Advertisement

Probabilistic Modeling of Discrete Images

  • Michael T. Chan
  • Gabor T. Herman
  • Emanuel Levitan
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

We present a methodology for constructing probabilistic models of discrete images using Gibbs distributions. The method differs from previous approaches in that the formulation is suited for the modeling of discrete images, and hence readily applicable to discrete tomography problems. Second, the distribution is “image-modeling” in the sense that random samples drawn from the distribution are likely to share important characteristics of the images in the application area. We propose a technique to estimate parameters of the model,based not only on local characteristics of the images, but also on their global properties. Using it as image priors in a Bayesian framework, we apply the model to the problems of recovering discrete images corrupted by additive Gaussian noise and discrete tomographic reconstruction. We demonstrate the usefulness of the proposed model and compare its effectiveness with previous approaches.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    J. Besag “On the statistical analysis of dirty pictures (with discussion).” Journal of the Royal Statistical Society Series B 48, 259–302 (1986).Google Scholar
  2. [2]
    S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Transactions on Pattern Analysis and Machine Intelligence 6, 721–741 (1984).CrossRefPubMedGoogle Scholar
  3. [3]
    A. J. Gray, J. W. Kay, and D. M. Titterington, “An empirical study of the simulation of various models used for images,” IEEE Transactions on Pattern Analysis and Machine Intelligence 16, 507–512, 1994.CrossRefGoogle Scholar
  4. [4]
    M. T. Chan, G. T. Herman, and E. Levitan, “A Bayesian approach to PET reconstruction using image-modeling Gibbs priors: Implementation and comparison,” IEEE Transactions on Nuclear Science 44, 1347–1354 (1997).CrossRefGoogle Scholar
  5. [5]
    S. Geman and D. McClure, “Statistical methods for tomographic image reconstruction,” In Proceedings of the.46th Session of the ISI, Bulletin of the ISI, volume 52 pp. 5–21, 1987.Google Scholar
  6. [6]
    P. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,” IEEE Transactions on Medical Imaging 9, 84–93 (1990).CrossRefPubMedGoogle Scholar
  7. [7]
    G. T. Herman, A. R. De Pierro, and N. Gai, “On methods for maximum a posteriori image reconstruction with a normal prior,” Journal of Visual Communication and Image Representation 3, 316–324 (1992).CrossRefGoogle Scholar
  8. [8]
    R. Leahy and X. Yan, “Incorporation of anatomical MR data for improved functional imaging with PET”In 12th International Conference on Information Processing in Medical Imaging, pp. 105–120, 1990.Google Scholar
  9. [9]
    M. Lee, A. Rangarajan, I. G. Zubal, and G. Gindi, “A continuation method for emission tomography,” IEEE Transactions on Nuclear Science 40, 2049–2058 (1993).CrossRefGoogle Scholar
  10. [10]
    E. Levitan and G. T. Herman, “A maximum a posteriori probability expectation maximization for image reconstruction in emission tomography,” IEEE Transactions on Medical Imaging 6, 185–192 (1987).CrossRefPubMedGoogle Scholar
  11. [11]
    J. Besag “Spatial interactions and the statistical analysis of lattice systems (with discussion),” Journal of the Royal Statistical Society Series B 36, 192–236 (1974).Google Scholar
  12. [12]
    V. E. Johnson, W. H. Wong, X. Hu, and C. T. Chen, “Image restoration using Gibbs priors: Boundary modeling, treatment of blurring, and selection of hyperparameter,” IEEE Transactions on Pattern Analysis and Machine Intelligence 13, 413–425 (1991).CrossRefGoogle Scholar
  13. [13]
    Z. Zhou, R.M. Leahy, and E.U. Mumcuoglu, “Maximum likelihood hyperparameter estimation for Gibbs priors from incomplete data with applications to PET,” In R. Di Paola Y. Bizais, and C. Barillot, Information Processing in Medical Imaging (Kluwer Academic Publishers, Dordrecht, The Netherlands) pp. 39–52, 1995.Google Scholar
  14. [14]
    E. Levitan, M. Chan, and G. T. Herman, “Image-modeling Gibbs priors,” Graphical Models and Image Processing 57, 117–130 (1995).CrossRefGoogle Scholar
  15. [15]
    H. Gould and J. Tobochnik, An Introduction to Computer Simulation Methods: Application to Physical Systems (Part 2) (Addison-Wesley Publishing Company, Reading, MA), 1988. Chapter 12.Google Scholar
  16. [16]
    A. Alavi, R. Dann, J. Chawluk, J. Alavi, M. Kushner, and M. Reivich, “Positron emission tomography imaging of regional cerebral glucose metabolism,” Seminars in Nuclear Medicine 16, 2–34 (1986).CrossRefPubMedGoogle Scholar
  17. [17]
    L. Shepp and Y. Vardi, “Maximum likelihood reconstruction in emission tomography,” IEEE Transactions on Medical Imaging 1, 113–121 (1982).CrossRefPubMedGoogle Scholar
  18. [18]
    D. Geiger and F. Girosi, “Parallel and deterministic algorithms from MRF’s: Surface reconstruction,” IEEE Transactions on Pattern Analysis and Machine Intelligence 13, 401–412 (1991).CrossRefGoogle Scholar
  19. [19]
    J. Zerubia and R. Chellappa, “Mean field annealing using compound Gauss-Markov random fields for edge detection and image estimation,” IEEE Transactions on Neural Networks 4, 703–709 (1993).CrossRefPubMedGoogle Scholar
  20. [20]
    G. T. Herman, M. Chan, Y. Censor, E. Levitan, R. M. Lewitt, and T. K. Narayan, “Maximum a posteriori image reconstruction from projections,” In S. E. Levinson and L. Shepp, Image Models (and their Speech Model Cousins), (Springer-Verlag, New York) pp. 53–89, 1996.Google Scholar
  21. [21]
    A. Papoulis, Probability, Random Variables, and Stochastic Processes, Second Edition (McGraw-Hill, New York), 1984.Google Scholar
  22. [22]
    D. W. Wilson and B. M. W. Tsui, “Noise properties of filteredbackprojection and ML-EM reconstructed emission tomographic images,” IEEE Transactions on Nuclear Science 40, 1198–1203 (1993).CrossRefGoogle Scholar
  23. [23]
    S. W. Rowland “Computer implementation of image reconstruction formulas,” In G. T. Herman, Image Reconstruction from Projections: Implementation and Applications, (Springer-Verlag, Berlin) pp. 9–79, 1979.CrossRefGoogle Scholar
  24. [24]
    M. T. Chan, G. T. Herman, and E. Levitan, “Bayesian image reconstruction using image-modeling Gibbs priors,” International Journal of Imaging Systems and Technology 9, 85–98 (1998).CrossRefGoogle Scholar
  25. [25]
    C. Yang “Efficient stochastic algorithms on locally bounded image space,” CVGIP: Graphical Models and Image Processing 55, 494–506 (1993).Google Scholar
  26. [26]
    J. Browne and G. T. Herman, “Computerized evaluation of image reconstruction algorithms,” International Journal of Imaging Systems and Technology 7, 256–267 (1996).CrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Michael T. Chan
    • 1
  • Gabor T. Herman
    • 2
  • Emanuel Levitan
    • 3
  1. 1.Rockwell Science CenterThousand OaksUSA
  2. 2.Department of RadiologyUniversity of PennsylvaniaPhiladelphiaUSA
  3. 3.Faculty of MedicineTechnionHaifaIsrael

Personalised recommendations