Smoothing and matching of 3-D space curves

  • André Guéziec
  • Nicholas Ayache
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 588)


We present a new approach to the problem of matching 3D curves. The approach has an algorithmic complexity sublinear with the number of models, and can operate in the presence of noise and partial occlusions.

Our method builds upon the seminal work of [9], where curves are first smoothed using B-splines, with matching based on hashing using curvature and torsion measures. However, we introduce two enhancements:

  • We make use of non-uniform B-spline approximations, which permits us to better retain information at high curvature locations. The spline approximations are controlled (i.e., regularized) by making use of normal vectors to the surface in 3-D on which the curves lie, and by an explicit minimization of a bending energy. These measures allow a more accurate estimation of position, curvature, torsion and Frénet frames along the curve;

  • The computational complexity of the recognition process is considerably decreased with explicit use of the Frénet frame for hypotheses generation. As opposed to previous approaches, the method better copes with partial occlusion. Moreover, following a statistical study of the curvature and torsion covariances, we optimize the hash table discretization and discover improved invariants for recognition, different than the torsion measure. Finally, knowledge of invariant uncertainties is used to compute an optimal global transformation using an extended Kalman filter.

We present experimental results using synthetic data and also using characteristic curves extracted from 3D medical images.


Extended Kalman Filter Partial Occlusion Rigid Transformation Basis Spline Curve Match 
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.


  1. 1.
    N. Ayache, J.D. Boissonnat, L. Cohen, B. Geiger, J. Levy-Vehel, O. Monga, and P. Sander. Steps toward the automatic interpretation of 3-d images. In H. Fuchs K. Hohne and S. Pizer, editors, 3D Imaging in Medicine, pages 107–120. NATO ASI Series, Springer-Verlag, 1990.Google Scholar
  2. 2.
    N. Ayache and O.D. Faugeras. Hyper: A new approach for the recognition and positioning of two-dimensional objects. IEEE Transactions on Pattern Analysis and Machine Intelligence, 8(1):44–54, January 1986.Google Scholar
  3. 3.
    R. Bartels, J. Beatty, and B. Barsky. An introduction to splines for use in computer graphics and geometric modeleling. Morgan Kaufmann publishers, 1987.Google Scholar
  4. 4.
    Court B. Cutting. Applications of computer graphics to the evaluation and treatment of major craniofacial malformation. In Jayaram K.Udupa and Gabor T. Herman, editors, 3D Imaging in Medicine. CRC Press, 1989.Google Scholar
  5. 5.
    W. Eric L. Grimson and Daniel P. Huttenlocher. On the verification of hypothesized matches in model-based recognition. IEEE Transactions on Pattern Analysis and Machine Intelligence, 13(12):1201–1213, December 1991.Google Scholar
  6. 6.
    W.E.L Grimson and T. Lozano-Perèz. Model-based recognition and localization from sparse range or tactile data. International Journal of Robotics Research, 3(3):3–35, 1984.Google Scholar
  7. 7.
    A. Guéziec and N. Ayache. Smoothing and matching of 3d-space curves. Technical Report 1544, Inria, 1991.Google Scholar
  8. 8.
    J.C. Holladay. Smoothest curve approximation. Math. Tables Aids Computation, 11:233–243, 1957.Google Scholar
  9. 9.
    E. Kishon, T. Hastie, and H. Wolfson. 3-d curve matching using splines. Technical report, AT&T, November 1989.Google Scholar
  10. 10.
    G. Malandain, G. Bertrand, and Nicholas Ayache. Topological segmentation of discrete surface structures. In Proc. International Conference on Computer Vision and Pattern Recognition, Hawai,USA, June 1991.Google Scholar
  11. 11.
    O. Monga, N. Ayache, and P. Sander. From voxels to curvature. In Proc. International Conference on Computer Vision and Pattern Recognition, Hawai,USA, June 1991.Google Scholar
  12. 12.
    Olivier Monga, Serge Benayoun, and Olivier D. Faugeras. Using third order derivatives to extract ridge lines in 3d images. In submitted to IEEE Conference on Vision and Pattern Recognition, Urbana Champaign, June 1992.Google Scholar
  13. 13.
    T. Pavlidis. Structural Pattern Recognition. Springer-Verlag, 1977.Google Scholar
  14. 14.
    M. Plass and M. Stone. Curve fitting with piecewise parametric cubics. In Siggraph, pages 229–239, July 1983.Google Scholar
  15. 15.
    P. Saint-Marc and G. Medioni. B-spline contour representation and symmetry detection. In First European Conference on Computer Vision (ECCV), Antibes, April 1990.Google Scholar
  16. 16.
    F. Stein. Structural hashing: Efficient 3-d object recognition. In Proc. International Conference on Computer Vision and Pattern Recognition, Hawai,USA, June 1991.Google Scholar
  17. 17.
    D. W. Thompson and J. L. Mundy. 3-d model matching from an unconstrained viewpoint. In Proc. International Conference on Robotics and Automation, pages 208–220, 1987.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1992

Authors and Affiliations

  • André Guéziec
    • 1
  • Nicholas Ayache
    • 1
  1. 1.INRIA BP 105Le Chesnay CédexFrance

Personalised recommendations