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Application of the DIRECT algorithm to searching for an optimal k-partition of the set \(\mathcal {A}\subset \mathbb {R}^n\) and its application to the multiple circle detection problem

  • Rudolf ScitovskiEmail author
  • Kristian Sabo
Article
  • 11 Downloads

Abstract

In this paper, we propose an efficient method for searching for a globally optimal k-partition of the set \(\mathcal {A}\subset \mathbb {R}^n\). Due to the property of the DIRECT global optimization algorithm to usually quickly arrive close to a point of global minimum, after which it slowly attains the desired accuracy, the proposed method uses the well-known k-means algorithm with a initial approximation chosen on the basis of only a few iterations of the DIRECT algorithm. In case of searching for an optimal k-partition of spherical clusters, the method is not worse than other known methods, but in case of solving the multiple circle detection problem, the proposed method shows remarkable superiority.

Keywords

Globally optimal partition k-means Incremental algorithm DIRECT Multiple circles detection problem 

Mathematics Subject Classification

65K05 90C26 90C27 90C56 90C57 05E05 

Notes

Acknowledgements

The author would like to thank the referees and the journal editors for their careful reading of the paper and insightful comments that helped us improve the paper. Especially, the author would like to thank Mrs. Katarina Moržan for significantly improving the use of English in the paper. This work was supported by the Croatian Science Foundation through research Grants IP-2016-06-6545 and IP-2016-06-8350

References

  1. 1.
    Ahn, S.J., Rauh, W., Warnecke, H.J.: Least-squares orthogonal distances fitting of circle, sphere, ellipse, hyperbola, and parabola. Pattern Recognit. 34, 2283–2303 (2001)CrossRefzbMATHGoogle Scholar
  2. 2.
    Akinlar, C., Topal, C.: Edcircles: a real-time circle detector with a false detection control. Pattern Recognit. 46, 725–740 (2013)CrossRefGoogle Scholar
  3. 3.
    Bagirov, A.M.: Modified global \(k\)-means algorithm for minimum sum-of-squares clustering problems. Pattern Recognit. 41, 3192–3199 (2008)CrossRefzbMATHGoogle Scholar
  4. 4.
    Bagirov, A.M., Ugon, J., Webb, D.: Fast modified global \(k\)-means algorithm for incremental cluster construction. Pattern Recognit. 44, 866–876 (2011)CrossRefzbMATHGoogle Scholar
  5. 5.
    Bezdek, J.C., Keller, J., Krisnapuram, R., Pal, N.R.: Fuzzy Models and Algorithms for Pattern Recognition and Image Processing. Springer, New York (2005)zbMATHGoogle Scholar
  6. 6.
    Butenko, S., Chaovalitwongse, W.A., Pardalos, P.M. (eds.): Clustering Challenges in Biological Networks. World Scientific Publishing Co, Singapore (2009)Google Scholar
  7. 7.
    Chernov, N.: Circular and Linear Regression: Fitting Circles and Lines by Least Squares. Monographs on Statistics and Applied Probability, vol. 117. Chapman & Hall, London (2010)CrossRefGoogle Scholar
  8. 8.
    Chung, K.L., Huang, Y.H., Shen, S.M., Yurin, A.S.K.D.V., Semeikina, E.V.: Efficient sampling strategy and refinement strategy for randomized circle detection. Pattern Recognit. 45, 252–263 (2012)CrossRefGoogle Scholar
  9. 9.
    Gablonsky, J.M.: DIRECT Version 2.0. Technical Report. Center for Research in Scientific Computation. North Carolina State University (2001)Google Scholar
  10. 10.
    Grbić, R., Grahovac, D., Scitovski, R.: A method for solving the multiple ellipses detection problem. Pattern Recognit. 60, 824–834 (2016)CrossRefGoogle Scholar
  11. 11.
    Grbić, R., Nyarko, E.K., Scitovski, R.: A modification of the DIRECT method for Lipschitz global optimization for a symmetric function. J. Glob. Optim. 57, 1193–1212 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  12. 12.
    Horst, R., Tuy, H.: Global Optimization: Deterministic Approach, 3rd Revised and Enlarged Edition. Springer, Berlin (1996)CrossRefGoogle Scholar
  13. 13.
    Hüllermeier, E., Rifqi, M., Henzgen, S., Senge, R.: Comparing fuzzy partitions: a generalization of the Rand index and related measures. EEE Trans. Fuzzy Syst. 20, 546–556 (2012)CrossRefGoogle Scholar
  14. 14.
    Jones, D.R.: The direct global optimization algorithm. In: Floudas, C.A., Pardalos, P.M. (eds.) The Encyclopedia of Optimization, pp. 431–440. Kluwer Academic Publishers, Dordrect (2001)CrossRefGoogle Scholar
  15. 15.
    Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Kogan, J.: Introduction to Clustering Large and High-dimensional Data. Cambridge University Press, New York (2007)zbMATHGoogle Scholar
  17. 17.
    Kvasov, D.E., Sergeyev, Y.D.: Lipschitz gradients for global optimization in a one-point-based partitioning scheme. J. Comput. Appl. Math. 236, 4042–4054 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Leisch, F.: A toolbox for k-centroids cluster analysis. Comput. Stat. Data Anal. 51, 526–544 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Likas, A., Vlassis, N., Verbeek, J.J.: The global \(k\)-means clustering algorithm. Pattern Recognit. 36, 451–461 (2003)CrossRefGoogle Scholar
  20. 20.
    Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications. SIAM, Philadelphia (2013)CrossRefzbMATHGoogle Scholar
  21. 21.
    Morales-Esteban, A., Martínez-Álvarez, F., Scitovski, S., Scitovski, R.: A fast partitioning algorithm using adaptive Mahalanobis clustering with application to seismic zoning. Comput. Geosci. 73, 132–141 (2014)CrossRefGoogle Scholar
  22. 22.
    Nievergelt, Y.: A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres. Numer. Math. 91, 257–303 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Paulavičius, R., Sergeyev, Y., Kvasov, D., Žilinskas, J.: Globally-biased DISIMPL algorithm for expensive global optimization. J. Glob. Optim. 59, 545–567 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. Springer, Berlin (2014a)CrossRefzbMATHGoogle Scholar
  25. 25.
    Paulavičius, R., Žilinskas, J.: Simplicial Lipschitz optimization without Lipschitz constant. J. Glob. Optim. 59, 23–40 (2014b)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Paulavičius, R., Žilinskas, J.: Advantages of simplicial partitioning for Lipschitz optimization problems with linear constraints. Optim. Lett. 10, 237–246 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Pelleg, D., Moore, A.W.: X-means: extending k-means with efficient estimation of the number of clusters, In: Proceedings of the Seventeenth International Conference on Machine Learning, ICML’00, Morgan Kaufmann Publishers Inc., San Francisco, CA, USA, pp. 727–734 (2000)Google Scholar
  28. 28.
    Qiao, Y., Ong, S.H.: Connectivity-based multiple-circle ftting. Pattern Recognit. 37, 755–765 (2004)CrossRefGoogle Scholar
  29. 29.
    Sabo, K., Scitovski, R., Vazler, I.: One-dimensional center-based \(l_1\)-clustering method. Optim. Lett. 7, 5–22 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Scitovski, R.: A new global optimization method for a symmetric Lipschitz continuous function and application to searching for a globally optimal partition of a one-dimensional set. J. Glob. Optim. 68, 713–727 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  31. 31.
    Scitovski, R., Marošević, T.: Multiple circle detection based on center-based clustering. Pattern Recognit. Lett. 52, 9–16 (2014)CrossRefGoogle Scholar
  32. 32.
    Scitovski, R., Sabo, K.: Analysis of the \(k\)-means algorithm in the case of data points occurring on the border of two or more clusters. Knowl. Based Syst. 57, 1–7 (2014)CrossRefGoogle Scholar
  33. 33.
    Scitovski, R., Scitovski, S.: A fast partitioning algorithm and its application to earthquake investigation. Comput. Geosci. 59, 124–131 (2013)CrossRefGoogle Scholar
  34. 34.
    Sergeyev, Y.D., Kvasov, D.E.: A deterministic global optimization using smooth diagonal auxiliary functions. Commun. Nonlinear Sci. Numer. Simul. 21, 99–111 (2015)MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Späth, H.: Cluster-Formation und Analyse. R. Oldenburg Verlag, München (1983)zbMATHGoogle Scholar
  36. 36.
    Thomas, J.C.R.: A new clustering algorithm based on k-means using a line segment as prototype. In: Martin, C.S., Kim, S.W. (eds.) Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, pp. 638–645. Springer, Berlin (2011)CrossRefGoogle Scholar
  37. 37.
    Tîrnăucă, C., Gómez-Pérez, D., Balcázar, J.L., Montaña, J.L.: Global optimality in k-means clustering. Inf. Sci. 439, 79–94 (2018)MathSciNetGoogle Scholar
  38. 38.
    Vidović, I., Scitovski, R.: Center-based clustering for line detection and application to crop rows detection. Comput. Electron. Agric. 109, 212–220 (2014)CrossRefGoogle Scholar
  39. 39.
    Weise, T.: Global Optimization Algorithms. Theory and Application. http://www.it-weise.de/projects/book.pdf (2008)

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of OsijekOsijekCroatia

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