Exact Euclidean Medial Axis in Higher Resolution

  • André Vital Saúde
  • Michel Couprie
  • Roberto Lotufo
Part of the Lecture Notes in Computer Science book series (LNCS, volume 4245)


The notion of skeleton plays a major role in shape analysis. Some usually desirable characteristics of a skeleton are: sufficient for the reconstruction of the original object, centered, thin and homotopic. The Euclidean Medial Axis presents all these characteristics in a continuous framework. In the discrete case, the Exact Euclidean Medial Axis (MA) is also sufficient for reconstruction and centered. It no longer preserves homotopy but it can be combined with a homotopic thinning to generate homotopic skeletons. The thinness of the MA, however, may be discussed. In this paper we present the definition of the Exact Euclidean Medial Axis on Higher Resolution which has the same properties as the MA but with a better thinness characteristic, against the price of rising resolution. We provide an efficient algorithm to compute it.


  1. 1.
    Blum, H.: An associative machine for dealing with the visual field and some of its biological implications. Biological prototypes and synthetic systems 1, 244–260 (1961)Google Scholar
  2. 2.
    Davies, E., Plummer, A.: Thinning algorithms: a critique and a new methodology. Pattern Recognition 14, 53–63 (1981)CrossRefMathSciNetGoogle Scholar
  3. 3.
    Talbot, H., Vincent, L.: Euclidean skeletons and conditional bisectors. In: Procs. VCIP 1992, SPIE, vol. 1818, pp. 862–876 (1992)Google Scholar
  4. 4.
    Couprie, M., Coeurjolly, D., Zrour, R.: Discrete bisector function and euclidean skeleton in 2d and 3d. Image and Vision Computing (accepted, 2006)Google Scholar
  5. 5.
    Bertrand, G.: Skeletons in derived grids. In: Procs. Int. Conf. Patt. Recogn., pp. 326–329 (1984)Google Scholar
  6. 6.
    Kovalevsky, V.: Finite topology as applied to image analysis. Computer Vision, Graphics and Image Processing 46, 141–161 (1989)CrossRefGoogle Scholar
  7. 7.
    Khalimsky, E., Kopperman, R., Meyer, P.: Computer graphics and connected topologies on finite ordered sets. Topology and its Applications 36, 1–17 (1990)zbMATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Kong, T.Y., Kopperman, R., Meyer, P.: A topological approach to digital topology. American Mathematical Monthly 38, 901–917 (1991)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Bertrand, G.: New notions for discrete topology. In: Bertrand, G., Couprie, M., Perroton, L. (eds.) DGCI 1999. LNCS, vol. 1568, pp. 218–226. Springer, Heidelberg (1999)CrossRefGoogle Scholar
  10. 10.
    Bertrand, G., Couprie, M.: New 3d parallel thinning algorithms based on critical kernels. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, Springer, Heidelberg (2006)CrossRefGoogle Scholar
  11. 11.
    Danielsson, P.: Euclidean distance mapping. Computer Graphics and Image Processing 14, 227–248 (1980)CrossRefGoogle Scholar
  12. 12.
    Meyer, F.: Cytologie quantitative et morphologie mathématique. PhD thesis, École des Mines de Paris, France (1979)Google Scholar
  13. 13.
    Saito, T., Toriwaki, J.: New algorithms for euclidean distance transformation of an n-dimensional digitized picture with applications. Pattern Recognition 27, 1551–1565 (1994)CrossRefGoogle Scholar
  14. 14.
    Hirata, T.: A unified linear-time algorithm for computing distance maps. Information Processing Letters 58(3), 129–133 (1996)zbMATHCrossRefMathSciNetGoogle Scholar
  15. 15.
    Meijster, A., Roerdink, J., Hesselink, W.: A general algorithm for computing distance transforms in linear time. In: Goutsias, J., Vincent, L., Bloomberg, D. (eds.) Mathematical morphology and its applications to image and signal processing 5th. Computational Imaging and Vision, vol. 18, pp. 331–340. Kluwer Academic Publishers, Dordrecht (2000)CrossRefGoogle Scholar
  16. 16.
    Remy, E., Thiel, E.: Look-up tables for medial axis on squared euclidean distance transform. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 224–235. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  17. 17.
    Coeurjolly, D.: d-dimensional reverse euclidean distance transformation and euclidean medial axis extraction in optimal time. In: Nyström, I., Sanniti di Baja, G., Svensson, S. (eds.) DGCI 2003. LNCS, vol. 2886, pp. 327–337. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  18. 18.
    Rémy, E., Thiel, E.: Exact medial axis with euclidean distance. Image and Vision Computing 23(2), 167–175 (2005)CrossRefGoogle Scholar
  19. 19.
    Saúde, A.V., Couprie, M., Lotufo, R.: Exact Euclidean medial axis in higher resolution. Technical Report IGM2006-5, IGM, Université de Marne-la-Vallée (2006)Google Scholar
  20. 20.
    Borgefors, G., Ragnemalm, I., di Baja, G.S.: The Euclidean Distance Transform: finding the local maxima and reconstructing the shape. In: Seventh Scandinavian Conference on Image Analysis, Aalborg, Denmark, vol. 2, pp. 974–981 (1991)Google Scholar
  21. 21.
    Hardy, G., Wright, E.: An Introduction to the Theory of Numbers, 5th edn. Oxford University Press, Oxford (1978)Google Scholar
  22. 22.
    Couprie, M., Saúde, A.V., Bertrand, G.: Euclidean homotopic skeleton based on critical kernels. In: Procs. SIBGRAPI (to appear, 2006)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2006

Authors and Affiliations

  • André Vital Saúde
    • 1
    • 2
  • Michel Couprie
    • 2
  • Roberto Lotufo
    • 1
  1. 1.School of Electrical and Computer Engineering, DCA-FEEC-UNICAMPState University of CampinasCampinas/SPBrazil
  2. 2.Laboratoire A2SIInstitut Gaspard-MongeNoisy-le-GrandFrance

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