Advertisement

Noise-Resistant Affine Skeletons of Planar Curves

  • Santiago Betelu
  • Guillermo Sapiro
  • Allen Tannenbaum
  • Peter J. Giblin
Conference paper
First Online:
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1842)

Abstract

A new definition of affine invariant skeletons for shape representation is introduced. A point belongs to the affine skeleton if and only if it is equidistant from at least two points of the curve, with the distance being a minima and given by the areas between the curve and its corresponding chords. The skeleton is robust, eliminating the need for curve denoising. Previous approaches have used either the Euclidean or affine distances, thereby resulting in a much less robust computation. We propose a simple method to compute the skeleton and give examples with real images, and show that the proposed definition works also for noisy data. We also demonstrate how to use this method to detect affine skew symmetry.

Keywords

Real Image Medial Axis Planar Curf Simple Closed Curve Euclidean Case 
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.
Download to read the full conference paper text

References

  1. 1.
    S. Betelu, G. Sapiro, A. Tannenbaum and P. Giblin, “On the computation of the affine skeletons of planar curves and the detection of skew symmetry,” submitted to Pattern Recognition, 1999.Google Scholar
  2. 2.
    H. Blum, “Biological shape and visual science,” J. Theor. Biology 38, pp. 205–287, 1973.CrossRefMathSciNetGoogle Scholar
  3. 3.
    J. M. Brady and H. Asada, “Smoothed local symmetries and their implementation,” Int. J. of Rob. Res. 3, 1984.Google Scholar
  4. 4.
    R. A. Brooks, “Symbolic reasoning among 3D models and 2D images,” Artificial Intelligence 17, pp. 285–348, 1981.CrossRefGoogle Scholar
  5. 5.
    J. W. Bruce and P. J. Giblin, Curves and Singularities, Cambridge University Press, Cambridge, 1984; second edition 1992.zbMATHGoogle Scholar
  6. 6.
    J. W. Bruce, P. J. Giblin and C. G. Gibson, “Symmetry sets,” Proc. Royal Soc. Edinburgh 101A pp. 163–186, 1985.MathSciNetGoogle Scholar
  7. 7.
    J.W. Bruce and P.J. Giblin, “Growth, motion and one-parameter families of symmetry sets,” Proc. Royal Soc. Edinburgh 104A, pp. 179–204, 1986.MathSciNetGoogle Scholar
  8. 8.
    V. Caselles, R. Kimmel, and G. Sapiro, “Geodesic active contours,” Int. J. Computer Vision, 22:1 pp. 61–79, 1997.zbMATHCrossRefGoogle Scholar
  9. 9.
    P. Giblin and G. Sapiro “Affine invariant symmetry sets and skew symmetry,” International Conference on Computer Vision, Bombay, India, January 1998.Google Scholar
  10. 10.
    P. J. Giblin and G. Sapiro, “Affine invariant distances, envelopes and symmetry sets”, Geom. Ded. 71, pp. 237–261, 1998.zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    P.J. Giblin and G. Sapiro, “Affine versions of the symmetry set,” in Real and Complex Singularities, Chapman and Hall/CRC Research Notes in Mathematics 412 (000), 173–187Google Scholar
  12. 12.
    Paul Holtom, Ph.D. Thesis, Liverpool University, 2000.Google Scholar
  13. 13.
    R. Jain, R. R. Kasturi, B. Schunck, Machine Vision, McGraw Hill, NY, 1995, p. 50.Google Scholar
  14. 14.
    I. Kovács and B. Julesz, “Perceptual sensitivity maps within globally defined visual shapes,” Nature 370 pp. 644–646, 1994.CrossRefGoogle Scholar
  15. 15.
    S. Kichenassamy, A. Kumar, P. Olver, A. Tannenbaum, and A. Yezzi, “Conformal curvature flows: from phase transitions to active vision,” Archive for Rational Mechanics and Analysis 134 (1996), pp. 275–301.zbMATHCrossRefMathSciNetGoogle Scholar
  16. 16.
    M. Leyton, Symmetry, Causality, Mind, MIT-Press, Cambridge, 1992.Google Scholar
  17. 17.
    L. Moisan, “Affine Plane Curve Evolution: A Fully Consistent Scheme,” IEEE Transactions on Image Processing, 7, pp. 411–420, (1998)zbMATHCrossRefMathSciNetGoogle Scholar
  18. 18.
    R. L. Ogniewicz, Discrete Voronoi skeletons, Hartung-Gorre Verlag Konstanz, Zurich, 1993.Google Scholar
  19. 19.
    J. Ponce, “On characterizing ribbons and finding skewed symmetries,” Proc. Int. Conf. on Robotics and Automation, pp. 49–54, 1989.Google Scholar
  20. 20.
    F. P. Preparata and M. I. Shamos, Computational Geometry, Texts and Monographs in Computer Science, Springer-Verlag, New York, 1990.Google Scholar
  21. 21.
    J. Serra, Image Analysis and Mathematical Morphology, Vol. 1, Academic Press, New York, 1982.zbMATHGoogle Scholar
  22. 22.
    L. Van Gool, T. Moons, and M. Proesmans, “Mirror and point symmetry under perspective skewing,” CVPR’ 96, pp. 285–292, San Francisco, June 1996.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2000

Authors and Affiliations

  • Santiago Betelu
    • 1
  • Guillermo Sapiro
    • 2
  • Allen Tannenbaum
    • 3
  • Peter J. Giblin
    • 4
  1. 1.Department of MathematicsUniversity of MinnesotaMinneapolis
  2. 2.Department of Electrical and Computer EngineeringUniversity of MinnesotaMinneapolis
  3. 3.Department of Electrical and Computer EngineeringGeorgia Institute of TechnologyAtlanta
  4. 4.Department of MathematicsU. of LiverpoolLiverpoolUK

Personalised recommendations

Cite paper