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Multiscale Bayesian Methods for Discrete Tomography

  • Thomas Frese
  • Charles A. Bouman
  • Ken Sauer
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)

Abstract

Statistical methods of discrete tomographic reconstruction pose new problems both in stochastic modeling to define an optimal reconstruction, and in optimization to find that reconstruction. Multiscale models have succeeded in improving representation of structure of varying scale in imagery, a chronic problem for common Markov random fields. This chapter shows that associated multiscale methods of optimization also avoid local minima of the log a posteriori probability better than single-resolution techniques. These methods are applied here to both segmentation/reconstruction of the unknown cross sections and estimation of unknown parameters represented by the discrete levels.

Keywords

Emission Rate Markov Random Field Coarse Scale Multiscale Method Discrete Tomography 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    G. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography (Academic Press, New York), 1980.Google Scholar
  2. [2]
    G. C. McKinnon and R. H. T. Bates “Towards imaging the beating heart usefully with a conventional CT scanner,” IEEE Trans. on Biomedical Engineering BME-28, 123–127 (1981).CrossRefGoogle Scholar
  3. [3]
    J. G. Sanderson “Reconstruction of fuel pin bundles by a maximum entropy method,” IEEE Trans. on Nuclear Science NS-26, 2685–2688 (1979).CrossRefGoogle Scholar
  4. [4]
    T. Inouye “Image reconstruction with limited-angle projection data,” IEEE Trans. on Nuclear Science NS-26, 2666–2684 (1979).Google Scholar
  5. [5]
    G. H. Glover and N. J. Pelc, “An algorithm for the reduction of metal clip artifacts in CT reconstruction,” Med. Phys. 8, 799–807 (1981).CrossRefPubMedGoogle Scholar
  6. [6]
    S. K. Chang “The reconstruction of binary patterns from their projections,” Communications of the ACM 14, 21–25 (1971).CrossRefGoogle Scholar
  7. [7]
    A. Shliferstein and Y. T. Chien, “Switching components and the ambiguity problem in the reconstruction of pictures from their projections,” Pattern Recognition 10, 327–340 (1978).CrossRefGoogle Scholar
  8. [8]
    A. Kuba “The reconstruction of two-directionally connected binary patterns from their two orthogonal projections,” Comput. Vision Graphics and Image Process. 27, 249–265 (1984).CrossRefGoogle Scholar
  9. [9]
    S. K. Chang and C. K. Chow, “The reconstruction of three-dimensional objects from two orthogonal projections and its application to cardiac cineangiography,” IEEE Trans. on Computers C-22, 18–28 (1973).CrossRefGoogle Scholar
  10. [10]
    M. Soumekh, “Binary image reconstruction from four projections,” Proc. of IEEE Int’l Conf. on Acoust., Speech and Sig. Proc., (IEEE, New York City NY) pp. 1280–1283, 1988.Google Scholar
  11. [11]
    D. J. Rossi and A. S. Willsky “Reconstruction from projections based on detection and estimation of objects — parts I and II: Performance analysis and robustness analysis,” IEEE Trans. on Acoustics, Speech, and Signal Processing ASSP-32 886–906 (1984).CrossRefGoogle Scholar
  12. [12]
    Y. Bresler, J. A. Fessier, and A. Macovski, “A Bayesian approach to reconstruction from incomplete projections of a multiple object 3D domain,” IEEE Trans. on Pattern Analysis and Machine Intelligence 11, 840–858 (1989).CrossRefGoogle Scholar
  13. [13]
    S. Geman and D. McClure, “Bayesian image analysis: An application to single photon emission tomography,” Proc. Statist. Comput. sect. Amer. Stat. Assoc., Washington, DC, pp. 12–18, 1985.Google Scholar
  14. [14]
    T. Hebert and R. Leahy, “A generalized EM algorithm for 3-D Bayesian reconstruction from Poisson data using Gibbs priors,” IEEE Trans. on Medical Imaging 8, 194–202 (1989).CrossRefGoogle Scholar
  15. [15]
    P. J. Green, “Bayesian reconstruction from emission tomography data using a modified EM algorithm,” IEEE Trans. on Medical Imaging 9, 84–93 (1990).CrossRefGoogle Scholar
  16. [16]
    K. Sauer and C. A. Bouman, “A local update strategy for iterative reconstruction from projections,” IEEE Trans. on Signal Processing 41, 534–548 (1993).CrossRefGoogle Scholar
  17. [17]
    B. Hunt, `Bayesian methods in nonlinear digital image restoration,“ IEEE Trans. on Computers C-26, 219–229 (1977).CrossRefGoogle Scholar
  18. [18]
    S. Geman and D. Geman “Stochastic relaxation, Gibbs distributions and the Bayesian restoration of images,” IEEE Trans. on Pattern Analysis and Machine Intelligence PAMI-6, 721–741 (1984).CrossRefGoogle Scholar
  19. [19]
    S. Geman and D. McClure “Statistical methods for tomographic image reconstruction,” Bull. Int. Stat. Inst. LII-4, 5–21 (1987).Google Scholar
  20. [20]
    C. A. Bouman and K. Sauer, “A generalized Gaussian image model for edge-preserving MAP estimation,” IEEE Trans. on Image Processing 2, 296–310 (1993).CrossRefGoogle Scholar
  21. [21]
    H. Derin, H. Elliot, R. Cristi, and D. Geman, “Bayes smoothing algorithms for segmentation of binary images modeled by Markov random fields,” IEEE Trans. on Pattern Analysis and Machine Intelligence PAMI-6 707–719 (1984).CrossRefGoogle Scholar
  22. [22]
    J. Besag “On the statistical analysis of dirty pictures,” Journal of the Royal Statistical Society B 48, 259–302 (1986).Google Scholar
  23. [23]
    L. Shepp and Y. Vardi “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. on Medical Imaging MI-1, 113–122 (1982).Google Scholar
  24. [24]
    H. Hart and Z. Liang, “Bayesian image processing in two dimensions,” IEEE Trans. on Medical Imaging MI-6 201–208 (1987).CrossRefGoogle Scholar
  25. [25]
    Z. Liang and H. Hart, `Bayesian image processing of data from constrained source distributions-I: Non-valued, uncorrelated and correlated constraints,“ Bull. Math. Biol. 49,51–74 (1987).Google Scholar
  26. [26]
    G. T. Herman and D. Odhner, “Performance evaluation of an iterative image reconstruction algorithm for positron emission tomography,” IEEE Trans. on Medical Imaging 10, 336–346 (1991).CrossRefGoogle Scholar
  27. [27]
    G. T. Herman, A. R. De Pierro, and N. Gai, “On methods for maximum a posteriori image reconstruction with normal prior,” J. Visual Comm. Image Rep. 3, 316–324 (1992).CrossRefGoogle Scholar
  28. [28]
    T. Hebert and S. Gopal, “The GEM MAP algorithm with 3-D SPECT system response,” IEEE Trans. on Medical Imaging 11, 81–90 (1992).CrossRefGoogle Scholar
  29. [29]
    A. De Pierro “A modified expectation maximization algorithm for penalized likelihood estimation in emission tomography,” IEEE Trans. on Medical Imaging 14, 132–137 (1995).CrossRefGoogle Scholar
  30. [30]
    C. A. Bouman and K. Sauer, “A unified approach to statistical tomography using coordinate descent optimization,” IEEE Trans. on Image Processing 5, 480–492 (1996).CrossRefGoogle Scholar
  31. [31]
    M. R. Luettgen, W. C. Karl, and A. S. Willsky, “Efficient multiscale regularization with applications to the computation of optical flow,” IEEE Trans. on Image Processing 3, 41–63 (1994).CrossRefGoogle Scholar
  32. [32]
    F. Heitz, P. Perez, and P. Bouthemy, “Multiscale minimization of global energy functions in some visual recovery problems,” Comput. Vision Graphics and Image Process. 59,125–134 (1994).CrossRefGoogle Scholar
  33. [33]
    C. A. Bouman and B. Liu, “Multiple resolution segmentation of textured images,” IEEE Trans. on Pattern Analysis and Machine Intelligence 13, 99–113 (1991).CrossRefGoogle Scholar
  34. [34]
    C. A. Bouman and M. Shapiro, “A multiscale random field model for Bayesian image segmentation,” IEEE Trans. on Image Processing 3, 162–177 (1994).CrossRefGoogle Scholar
  35. [35]
    S. S. Saquib, C. A. Bouman, and K. Sauer, “A non-homogeneous MRF model for multiresolution Bayesian estimation,” Proc. of IEEE Int’l Conf. on Image Proc., (IEEE, Lausanne Switzerland) pp. 445–448, 1996.Google Scholar
  36. [36]
    D. Snyder and M. Miller “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. on Nuclear Science NS-32, 3864–3871 (1985).CrossRefGoogle Scholar
  37. [37]
    E. Veklerov and J. Llacer “Stopping rule for the MLE algorithm based on statistical hypothesis testing,” IEEE Trans. on Medical Imaging MI-6, 313–319 (1987).CrossRefGoogle Scholar
  38. [38]
    T. Hebert, R. Leahy, and M. Singh, “Fast MLE for SPECT using an intermediate polar representation and a stopping criterion,” IEEE Trans. on Nuclear Science 35, 615–619 (1988).CrossRefGoogle Scholar
  39. [39]
    A. Tikhonov and V. Arsenin, Solutions of Ill-Posed Problems (Winston and Sons, New York), 1977.Google Scholar
  40. [40]
    J. Besag “Spatial interaction and the statistical analysis of lattice systems,” Journal of the Royal Statistical Society B 36, 192–236 (1974).Google Scholar
  41. [41]
    R. Kindermann and J. Snell, Markov Random Fields and their Applications (American Mathematical Society, Providence), 1980.Google Scholar
  42. [42]
    R. Kashyap and R. Chellappa “Estimation and choice of neighbors in spatial-interaction models of images,” IEEE Trans. on Information Theory IT-29, 60–72 (1983).CrossRefGoogle Scholar
  43. [43]
    D. Pickard “Inference for discrete Markov fields: The simplest nontrivial case,” Journal of the American Statistical Association 82, 90–96 (1987).CrossRefGoogle Scholar
  44. [44]
    R. Dubes and A. Jain, “Random field models in image analysis,” Journal of Applied Statistics 16, 131–164 (1989).CrossRefGoogle Scholar
  45. [45]
    K. Sauer and C. Bouman, “Bayesian estimation of transmission tomograms using segmentation based optimization,” IEEE Trans. on Nuclear Science 39, 1144–1152 (1992).CrossRefGoogle Scholar
  46. [46]
    C. A. Bouman, “Cluster: an unsupervised algorithm for modeling Gaussian mixtures.” Available from http://www.ece.purdue.edu/ “bouman, 1997.

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Thomas Frese
    • 1
  • Charles A. Bouman
    • 2
  • Ken Sauer
    • 3
  1. 1.Department of Electrical and Computer EngineeringPurdue UniversityWest LafayetteUSA
  2. 2.Department of Electrical and Computer EngineeringPurdue UniversityWest LafayetteUSA
  3. 3.Department of Electrical EngineeringUniversity of Notre DameNotre DameUSA

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