Spherical Flattening of the Cortex Surface

  • A. Elad
  • R. Kimmel
Part of the Mathematics and Visualization book series (MATHVISUAL)


We present a novel technique to ‘unfold’ the curved convoluted outer surface of the brain known as the cortex and map it onto a sphere. The mapping procedure is constructed by first measuring the inter geodesic distances between points on the cortical surface. Next, a multi-dimensional scaling (MDS) technique is applied to map the whole or a section of the surface onto the sphere. The geodesic distances on the cortex are measured by the ‘fast marching on triangulated domains’ algorithm. It calculates the geodesic distances from a vertex on a triangulated surface to the rest of the vertices in O(n) operations, where n is the number of vertices that represent the surface. Using this procedure, a matrix of the geodesic distances between every two vertices on the surface is computed. Next, a constrained MDS procedure finds the coordinates of points on a sphere such that the inter geodesic distances between points on the sphere are as close as possible to the geodesic distances measured between the corresponding points on the cortex. Thereby, our approach maximizes the goodness of fit of distances on the cortex surface to distances on the sphere. We apply our algorithm to sections of the human cortex, which is an extremely complex folded surface.


Multidimensional Scaling Geodesic Distance Cortical Surface Texture Mapping Eikonal Equation 
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.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    S. Angenent, S. Haker, A. Tannenbaum, and R. Kikinis. On the laplacebeltrami operator and brain surface flattening. IEEE Trans. on Medical Imaging, 18(8):700–711, August 1999.CrossRefGoogle Scholar
  2. 2.
    I. Borg and P. Groenen. Modern Multidimensional Scaling — Theory and Applications. Springer, 1997.Google Scholar
  3. 3.
    M. Cox and T. Cox. Multidimensional scaling on a sphere. Commun. Statist, 20(9):2943–2953, 1991.CrossRefGoogle Scholar
  4. 4.
    M. Cox and T. Cox. Multidimensional Scaling. Chapman and Hall, 1994.Google Scholar
  5. 5.
    C. Frederick and E. L. Schwartz. Confromal image warping. IEEE Trans. Pattern Anal. Machine Intell, 11(9):1005–1008, 1989.CrossRefGoogle Scholar
  6. 6.
    R. Grossman, N. Kiryati, and R. Kimmel. Computational surface flattening: A voxel-based approach. Lecture Notes in Computer Science, 2059:196–204, 2001.CrossRefGoogle Scholar
  7. 7.
    S. Haker, S. Angenent, A. Tannenbaum, R. Kikinis, G. Sapiro, and M. Halle. Conformal surface parameterization for texture mapping. IEEE Trans. on Visualization and Computer Graphics, 6:181–189, 2000.CrossRefGoogle Scholar
  8. 8.
    R. Kimmel and J. Sethian. Computing geodesic on manifolds. Proc. of National Academy of Science, 95:8431–8435, 1998.MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    J. Kruskal. Multidimensional scaling: anumerical method. Psychometrika, 36:57–62, 1964.MathSciNetCrossRefGoogle Scholar
  10. 10.
    J. Kruskal. Multidimensional scaling by optimizinggoodness of-fit to a nonmetric hypothesis. Psychometrika, 29:1–27, 1964.MathSciNetzbMATHCrossRefGoogle Scholar
  11. 11.
    J. B. Kruskal and M. Wish. Multidimensional Scaling. Sage, 1978.Google Scholar
  12. 12.
    S. Melax. A simple, fast and effective polygon reduction algorithm. Game Developer Journal, November 1998.Google Scholar
  13. 13.
    E. L. Schwartz, A. Shaw, and E. Wolfson. A numerical solution to the generalized mapmaker’s problem: Flattening nonconvex polyhedral surfaces. IEEE Trans. Pattern Anal. Machine Intell, 11(9):1005–1008, 1989.CrossRefGoogle Scholar
  14. 14.
    J. Sethian. A review of the theory, algorithms, and applications of level set method for propagating surfaces. Acta Numerica, Cambridge University Press, 1996.Google Scholar
  15. 15.
    J. N. Tsitsiklis. Efficient algorithms for globally optimal trajectories. IEEE Trans. on Automatic Control, 40:1528–1538, 1995.MathSciNetzbMATHCrossRefGoogle Scholar
  16. 16.
    B. A. Wandell, S. Chial, and B. Backus. Visualization and measurements of the cortical surface. Journal of Cognitive Neuroscience, January 2000.Google Scholar
  17. 17.
    E. Wolfson and E. L. Schwartz. Computing minimal distances on arbitrary twodimensional polyhedral surfaces. IEEE Computer Graphics and Applications, 1990.Google Scholar
  18. 18.
    G. Zigelman and R. Kimmel. Texture mapping using surface flattening via MDS. Accepted to IEEE Trans. on Visualization and Computer Graphics, 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • A. Elad
    • 1
  • R. Kimmel
    • 1
  1. 1.Technion-Israel Institute of TechnologyHaifaIsrael

Personalised recommendations