Nonparametric Bayesian Image Segmentation

  • Peter Orbanz
  • Joachim M. Buhmann


Image segmentation algorithms partition the set of pixels of an image into a specific number of different, spatially homogeneous groups. We propose a nonparametric Bayesian model for histogram clustering which automatically determines the number of segments when spatial smoothness constraints on the class assignments are enforced by a Markov Random Field. A Dirichlet process prior controls the level of resolution which corresponds to the number of clusters in data with a unique cluster structure. The resulting posterior is efficiently sampled by a variant of a conjugate-case sampling algorithm for Dirichlet process mixture models. Experimental results are provided for real-world gray value images, synthetic aperture radar images and magnetic resonance imaging data.


Markov random fields Nonparametric Bayesian methods Dirichlet process mixtures Image segmentation Clustering Spatial statistics Markov chain Monte Carlo 


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  1. Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric estimation. Annals of Statistics, 2(6), 1152–1174. zbMATHCrossRefMathSciNetGoogle Scholar
  2. Bach, F., & Jordan, M. I. (2004). Learning spectral clustering. In S. Thrun, L. K. Saul, & B. Schölkopf (Eds.), Advances in neural information processing system 16 (pp. 305–312). Cambridge: MIT. Google Scholar
  3. Besag, J. (1986). On the statistical analysis of dirty pictures (with discussion). Journal of the Royal Statistical Society Series B, 48(3), 259–302. zbMATHMathSciNetGoogle Scholar
  4. Besag, J., Green, P., Higdon, D., & Mengersen, K. (1995). Bayesian computation and stochastic systems. Statistical Science, 10(1), 3–66. zbMATHCrossRefMathSciNetGoogle Scholar
  5. Blei, D. M. (2004). Probabilistic models for text and images. PhD thesis, U. C. Berkeley. Google Scholar
  6. Blei, D. M., & Jordan, M. I. (2006). Variational methods for the Dirichlet process. Bayesian Analysis, 1(1), 121–144. MathSciNetGoogle Scholar
  7. Breckenridge, J. (1989). Replicating cluster analysis: Method, consistency and validity. Multivariate Behavioral Research, 24, 147–161. CrossRefGoogle Scholar
  8. Devroye, L. (1986). Non-uniform random variate generation. Berlin: Springer. zbMATHGoogle Scholar
  9. Dudoit, S., & Fridyland, J. (2002). A prediction-based resampling method for estimating the number of clusters in a dataset. Genome Biology, 3(7). Google Scholar
  10. Escobar, M. D. (1994). Estimating normal means with a Dirichlet process prior. Journal of the American Statistical Association, 89(425), 268–277. zbMATHCrossRefMathSciNetGoogle Scholar
  11. Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1(2). Google Scholar
  12. Fischer, B., & Buhmann, J. M. (2003). Bagging for path-based clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(11), 1411–1415. CrossRefGoogle Scholar
  13. Geman, S., & Geman, D. (1984). Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images. IEEE Transactions on Pattern Analysis and Machine Intelligence, 6(6), 721–741. zbMATHCrossRefGoogle Scholar
  14. Geman, D., Geman, S., Graffigne, C., & Dong, P. (1990). Boundary detection by constrained optimization. IEEE Transactions on Pattern Analysis and Machine Intelligence, 12(7), 609–628. CrossRefGoogle Scholar
  15. Hermes, L., Zöller, T., & Buhmann, J. M. (2002). Parametric distributional clustering for image segmentation. In Lecture notes in computer science : Vol. 2352. Computer vision-ECCV ’02 (pp. 577–591). Berlin: Springer. CrossRefGoogle Scholar
  16. Hofmann, T., Puzicha, J., & Buhmann, J. M. (1998). Unsupervised texture segmentation in a deterministic annealing framework. IEEE Transactions on Pattern Analysis and Machine Intelligence, 20(8), 803–818. CrossRefGoogle Scholar
  17. Kotz, S., Balakrishnan, N., & Johnson, N. L. (2000). Continuous multivariate distributions (Vol. 1, 2nd ed.). New York: Wiley. zbMATHGoogle Scholar
  18. Lange, T., Roth, V., Braun, M., & Buhmann, J. M. (2004). Stability-based validation of clustering solutions. Neural Computation, 16(6), 1299–1323. zbMATHCrossRefGoogle Scholar
  19. Lehmann, E. L., & Romano, J. P. (2005). Testing statistical hypotheses (3rd ed.). Berlin: Springer. zbMATHGoogle Scholar
  20. Lin, J. (1991). Divergence measures based on the Shannon entropy. IEEE Transactions on Information Theory, 37, 145–151. zbMATHCrossRefGoogle Scholar
  21. MacEachern, S. N. (1994). Estimating normal means with a conjugate style Dirichlet process prior. Communications in Statistics: Simulation and Computation, 23, 727–741. zbMATHCrossRefMathSciNetGoogle Scholar
  22. MacEachern, S. N. (1998). Computational methods for mixture of Dirichlet process models. In D. Dey, P. Müller, & D. Sinha (Eds.), Lecture notes in statistics : Vol. 133. Practical nonparametric and semiparametric Bayesian statistics (pp. 23–43). Berlin: Springer. Google Scholar
  23. MacEachern, S. N., & Müller, P. (2000). Efficient MCMC schemes for robust model extensions using encompassing Dirichlet process mixture models. In D. Rios Insua & F. Ruggeri (Eds.), Robust Bayesian analysis (pp. 295–315). Berlin: Springer. Google Scholar
  24. Martin, D., Fowlkes, C., & Malik, J. (2004). Learning to detect natural image boundaries using local brightness, color, and texture cues. IEEE Transactions on Pattern Analysis and Machine Intelligence, 26(5), 530–549. CrossRefGoogle Scholar
  25. McAuliffe, J. D., Blei, D. M., & Jordan, M. I. (2004). Nonparametric empirical Bayes for the Dirichlet process mixture model. Technical report, UC Berkeley. Google Scholar
  26. Morel, J.-M., & Solimini, S. (1995). Variational methods for image segmentation. Basel: Birkhäuser. Google Scholar
  27. Müller, P., & Quintana, F. A. (2004). Nonparametric Bayesian data analysis. Statistical Science, 19(1), 95–111. zbMATHCrossRefMathSciNetGoogle Scholar
  28. Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9, 249–265. CrossRefMathSciNetGoogle Scholar
  29. Puzicha, J., Hofmann, T., & Buhmann, J. M. (1999). Histogram clustering for unsupervised segmentation and image retrieval. Pattern Recognition Letters, 20, 899–909. CrossRefGoogle Scholar
  30. Rissanen, J. (1983). A universal prior for integers and estimation by minimum description length. Annals of Statistics, 11, 416–431. zbMATHCrossRefMathSciNetGoogle Scholar
  31. Roberts, C. P. (1996). Mixtures of distributions: inference and estimation. In W. R. Gilks, S. Richardson, & D. J. Spiegelhalter (Eds.), Markov chain Monte Carlo in practice (pp. 441–464). London: Chapman & Hall. Google Scholar
  32. Roth, V., Laub, J., Motoaki, K., & Buhmann, J. M. (2003). Optimal cluster preserving embedding of non-metric proximity data. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(12), 1540–1551. CrossRefGoogle Scholar
  33. Samson, C., Blanc-Féraud, L., Aubert, G., & Zerubia, J. (2000). A variational model for image classification and restoration. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(5), 460–472. CrossRefGoogle Scholar
  34. Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8), 888–905. CrossRefGoogle Scholar
  35. Tu, Z., & Zhu, S.-C. (2002). Image segmentation by data-driven Markov chain Monte Carlo. IEEE Transactions on Pattern Analysis and Machine Intelligence, 24(5), 657–673. CrossRefGoogle Scholar
  36. Walker, S. G., Damien, P., Laud, P. W., & Smith, A. F. M. (1999). Bayesian nonparametric inference for random distributions and related functions. Journal of the Royal Statistical Society B, 61(3), 485–527. zbMATHCrossRefMathSciNetGoogle Scholar
  37. Winkler, G. (2003). Image analysis, random fields and Markov chain Monte Carlo methods. Berlin: Springer. zbMATHGoogle Scholar
  38. Zaragoza, H., Hiemstra, D., Tipping, D., & Robertson, S. (2003). Bayesian extension to the language model for ad hoc information retrieval. In Proceedings of SIGIR 2003. Google Scholar

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© Springer Science+Business Media, LLC 2007

Authors and Affiliations

  1. 1.Institute of Computational ScienceETH ZürichZurichSwitzerland

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