Transitions of the 3D Medial Axis under a One-Parameter Family of Deformations

  • Peter Giblin
  • Benjamin B. Kimia
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 2351)


The instabilities of the medial axis of a shape under deformations have long been recognized as a major obstacle to its use in recognition and other applications. These instabilities, or transitions, occur when the structure of the medial axis graph changes abruptly under deformations of shape. The recent classification of these transitions in 2D for the medial axis and for the shock graph, was a key factor both in the development of an object recognition system and an approach to perceptual organization. This paper classifies generic transitions of the 3D medial axis, by examining the order of contact of spheres with the surface, leading to an enumeration of possible transitions, which are then examined on a case by case basis. Some cases are ruled out as never occurring in any family of deformations, while others are shown to be non-generic in a one-parameter family of deformations. Finally, the remaining cases are shown to be viable by developing a specific example for each. We relate these transitions to a classification by Bogaevsky of singularities of the viscosity solutions of the Hamilton-Jacobi equation. We believe that the classification of these transitions is vital to the successful regularization of the medial axis and its use in real applications.


Viscosity Solution Medial Axis Double Arrow Deformation Path Circle Tangent 
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.
    I. A. Bogaevski. Singularities of viscosity solutions of Hamilton-Jacobi equations. 2001.Google Scholar
  2. 2.
    I. A. Bogavski. Metamorphoses of singularities of minimum functions and bifurcations of shock waves of the Burgers Eq. with vanishing viscosity. Math. J., 1(4):807–823, 1990.MathSciNetGoogle Scholar
  3. 3.
    J. Bruce and P. Giblin. Curves and Singularities. Cambridge University Press, 1984.Google Scholar
  4. 4.
    J. Bruce, P. Giblin, and C. Gibson. Symmetry sets. Proceedings of the Royal Society of Edinburgh, 101A:163–186, 1985.MathSciNetGoogle Scholar
  5. 5.
    P. J. Giblin and B. B. Kimia. On the local form of symmetry sets, and medial axes, and shocks in 3D. Technical Report LEMS-171, LEMS, Brown University, May 1998.Google Scholar
  6. 6.
    P. J. Giblin and B. B. Kimia. On the local form and transitions of symmetry sets, and medial axes, and shocks in 2D. In ICCV, pages 385–391, KerKyra, Greece, Sept. 1999.Google Scholar
  7. 7.
    P. J. Giblin and B. B. Kimia. On the local form of symmetry sets, and medial axes, and shocks in 3D. In Proceedings of CVPR, pages 566–573, Hilton Head Island, South Carolina, USA, June 13–15 2000. IEEE Computer Society Press.Google Scholar
  8. 8.
    P. J. Giblin and B. B. Kimia. On the local form and transitions of symmetry sets, medial axes, and shocks. IJCV, Submitted, March, 2001.Google Scholar
  9. 9.
    P. L. Halliman, G. G. Gordon, A. L. Yuille, P. Giblin, and D. Mumford. Two-and Three-Dimensional Patterns of the Face. A. K. Peters, 1999.Google Scholar
  10. 10.
    M. S. Johannes, T. B. Sebastian, H. Tek, and B. B. Kimia. Perceptual organization as object recognition divided by two. In Workshop on POCV, pages 41–46, 2001.Google Scholar
  11. 11.
    L. Lam, S.-W. Lee, and C. Y. Suen. Thinning methodologies-a comprehensive survey. IEEE Trans. on PAMI, 14(9):869–885, September 1992.Google Scholar
  12. 12.
    F. F. Leymarie. 3D Shape Representation via Shock Flows. PhD thesis, Brown University, 2002.Google Scholar
  13. 13.
    R. L. Ogniewicz and O. Kubler. Hierarchic voronoi skeletons. Pattern Recognition, 28(3):343–359, 1995.CrossRefGoogle Scholar
  14. 14.
    T. B. Sebastian, P. N. Klein, and B. B. Kimia. Recognition of shapes by editing shock graphs. In Proceedings of the Eighth International Conference on Computer Vision, pages 755–762, Vancouver, Canada, July 9–12 2001. IEEE Computer Society Press.Google Scholar
  15. 15.
    D. Shaked and A. M. Bruckstein. Pruning medial axes. Computer Vision and Image Understanding, 69:156–169, 1998.CrossRefGoogle Scholar
  16. 16.
    S. Tari and J. Shah. Extraction of shape skeletons from grayscale images. Computer Vision Image Understanding, 66(2):133–146, 1997.CrossRefGoogle Scholar
  17. 17.
    H. Tek and B. B. Kimia. Boundary smoothing via symmetry transforms. Journal of Mathematical Imaging and Vision, 14(3):211–223, May 2001.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2002

Authors and Affiliations

  • Peter Giblin
    • 1
  • Benjamin B. Kimia
    • 2
  1. 1.Department of Mathematical SciencesThe University of LiverpoolLiverpoolEngland
  2. 2.Division of EngineeringBrown UniversityProvidenceUSA

Personalised recommendations