Medial Axis LUT Computation for Chamfer Norms Using \(\mathcal{H}\)-Polytopes

  • Nicolas Normand
  • Pierre Évenou
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4992)


Chamfer distances are discrete distances based on the propagation of local distances, or weights defined in a mask. The medial axis, i.e. the centers of the maximal disks (disks which are not contained in any other disk), is a powerful tool for shape representation and analysis. The extraction of maximal disks is performed in the general case with comparison tests involving look-up tables representing the covering relation of disks in a local neighborhood. Although look-up table values can be computed efficiently [1], the computation of the look-up table neighborhood tend to be very time-consuming. By using polytope [2] descriptions of the chamfer disks, the necessary operations to extract the look-up tables are greatly reduced.


Medial Axis Minimal Representation Distance Transformation Neighborhood Sequence Pattern Recognition Letter 
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.
    Rémy, É., Thiel, É.: Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2d or 3d. Pattern Recognition Letters 23(6), 649–661 (2002)CrossRefzbMATHGoogle Scholar
  2. 2.
    Ziegler, G.M.: Lectures on Polytopes (Graduate Texts in Mathematics). Springer, Heidelberg (2001)Google Scholar
  3. 3.
    Rosenfeld, A., Pfaltz, J.L.: Sequential operations in digital picture processing. Journal of the ACM 13(4), 471–494 (1966)CrossRefzbMATHGoogle Scholar
  4. 4.
    Borgefors, G.: Distance transformations in arbitrary dimensions. Computer Vision, Graphics, and Image Processing 27(3), 321–345 (1984)CrossRefGoogle Scholar
  5. 5.
    Coeurjolly, D., Montanvert, A.: Optimal separable algorithms to compute the reverse euclidean distance transformation and discrete medial axis in arbitrary dimension. IEEE Transactions on Pattern Analysis and Machine Intelligence 29(3), 437–448 (2007)CrossRefGoogle Scholar
  6. 6.
    Strand, R.: Weighted distances based on neighborhood sequences. Pattern Recognition Letters 28, 2029–2036 (2007)CrossRefGoogle Scholar
  7. 7.
    Borgefors, G.: Centres of maximal discs in the 5-7-11 distance transforms. In: Proc. 8th Scandinavian Conf. on Image Analysis, Tromsø, Norway (1993)Google Scholar
  8. 8.
    Rosenfeld, A., Pfaltz, J.L.: Distances functions on digital pictures. Pattern Recognition Letters 1(1), 33–61 (1968)Google Scholar
  9. 9.
    Thiel, É.: Géométrie des distances de chanfrein. In: mémoire d’habilitation à diriger des recherches (2001),
  10. 10.
    Verwer, B.J.H.: Local distances for distance transformations in two and three dimensions. Pattern Recognition Letters 12(11), 671–682 (1991)CrossRefGoogle Scholar
  11. 11.
    Rémy, É.: Normes de chanfrein et axe médian dans le volume discret. Thèse de doctorat, Université de la Méditerranée (2001)Google Scholar
  12. 12.
    Thiel, É.: Les distances de chanfrein en analyse d’images: fondements et applications. Thèse de doctorat, Université Joseph Fourier, Grenoble 1 (1994),
  13. 13.
    Pfaltz, J.L., Rosenfeld, A.: Computer representation of planar regions by their skeletons. Communications of the ACM 10(2), 119–122 (1967)CrossRefGoogle Scholar
  14. 14.
    Arcelli, C., di Baja, G.S.: Finding local maxima in a pseudo-Euclidian distance transform. Computer Vision, Graphics, and Image Processing 43(3), 361–367 (1988)CrossRefGoogle Scholar
  15. 15.
    Svensson, S., Borgefors, G.: Digital distance transforms in 3d images using information from neighbourhoods up to 5×5×5. Computer Vision and Image Understanding 88(1), 24–53 (2002)CrossRefzbMATHGoogle Scholar
  16. 16.
    Rémy, É., Thiel, É.: Medial axis for chamfer distances: computing look-up tables and neighbourhoods in 2d or 3d. Pattern Recognition Letters 23(6), 649–661 (2002)CrossRefzbMATHGoogle Scholar
  17. 17.
    Remy, E., Thiel, E.: Computing 3D Medial Axis for Chamfer Distances. In: Nyström, I., Sanniti di Baja, G., Borgefors, G. (eds.) DGCI 2000. LNCS, vol. 1953, pp. 418–430. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  18. 18.
    Borgefors, G.: Weighted digital distance transforms in four dimensions. Discrete Applied Mathematics 125(1), 161–176 (2003)CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Nicolas Normand
    • 1
  • Pierre Évenou
    • 1
  1. 1.IRCCyN UMR CNRS 6597École polytechnique de l’Université de NantesNantes Cedex 3France

Personalised recommendations