Skip to main content

Nonparametric Bayesian Image Segmentation

Abstract

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.

This is a preview of subscription content, access via your institution.

References

  1. Antoniak, C. E. (1974). Mixtures of Dirichlet processes with applications to Bayesian nonparametric estimation. Annals of Statistics, 2(6), 1152–1174.

    MATH  Article  MathSciNet  Google 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.

    MATH  MathSciNet  Google Scholar 

  4. Besag, J., Green, P., Higdon, D., & Mengersen, K. (1995). Bayesian computation and stochastic systems. Statistical Science, 10(1), 3–66.

    MATH  Article  MathSciNet  Google Scholar 

  5. Blei, D. M. (2004). Probabilistic models for text and images. PhD thesis, U. C. Berkeley.

  6. Blei, D. M., & Jordan, M. I. (2006). Variational methods for the Dirichlet process. Bayesian Analysis, 1(1), 121–144.

    MathSciNet  Google Scholar 

  7. Breckenridge, J. (1989). Replicating cluster analysis: Method, consistency and validity. Multivariate Behavioral Research, 24, 147–161.

    Article  Google Scholar 

  8. Devroye, L. (1986). Non-uniform random variate generation. Berlin: Springer.

    MATH  Google 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).

  10. Escobar, M. D. (1994). Estimating normal means with a Dirichlet process prior. Journal of the American Statistical Association, 89(425), 268–277.

    MATH  Article  MathSciNet  Google Scholar 

  11. Ferguson, T. S. (1973). A Bayesian analysis of some nonparametric problems. Annals of Statistics, 1(2).

  12. Fischer, B., & Buhmann, J. M. (2003). Bagging for path-based clustering. IEEE Transactions on Pattern Analysis and Machine Intelligence, 25(11), 1411–1415.

    Article  Google 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.

    MATH  Article  Google 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.

    Article  Google 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.

    Chapter  Google 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.

    Article  Google Scholar 

  17. Kotz, S., Balakrishnan, N., & Johnson, N. L. (2000). Continuous multivariate distributions (Vol. 1, 2nd ed.). New York: Wiley.

    MATH  Google Scholar 

  18. Lange, T., Roth, V., Braun, M., & Buhmann, J. M. (2004). Stability-based validation of clustering solutions. Neural Computation, 16(6), 1299–1323.

    MATH  Article  Google Scholar 

  19. Lehmann, E. L., & Romano, J. P. (2005). Testing statistical hypotheses (3rd ed.). Berlin: Springer.

    MATH  Google Scholar 

  20. Lin, J. (1991). Divergence measures based on the Shannon entropy. IEEE Transactions on Information Theory, 37, 145–151.

    MATH  Article  Google 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.

    MATH  Article  MathSciNet  Google 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.

    Article  Google 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.

  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.

    MATH  Article  MathSciNet  Google Scholar 

  28. Neal, R. M. (2000). Markov chain sampling methods for Dirichlet process mixture models. Journal of Computational and Graphical Statistics, 9, 249–265.

    Article  MathSciNet  Google Scholar 

  29. Puzicha, J., Hofmann, T., & Buhmann, J. M. (1999). Histogram clustering for unsupervised segmentation and image retrieval. Pattern Recognition Letters, 20, 899–909.

    Article  Google Scholar 

  30. Rissanen, J. (1983). A universal prior for integers and estimation by minimum description length. Annals of Statistics, 11, 416–431.

    MATH  Article  MathSciNet  Google 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.

    Article  Google 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.

    Article  Google Scholar 

  34. Shi, J., & Malik, J. (2000). Normalized cuts and image segmentation. IEEE Transactions on Pattern Analysis and Machine Intelligence, 22(8), 888–905.

    Article  Google 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.

    Article  Google 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.

    MATH  Article  MathSciNet  Google Scholar 

  37. Winkler, G. (2003). Image analysis, random fields and Markov chain Monte Carlo methods. Berlin: Springer.

    MATH  Google 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.

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Peter Orbanz.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Orbanz, P., Buhmann, J.M. Nonparametric Bayesian Image Segmentation. Int J Comput Vis 77, 25–45 (2008). https://doi.org/10.1007/s11263-007-0061-0

Download citation

Keywords

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