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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

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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.

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Notes

  1. All evaluations were done on the basis of our own Mathematica-modules freely available at: https://www.mathos.unios.hr/images/homepages/scitowsk/GOPart.rar, and were performed on the computer with a 2.90 GHz Intel(R) Core(TM)i7-75000 CPU with 16GB of RAM.

References

  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)

    Article  MATH  Google Scholar 

  2. Akinlar, C., Topal, C.: Edcircles: a real-time circle detector with a false detection control. Pattern Recognit. 46, 725–740 (2013)

    Article  Google Scholar 

  3. Bagirov, A.M.: Modified global \(k\)-means algorithm for minimum sum-of-squares clustering problems. Pattern Recognit. 41, 3192–3199 (2008)

    Article  MATH  Google Scholar 

  4. Bagirov, A.M., Ugon, J., Webb, D.: Fast modified global \(k\)-means algorithm for incremental cluster construction. Pattern Recognit. 44, 866–876 (2011)

    Article  MATH  Google Scholar 

  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)

    MATH  Google Scholar 

  6. Butenko, S., Chaovalitwongse, W.A., Pardalos, P.M. (eds.): Clustering Challenges in Biological Networks. World Scientific Publishing Co, Singapore (2009)

    Google Scholar 

  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)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  9. Gablonsky, J.M.: DIRECT Version 2.0. Technical Report. Center for Research in Scientific Computation. North Carolina State University (2001)

  10. Grbić, R., Grahovac, D., Scitovski, R.: A method for solving the multiple ellipses detection problem. Pattern Recognit. 60, 824–834 (2016)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  12. Horst, R., Tuy, H.: Global Optimization: Deterministic Approach, 3rd Revised and Enlarged Edition. Springer, Berlin (1996)

    Book  Google Scholar 

  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)

    Article  Google Scholar 

  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)

    Chapter  Google Scholar 

  15. Jones, D.R., Perttunen, C.D., Stuckman, B.E.: Lipschitzian optimization without the Lipschitz constant. J. Optim. Theory Appl. 79, 157–181 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  16. Kogan, J.: Introduction to Clustering Large and High-dimensional Data. Cambridge University Press, New York (2007)

    MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  18. Leisch, F.: A toolbox for k-centroids cluster analysis. Comput. Stat. Data Anal. 51, 526–544 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  19. Likas, A., Vlassis, N., Verbeek, J.J.: The global \(k\)-means clustering algorithm. Pattern Recognit. 36, 451–461 (2003)

    Article  Google Scholar 

  20. Locatelli, M., Schoen, F.: Global Optimization: Theory, Algorithms, and Applications. SIAM, Philadelphia (2013)

    Book  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  22. Nievergelt, Y.: A finite algorithm to fit geometrically all midrange lines, circles, planes, spheres, hyperplanes, and hyperspheres. Numer. Math. 91, 257–303 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  23. Paulavičius, R., Sergeyev, Y., Kvasov, D., Žilinskas, J.: Globally-biased DISIMPL algorithm for expensive global optimization. J. Glob. Optim. 59, 545–567 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  24. Paulavičius, R., Žilinskas, J.: Simplicial Global Optimization. Springer, Berlin (2014a)

    Book  MATH  Google Scholar 

  25. Paulavičius, R., Žilinskas, J.: Simplicial Lipschitz optimization without Lipschitz constant. J. Glob. Optim. 59, 23–40 (2014b)

    Article  MathSciNet  MATH  Google Scholar 

  26. Paulavičius, R., Žilinskas, J.: Advantages of simplicial partitioning for Lipschitz optimization problems with linear constraints. Optim. Lett. 10, 237–246 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  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)

  28. Qiao, Y., Ong, S.H.: Connectivity-based multiple-circle ftting. Pattern Recognit. 37, 755–765 (2004)

    Article  Google Scholar 

  29. Sabo, K., Scitovski, R., Vazler, I.: One-dimensional center-based \(l_1\)-clustering method. Optim. Lett. 7, 5–22 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  31. Scitovski, R., Marošević, T.: Multiple circle detection based on center-based clustering. Pattern Recognit. Lett. 52, 9–16 (2014)

    Article  Google Scholar 

  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)

    Article  Google Scholar 

  33. Scitovski, R., Scitovski, S.: A fast partitioning algorithm and its application to earthquake investigation. Comput. Geosci. 59, 124–131 (2013)

    Article  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  35. Späth, H.: Cluster-Formation und Analyse. R. Oldenburg Verlag, München (1983)

    MATH  Google Scholar 

  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)

    Chapter  Google Scholar 

  37. Tîrnăucă, C., Gómez-Pérez, D., Balcázar, J.L., Montaña, J.L.: Global optimality in k-means clustering. Inf. Sci. 439, 79–94 (2018)

    MathSciNet  Google Scholar 

  38. Vidović, I., Scitovski, R.: Center-based clustering for line detection and application to crop rows detection. Comput. Electron. Agric. 109, 212–220 (2014)

    Article  Google Scholar 

  39. Weise, T.: Global Optimization Algorithms. Theory and Application. http://www.it-weise.de/projects/book.pdf (2008)

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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

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  1. Department of Mathematics, University of Osijek, Trg Ljudevita Gaja 6, 31 000, Osijek, Croatia

    Rudolf Scitovski & Kristian Sabo

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  1. Rudolf Scitovski
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Correspondence to Rudolf Scitovski.

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Scitovski, R., Sabo, K. 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. J Glob Optim 74, 63–77 (2019). https://doi.org/10.1007/s10898-019-00743-8

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