MATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Gu, X., Wang, Y., Yau, S.T.: Computing Conformal Invariants: Period Matrices. Communications In Information and Systems 2(2), 121–146 (2003)Google Scholar
  7. 7.
    Gu, X., Yau, S.T.: Computing Conformal Structure of Surfaces. Communications In Information and Systems 2(2), 121–146 (2002)MATHMathSciNetGoogle Scholar
  8. 8.
    Gu, X., Yau, S.T.: Global Conformal Surface Parameterization. In: Eurographics Symposium on Geometry Processing (2003)Google Scholar
  9. 9.
    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
  10. 10.
    Haker, S., Angenet, S., Tannenbaum, A., Kikinis, R., Sapiro, G., Halle, M.: Conformal Surface Parametrization for Texture Mapping. IEEE Transactions on Visualization and Computer Graphics 6(2) (June 2000)Google Scholar
  11. 11.
    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 Cerebellum. In: Taylor, C., Colchester, A. (eds.) MICCAI 1999. LNCS, vol. 1679, pp. 279–286. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  12. 12.
    Stephenson, K.: Personal communicationGoogle Scholar
  13. 13.
    Sheffer, A., de Stuler, E.: Parametrization of Faceted Surfaces for Meshing Using Angle Based Flattening. Enginneering with Computers 17, 326–337 (2001)MATHCrossRefGoogle Scholar
  14. 14.
    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
  15. 15.
    Thurston, W.: Three-Dimensional Geometry and Topology. Levy, S. (ed.), vol. 1. Princeton University Press, Princeton (1997)Google Scholar
  16. 16.
    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
  1. 1.Electrical Engineering DepartmentTechnionHaifaIsrael

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