Optimal multilevel thresholding is a quite important problem in image segmentation and pattern recognition. Although efficient algorithms have been proposed recently, they do not address the issue of irregularly sampled histograms. A polynomial-time algorithm for multilevel thresholding of irregularly sampled histograms is proposed. The algorithm is polynomial not just on the number of bins of the histogram, n, but also on the number of thresholds, k, i.e. it runs in Θ(kn 2). The proposed algorithm is general enough for a wide range of thresholding and clustering criteria, and has the capability of dealing with irregularly sampled histograms. This implies important consequences on pattern recognition, since optimal clustering in the one-dimensional space can be obtained in polynomial time. Experiments on synthetic and real-life histograms show that for typical cases, the proposed algorithm can find the optimal thresholds in a fraction of a second.


Particle Swarm Optimization Optimal Threshold Optimal Cluster Pattern Recognition Letter Algorithm Multilevel 
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  1. 1.
    Beyenal, H., Donovan, C., Lewandowski, Z., Harkin, G.: Three-dimensional Biofilm Structure Quantification. Journal of Microbiological Methods 59(3), 395–413 (2004)CrossRefGoogle Scholar
  2. 2.
    Fan, S., Lin, Y.: A Multi-level Thresholding Approach Using a Hybrid Optimal Estimation Algorithm. Pattern Recognition Letters 28, 662–669 (2007)CrossRefGoogle Scholar
  3. 3.
    Kapur, J., Sahoo, P., Wong, A.: A New Method for Gray-level Picture Thresholding Using the Entropy of the Histogram. Computer Vision Graphics and Image Processing 29, 273–285 (1985)CrossRefGoogle Scholar
  4. 4.
    Kittler, J., Illingworth, J.: Minimum Error Thresholding. Pattern Recognition 19(1), 41–47 (1986)CrossRefGoogle Scholar
  5. 5.
    Liao, P., Chen, T., Chung, P.: A Fast Algorithm for Multilevel Thresholding. Journal of Information Science and Engineering 17, 713–727 (2001)Google Scholar
  6. 6.
    Luessi, M., Eichmann, M., Shuster, M., Katsaggelos, A.: New results on efficient optimal multilevel image thresholding. In: Proc. of the IEEE International Conference on Image Processing, Atlanta, USA, pp. 773–776. IEEE Press, Los Alamitos (2006)Google Scholar
  7. 7.
    Maulik, U., Bandyopadhyay, S.: Performance Evaluation of Some Clustering Algorithms and Validity Indices. IEEE Trans. on Pattern Analysis and Machine Intelligence 24(12), 1650–1655 (2002)CrossRefGoogle Scholar
  8. 8.
    Otsu, N.: A Threshold Selection Method from Gray-level Histograms. IEEE Trans. on Systems, Man and Cybernetics SMC-9, 62–66 (1979)CrossRefGoogle Scholar
  9. 9.
    Rueda, L.: A Polynomial-time Algorithm for Optimal Multilevel Thresholding (submitted, 2008),
  10. 10.
    Sahoo, P., Wilkins, C., Yeager, J.: Threshold Selection Using Renyi’s Entropy. Pattern Recognition 30(1), 71–84 (1997)CrossRefzbMATHGoogle Scholar
  11. 11.
    Wang, S., Chung, F., Xiong, F.: A Novel Image Thresholding Method Based on Parzen Window Estimate. Pattern Recognition 41(1), 117–129 (2008)CrossRefzbMATHGoogle Scholar
  12. 12.
    Wu, B., Chen, Y., Chiu, C.: Efficient Implementation of Several Multilevel Thresholding Algorithms Using a Combinatorial Scheme. International Journal of Computers and Applications 28(3), 259–269 (2006)CrossRefGoogle Scholar
  13. 13.
    Yan, H.: Unified Formulation of a Class of Optimal Image Thresholding Techniques. Pattern Recognition 29(12), 2025–2032 (1996)CrossRefGoogle Scholar
  14. 14.
    Yen, J., Chang, F., Chang, S.: A New Criterion for Automatic Multilevel Thresholding. IEEE Trans. on Image Processing 4(3), 370–378 (1995)CrossRefGoogle Scholar
  15. 15.
    Yin, P.: A Fast Scheme for Optimal Thresholding Using Genetic Algorithms. Signal Processing 72, 85–95 (1999)CrossRefzbMATHGoogle Scholar
  16. 16.
    Zahara, E., Fan, S., Tsai, D.: Optimal Multi-thresholding Using a Hybrid Optimization Approach. Pattern Recognition Letters 26, 1082–1095 (2005)CrossRefGoogle Scholar
  17. 17.
    Zhou, Y., Yang, K.: New 2D Adaptive Image Thresholding Method Based on Within and Between Cluster Scatter. In: Proc. of the 27th International Congress on High-Speed Photography and Photonics, SPIE, vol. 6279, p. 62793N (2007)Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2008

Authors and Affiliations

  • Luis Rueda
    • 1
  1. 1.Department of Computer ScienceUniversity of WindsorWindsorCanada

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