Discrete Approach to Curve Evolution

  • Longin Jan Latecki
  • Rolf Lakämper
Part of the Informatik aktuell book series (INFORMAT)


We propose a simple approach to evolution of digital planar curves that is specially designed to fit discrete nature of curves in digital images. The obtained curve evolution method by digital linearization has many advantages in comparison to curve evolutions in scale-space theories that are usually guided by diffusion equations. We will show that it leads to simplification of shape complexity, analog to evolutions guided by diffusion equations, but with no blurring (i.e., shape rounding) effects and no dislocation of relevant features. Moreover, in our approach the problem to determine the size of discrete steps for numerical implementations does not occur, since our evolution method leads in a natural way to a finite number of discrete evolution steps which are just the iterations of a basic procedure of digital linearization.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    A. Bengtsson and J.-O. Eklundh. Shape Representation by Mutliscale Contour Approximation. IEEE Trans. PAMI 13, 85–93, 1991.CrossRefGoogle Scholar
  2. 2.
    A. M. Bruckstein, G. Shapiro, and C. Shaked. Evolutions of Planer Polygons. Int. J. of of Pattern Recognition and AI9, 991–1014, 1995.Google Scholar
  3. 3.
    A. Brunn, U. Weidner, and W. Forstner. Model-based 2D-Shape Recovery. Proc. of 17. DAGM Conf. on Pattern Recognition (Mustererkennung), Bielefeld, Springer- Verlag, Berlin, 260–268, 1995.Google Scholar
  4. 4.
    I. Debled-Rennesson and J.-P. Reveilles. A Linear Algorithm for Segmentation of Digital Curves. Intl. J. of Pattern Recognition and AI9, 635–662, 1995.Google Scholar
  5. 5.
    M. A. Grayson. The Heat Equation Shrinks Embedded Plane Curves to Round Points. Journal of Differential Geometry 26, 285–314, 1987.MathSciNetzbMATHGoogle Scholar
  6. 6.
    A. Gross and L. J. Latecki, A realistic digitization model of straight lines, Computer Vision and Image Understanding 67, 131–142, 1997.CrossRefGoogle Scholar
  7. 7.
    T. Iijima. Basic theory of pattern observation, (in Japanese) IECE, 1959.Google Scholar
  8. 8.
    B. B. Kimia, A. R. Tannenbaum, and S. W. Zucker. Shapes, shocks, and deformations, I: The components of shape and the reaction-diffusion space. Int. J. of Computer Vision 15, 189–224, 1995.CrossRefGoogle Scholar
  9. 9.
    B. B. Kimia and K. Siddiqi. Geometric Heat Equation and Nonlinear Diffusion of Shapes and Images. Computer Vision and Image Understanding 64, 305–322, 1996.CrossRefGoogle Scholar
  10. 10.
    L. J. Latecki. Discrete Representation of Spatial Objects in Computer Vision Kluwer Academic Publisher, Dordrecht, 1998.Google Scholar
  11. 11.
    L. J. Latecki and R. Lakamper. Scholar
  12. 12.
    L. J. Latecki and R. Lakamper. Hierarchical Shape Decomposition Based on Contour Evolution. Proc. DAGM Mustererkennung, Stuttgart 1998.Google Scholar
  13. 13.
    F. Mokhtarian and A. K. Mackworth. A Theory of Multiscale, Curvature-Based Shape Representation for Planar Curves. IEEE Trans. PAMI 14, 789–805, 1992.CrossRefGoogle Scholar
  14. 14.
    U. Ramer. An Iterative Procedure for the Polygonal Approximation of Plane Curves. Computer Graphics and Image Processing 1, 244–256, 1972.CrossRefGoogle Scholar
  15. 15.
    J. A. Sethian. Level Set Methods. Cambridge University Press, Cambridge, 1996.zbMATHGoogle Scholar
  16. 16.
    J. Weickert. A Review of Nonlinear Diffusion Filtering. In B. M. ter Haar Romeny et al. (Eds.) Scale-Space Theory in Computer Vision, Springer, Berlin, 3–28, 1997.Google Scholar
  17. 17.
    J. Weickert, S. Ishikawa, and A. Imiya. Scale-Space has been Discovered in Japan. J. of Mathematical Imaging and Vision, submitted.Google Scholar
  18. 18.
    A. P. Witkin. Scale-space filtering. Proc. IJCAI, Vol. 2, 1019–1022, 1983.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1998

Authors and Affiliations

  • Longin Jan Latecki
    • 1
  • Rolf Lakämper
    • 1
  1. 1.Institut für Angewandte MathematikUniversität HamburgGermany

Personalised recommendations