O(n3logn) Time Complexity for the Optimal Consensus Set Computation for 4-Connected Digital Circles

  • Gaelle Largeteau-Skapin
  • Rita Zrour
  • Eric Andres
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7749)


This paper presents a method for fitting 4-connected digital circles to a given set of points in 2D images in the presence of noise by maximizing the number of inliers, namely the optimal consensus set, while fixing the thickness. Our approach has a O(n 3 log n) time complexity and O(n) space complexity, n being the number of points, which is lower than previous known methods while still guaranteeing optimal solution(s).


Shape fitting consensus set inliers outliers digital circle 4-connected digital circle 0-Flake digital circle 


  1. 1.
    Zrour, R., Largeteau-Skapin, G., Andres, E.: Optimal Consensus Set for Annulus Fitting. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 358–368. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  2. 2.
    Andres, E., Largeteau-Skapin, G., Zrour, R., Sugimoto, A., Kenmochi, Y.: Optimal Consensus Set and Preimage of 4-connected circles in a noisy environment. In: 21st International Conference on Pattern Recognition. IEEE Xplore, Tsukuba Science City (2012)Google Scholar
  3. 3.
    Andres, E., Roussillon, T.: Analytical Description of Digital Circles. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 235–246. Springer, Heidelberg (2011)CrossRefGoogle Scholar
  4. 4.
    Phan, M.S., Kenmochi, Y., Sugimoto, A., Talbot, H., Andres, E., Zrour, R.: Efficient Robust Digital Annulus Fitting with Bounded Error. In: Gonzalez-Diaz, R., Jimenez, M.-J., Medrano, B. (eds.) DGCI 2013. LNCS, vol. 7749, pp. 253–264. Springer, Heidelberg (2013)Google Scholar
  5. 5.
    Fiorio, C., Toutant, J.-L.: Arithmetic Discrete Hyperspheres and Separatingness. In: Kuba, A., Nyúl, L.G., Palágyi, K. (eds.) DGCI 2006. LNCS, vol. 4245, pp. 425–436. Springer, Heidelberg (2006)CrossRefGoogle Scholar
  6. 6.
    Fischler, M.A., Bolles, R.C.: Random Sample Consensus: A Paradigm for Model Fitting with Applications to Image Analysis and Automated Cartography. Communications of the ACM 24, 381–395 (1981)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Har-Peled, S., Wang, Y.: Shape Fitting with Outliers. SIAM Journal on Computing 33(2), 269–285 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Rivlin, T.J.: Approximation by circles. Computing 21(2), 93–104 (1979)MathSciNetzbMATHCrossRefGoogle Scholar
  9. 9.
    Toutant, J.-L., Andres, E., Roussillon, T.: Digital Circles, Spheres and Hyperspheres: From Morphological Models to Analytical Characterizations and Topological Properties. Submitted to Discrete Applied Mathematic Journal (2012)Google Scholar
  10. 10.
    Andres, E., Jacob, M.-A.: The Discrete analytical hyperspheres. IEEE Transactions on Visualization and Computer Graphics 3, 75–86 (1997)CrossRefGoogle Scholar
  11. 11.
    Diaz-Banez, J.M., Hurtado, F., Meijer, H., Rappaport, D., Sellarès, T.: The largest empty annulus problem. International Journal of Computational Geometry and Applications 13(4), 317–325 (2003)MathSciNetzbMATHCrossRefGoogle Scholar
  12. 12.
    Dunagan, J., Vempala, S.: Optimal outlier removal in high-dimensional spaces. Journal of Computer and System Sciences 68(2), 335–373 (2004)MathSciNetzbMATHCrossRefGoogle Scholar
  13. 13.
    Matousek, J.: On enclosing k points by a circle. Information Processing Letters 53(4), 217–221 (1995)MathSciNetzbMATHCrossRefGoogle Scholar
  14. 14.
    O’Rourke, J., Kosaraju, S.R., Megiddo, N.: Computing circular separability. Discrete and Computational Geometry 1, 105–113 (1986)MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Roussillon, T., Tougne, L., Sivignon, I.: On Three Constrained Versions of the Digital Circular Arc Recognition Problem. In: Brlek, S., Reutenauer, C., Provençal, X. (eds.) DGCI 2009. LNCS, vol. 5810, pp. 34–45. Springer, Heidelberg (2009)CrossRefGoogle Scholar
  16. 16.
    Aiger, D., Kenmochi, Y., Talbot, H., Buzer, L.: Efficient Robust Digital Hyperplane Fitting with Bounded Error. In: Debled-Rennesson, I., Domenjoud, E., Kerautret, B., Even, P. (eds.) DGCI 2011. LNCS, vol. 6607, pp. 223–234. Springer, Heidelberg (2011)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2013

Authors and Affiliations

  • Gaelle Largeteau-Skapin
    • 1
  • Rita Zrour
    • 1
  • Eric Andres
    • 1
  1. 1.Laboratoire XLIM-SIC UMR CNRS 7252Université de PoitiersFuturoscope Chasseneuil CedexFrance

Personalised recommendations