A Discrete Bisector Function Based on Annulus

  • Sangbé Sidibe
  • Rita ZrourEmail author
  • Eric Andres
  • Gaelle Largeteau-Skapin
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11414)


In this paper we are proposing a new way to compute a discrete bisector function, which is an important tool for analyzing and filtering Euclidean skeletons. From a continuous point of view, a point that belongs to the medial axis is the center of a maximal ball that hits the background in more than one point. The maximal angle between those points is expected to be high for most of the object points and corresponds to the bisector angle. This logic is not really applicable in the discrete space since in some configurations we miss some background points leading sometimes to small bisector angles. In this work we use annuli to find the background points in order to compute the bisector angle. The main advantage of this approach is the possibility to change the thickness and therefore to be more flexible while computing the bisector angle.


Bisector function Euclidean distance transform Skeleton Digital geometry Digital annulus 


  1. 1.
    DGtal: Digital geometry tools & algorithms library.
  2. 2.
    Andres, E.: Discrete circles, rings and spheres. Comput. Graph. 18(5), 695–706 (1994)CrossRefGoogle Scholar
  3. 3.
    Andres, E., Jacob, M.: Discrete analytical hyperspheres. IEEE Trans. Vis. Comput. Graph 3, 75–86 (1997)CrossRefGoogle Scholar
  4. 4.
    Attali, D., di Baja, G.S., Thiel, E.: Pruning discrete and semicontinuous skeletons. In: Braccini, C., DeFloriani, L., Vernazza, G. (eds.) ICIAP 1995. LNCS, vol. 974, pp. 488–493. Springer, Heidelberg (1995). Scholar
  5. 5.
    Attali, D., Lachaud, J.O.: Delaunay conforming iso-surface, skeleton extraction and noise removal. Comput. Geom.: Theory Appl. 19, 175–189 (2001)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Attali, D., Montanvert, A.: Modeling noise for a better simplification of skeletons. In: ICIP, vol. 3, pp. 13–16 (1996)Google Scholar
  7. 7.
    Blum, H.: A transformation for extracting new descriptors of shape. In: Models for the Perception of Speech and Visual Form, pp. 362–380 (1967)Google Scholar
  8. 8.
    Chaussard, J., Couprie, M., Talbot, H.: Robust skeletonization using the discrete \(\lambda \)-medial axis. Pattern Recognit. Lett. 32(9), 1384–1394 (2011)CrossRefGoogle Scholar
  9. 9.
    Chazal, F., Lieutier, A.: The \(\lambda \)-medial axis. Graph. Model. 67(4), 304–331 (2005)CrossRefGoogle Scholar
  10. 10.
    Couprie, M., Coeurjolly, D., Zrour, R.: Discrete bisector function and Euclidean skeleton in 2D and 3D. Image Vis. Comput. 25(10), 1519–1698 (2007)CrossRefGoogle Scholar
  11. 11.
    Couprie, M., Zrour, R.: Discrete bisector function and Euclidean skeleton. In: Andres, E., Damiand, G., Lienhardt, P. (eds.) DGCI 2005. LNCS, vol. 3429, pp. 216–227. Springer, Heidelberg (2005). Scholar
  12. 12.
    Danielsson, P.E.: Euclidean distance mapping. Comput. Graph. Image Process. 14, 227–248 (1980)CrossRefGoogle Scholar
  13. 13.
    Hirata, T.: A unified linear-time algorithm for computing distance maps. Inf. Process. Lett. 58(3), 129–133 (1996)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Le, B.H., Hodgins, J.K.: Real-time skeletal skinning with optimized centers of rotation. ACM Trans. Graph. 35(4), 37:1–37:10 (2016)CrossRefGoogle Scholar
  15. 15.
    Malandain, G., Fernandez-Vidal, S.: Euclidean skeletons. Image Vis. Comput. 16(5), 317–328 (1998)CrossRefGoogle Scholar
  16. 16.
    Marie, R., Labbani-Igbida, O., Mouaddib, E.M.: The delta medial axis: a fast and robust algorithm for filtered skeleton extraction. Pattern Recognit. 56, 26–39 (2016)CrossRefGoogle Scholar
  17. 17.
    Meijster, A., Roerdink, J.B.T.M., Hesselink, W.H.: A general algorithm for computing distance transforms in linear time. In: Goutsias, J., Vincent, L., Bloomberg, D.S. (eds.) Mathematical Morphology and its Applications to Image and Signal Processing. CIVI, vol. 18, pp. 331–340. Springer, Heidelberg (2002). Scholar
  18. 18.
    Meyer, F.: Cytologie quantitative et morphologie mathématique. Ph.D. thesis, Ecole des mines de Paris (1979)Google Scholar
  19. 19.
    Remy, E., Thiel, E.: Exact medial axis with Euclidean distance. Image Vis. Comput. 23(2), 167–175 (2005)CrossRefGoogle Scholar
  20. 20.
    Saito, T., Toriwaki, J.I.: New algorithms for Euclidean distance transformation of an n-dimensional digitized picture with applications. Pattern Recognit. 27(11), 1551–1565 (1994)CrossRefGoogle Scholar
  21. 21.
    Shamos, M.I.: Computational geometry. Ph.D. thesis, Yale University (1978)Google Scholar
  22. 22.
    Talbot, H., Vincent, L.M.: Euclidean skeletons and conditional bisectors. In: Proceedings of Visual Communications and Image Processing, vol. 1818, pp. 862–877. International Society for Optics and Photonics (1992)Google Scholar
  23. 23.
    Xu, Y., Mattikalli, R., Khosla, P.: Motion planning using medial axis. IFAC Proc. 25(28), 135–140 (1992)CrossRefGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Sangbé Sidibe
    • 1
  • Rita Zrour
    • 1
    Email author
  • Eric Andres
    • 1
  • Gaelle Largeteau-Skapin
    • 1
  1. 1.Laboratoire XLIM UMR CNRS 7252, ASALI, Université de PoitiersFuturoscope Chasseneuil CedexFrance

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