Reconstruction of Binary Images via the EM Algorithm

  • Yehuda Vardi
  • Cun-Hui Zhang
Part of the Applied and Numerical Harmonic Analysis book series (ANHA)


The problem of reconstructing a binary function x defined on a finite subset of a lattice ℤ, from an arbitrary collection of its partial sums is considered. The approach is based on (a) relaxing the binary constraints \(x\left( i \right) = 0\) or 1 to interval constraints \(0x\left( i \right)1,i \in {\Bbb Z}\),and (b) applying a minimum distance method (using Kullback-Leibler’s information divergence index as our distance function) to find such an x — say, \(\hat x\)— for which the distance between the observed and the theoretical partial sums is as small as possible (Turning this \(\hat x\) into a binary function can be done as a separate postprocessing step: for instance,through thresholding, or through some additional Bayes modeling.) This minimum-distance solution is derived via a new EM algorithm that extends the often-studied EM/maximum likelihood (EM/ML) algorithm in emission tomography and certain linear-inverse problems to include lower-and upper-bound constraints on the function x. Properties of the algorithm including convergence and uniqueness conditions on the solution (or parts of it) are described.


Binary Image Maximum Likelihood Estimator Probability Vector Binary Function Binary Constraint 
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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  1. [1]
    Y. Vardi and D. Lee, “The discrete Radon transform and its approximate inversion via the EM algorithm,” International J. Imaging Systems and Techn. 9, 155–173 (1998).CrossRefGoogle Scholar
  2. [2]
    Y. Vardi and C.-H. Zhang, “Estimating mixing probabilities subject to lower and upper bound constraints with applications,” Technical Report #97–006, Department of Statistics, Rutgers University, Piscataway, NJ (1997).Google Scholar
  3. [3]
    U. Rothblum and Y. Vardi “Maximum likelihood estimation of cell probabilities in constrained multinomial models,” J. Statist. Computation and Simulation, 61, 141–161 (1998).CrossRefGoogle Scholar
  4. [4]
    A. P. Dempster, N. M. Laird, and R. B. Rubin “Maximum likelihood from incomplete data via the EM algorithm (with Discussion),” J. R. Statist. Soc. B 39, 1–38 (1977).Google Scholar
  5. [5]
    G. T. Herman “Reconstruction of binary patterns from a few projections,” In A. Günther, B. Levrat, and H. Lipps, International Computing Symposium 1973, (North-Holland Publ. Co., Amsterdam) pp. 371–378,1974.Google Scholar
  6. [6]
    L. A. Shepp, and Y. Vardi, “Maximum likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imaging 1, 113–122 (1982).CrossRefPubMedGoogle Scholar
  7. [7]
    Y. Vardi, L. A. Shepp and L. Kaufman “A statistical model for positron emission tomography (with Discussion),” J. Amer. Statist. Assoc. 80, 8–37 (1985).CrossRefGoogle Scholar
  8. [8]
    Y. Vardi and D. Lee, “From image deblurring to optimal investments: Maximum likelihood solutions for positive linear inverse problems (with Discussion),” J. R. Statist. Soc. B 55, 569–612 (1993).Google Scholar
  9. [9]
    T. M. Cover “An algorithm for maximizing expected log investment return,” IEEE Trans. Inform. Theory 30, 369–373 (1984).CrossRefGoogle Scholar
  10. [10]
    I. Csiszár and G. Tusnády, “Information geometry and alternating minimization procedures,” Statist. Decis., Suppl. 1, 205–237 (1984).Google Scholar
  11. [11]
    P. Fishburn, P. Schwander, L. Shepp, and R. Vanderbei, “The discrete Radon transform and its approximate inversion via linear programming,” Discrete Applied Mathematics 75, 39–61 (1997).CrossRefGoogle Scholar
  12. [12]
    R. Gordon, R. Bender, and G. T. Herman, “Algebraic reconstruction techniques (ART) for three-dimensional electron microscopy and x-ray photography,” J. Theoret. Biol. 29, 471–481 (1970).CrossRefGoogle Scholar
  13. [13]
    L. M. Bregman “The relaxation method of finding the common point of convex sets and its application to the solution of problems in convex programming,” U.S.S.R. Comput. Math. and Math. Phys. 7, 200–217 (1967).CrossRefGoogle Scholar
  14. [14]
    Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications (Oxford University Press, New York), 1997.Google Scholar
  15. [15]
    Y. Censor, A. R. De Pierro, T. Elfving, G. T. Herman, and A. N. Iusem, “On iterative methods for linearly constrained entropy maximization,” In A. Wakulicz, Numerical Analysis and Mathematical Modeling, (PWN - Polish Scientific Publishers, Warsaw) pp. 145–163, 1990.Google Scholar
  16. [16]
    Y. Censor “Finite series-expansion reconstruction methods,” Proc. IEEE 71, 409–419 (1983).CrossRefGoogle Scholar
  17. [17]
    G. T. Herman “Applications to maximum entropy and Bayesian optimization methods to image reconstruction from projections” In C. P. Smith and W. T. Grandy, Jr., Maximum-Entropy and Bayesian Methods in Inverse Problems, (D. Reidel Publishing Company, Dordrecht The Netherlands), pp. 319–338, 1985.CrossRefGoogle Scholar
  18. [18]
    C. L. Byrne, “Iterative image reconstruction algorithms based on cross-entropy minimization,” IEEE Trans. on Image Processing 2, 96103 (1993).CrossRefGoogle Scholar
  19. [19]
    R. Aharoni, G. T. Herman, and A. Kuba, “Binary vectors partially determined by linear equation systems,” Discrete Math. 171, 1–16 (1997).CrossRefGoogle Scholar
  20. [20]
    M. T. Chan, G. T. Herman, and E. Levitan, “Bayesian image reconstruction using image-modeling Gibbs priors,” Int. J. Imaging Sys. Tech. 9, 85–98 (1997).CrossRefGoogle Scholar
  21. [21]
    S-K. Chang, “The reconstruction of binary patterns from their projections,” Comm. of the ACM 14, 21–25 (1971).CrossRefGoogle Scholar
  22. [22]
    H. W. Kuhn and A. W. Tucker “Nonlinear programming.” Proc. Second Berkeley Symp. Math. Statist. Probab. 1, 481–492 (1951).Google Scholar

Copyright information

© Springer Science+Business Media New York 1999

Authors and Affiliations

  • Yehuda Vardi
    • 1
  • Cun-Hui Zhang
    • 1
  1. 1.Department of StatisticsRutgers UniversityPiscatawayUSA

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