Fast estimation of mean curvature on the surface of a 3D discrete object

  • Alexandre Lenoir
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1347)


Discrete surfaces composed of surfels (a surfel is a facet of a voxel) have interesting features. They represent the border of a discrete object and possess the Jordan property. These surfaces, although discrete by nature, represent most of the time real world continuous surfaces for which local geometrical characteristics are useful for registration, segmentation, recognition and measure. We propose a technique designed to estimate the mean curvature field of such a surface. Our approach depends on a scale parameter and has a low computational complexity. It is evaluated on synthetic surfaces, and an application is presented the extraction of sulci on a brain surface.

Index terms

Discrete surfaces surfels geometric invariant mean curvature 


  1. 1.
    E. Artzy, G. Frieder, and G. T. Herman. The theory, design, implementation and evaluation of a three-dimensional surface detection algorithm. CVGIP, 15(6):1–24, 1981.Google Scholar
  2. 2.
    G. Bertrand and R. Malgouyres. Some topological properties of discrete surfaces. In Proceedings of DGCI'96, volume 1176 of Lecture Notes in Computer Sciences. Springer, 1996.Google Scholar
  3. 3.
    P. J. Besl and R.C. Jain. Invariant surface characteristic for 3d object recognition in range images. CVGIP, 33(999):33–80, 1986.Google Scholar
  4. 4.
    M. Do Carmo. Differential geometry of curves and surfaces. Prentice Hall, 1976.Google Scholar
  5. 5.
    R. Deriche. Recursively implementing the gaussian and its derivatives. Rapport de Recherche 1893, INRIA, April 1993.Google Scholar
  6. 6.
    T. J. Fan, G. Medioni, and R. Nevata. Recognising 3d objects using surface descriptions. IEEE Transactions on Pattern Analysis and Machine Intelligence, 11(11):1140–1157, November 1989.Google Scholar
  7. 7.
    G.T. Herman. Discrete multidimensional jordan surfaces. CVGIP: Graphical Models and Image Processing, 54(6):507–515, November 1992.Google Scholar
  8. 8.
    Y. Kerbrat and J.M. Braemer. Géométrie des courbes et des surfaces. Collection méthodes. Hermann, 1976.Google Scholar
  9. 9.
    A. Lenoir, R. Malgouyres, and M. Revenu. Fast computation of the normal vector field of the surface of a 3d discrete object. In Proceedings of DGCI'96, volume 1176 of Lecture Notes in Computer Sciences, pages 101–112. Springer, 1996.Google Scholar
  10. 10.
    P. Liang and C. H. Taubes. Orientation based differential geometric representations for computer vision applications. IEEE Transactions on Pattern Analysis and Machine Intelligence, 16(3), March 1994.Google Scholar
  11. 11.
    R. Malgouyres. A definition of surfaces in ℤ3. Theorical Computer Sciences. to appear.Google Scholar
  12. 12.
    O. Monga, N. Ayache, and P. T Sander. From voxel to curvature. Rapport de Recherche 1356, INRIA, December 1990.Google Scholar
  13. 13.
    O. Monga and S. Benayoun. Using partial derivatives of 3d images to extract typical surface features, CVIU, 61:171–189, 1995.Google Scholar
  14. 14.
    D.G. Morgenthaler and A. Rosenfeld. Surfaces in three-dimensional digital images. Information and Control, 51(3):227–247, 1981.Google Scholar
  15. 15.
    J. P. Thirion and A. Gourdon. Computing the differential characteristics of isointensity surfaces. CVIU, 61:190–202, 1995.Google Scholar
  16. 16.
    L. Thurfjellj, E. Bengtsson, and Bo Nordin. A boundary-approach for fast neighbourhood operations on three-dimensional binary data. CVGIP, 57(1):13–19, 1995.Google Scholar
  17. 17.
    J. K. Udupa. Determination of 3d shape parameters from boundary information. Computer Graphics and Image Processing, 17:52–59, 1931.Google Scholar
  18. 18.
    J. K. Udupa. Multidimensional digital boundaries. CVGIP: Graphical Models and Image Processing, 56(4):311–323, July 1994.Google Scholar
  19. 19.
    J. K. Udupa and V. G. Ajjanagadde. Boundary and objet labelling in threedimensional images. CVGIP, 51:355–369; 1990.Google Scholar
  20. 20.
    M. Worring and A.W.M. Smeulders. Digital curvature estimation. CVGIP IU, 58(3):366–382, 1993.Google Scholar
  21. 21.
    L. Yang, F. Albregtsen, and T. Taxt. Fast computation of three-dimensional geometric moments using a discrete divergence theorem and a generalisation to higher dimensions. CVGIP, 59(2):97–108, March 1997.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1997

Authors and Affiliations

  • Alexandre Lenoir
    • 1
  1. 1.GREYC UPRESA 6072CAEN CedexFrance

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