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Multiplicative iterative methods in computed tomography

  • Inverse Problems And Optimization
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Mathematical Methods in Tomography

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References

  1. D.L. Anderson and A.M. Dziewonski, Seismic tomography, Scientific American 251 (1984), pp. 58–66.

    Article  Google Scholar 

  2. M. Avriel, Nonlinear Programming, Analysis and Methods, Prentice-Hall, New Jersey, 1976.

    Google Scholar 

  3. L.M. Bregman, The relaxation method of finding the common point of convex sets and its applications to the solution of problems in convex programming, U.S.S.R Computational Mathematics and Mathematical Physics, vol. 7 (1967), pp. 200–217.

    Article  Google Scholar 

  4. J.P. Burg, Maximum entropy spectral analysis, in: Proceedings of the 37th Annual Meeting of the Society of Exploration Geophysicists, Oklahoma City, Oklahoma, 1967.

    Google Scholar 

  5. Y. Censor, Row-action methods for huge and sparse systems and their applications, SIAM Review 23 (1981), pp. 444–466.

    Article  Google Scholar 

  6. Y. Censor, On the selective use of iterative algorithms for inversion problems in image reconstruction and radiotherapy, in: Proceedings of the 12th World Congress on Scientific Computation-IMACS 1988 (R. Vichnevetsky, P. Borne and J. Vignes, Editors), Gerfdin, Paris, France, vol. 4 (1988), pp. 563–565.

    Google Scholar 

  7. Y. Censor, M.D. Altschuler and W.D. Powlis, A computational solution of the inverse problem in radiation therapy treatment planning, Applied Mathematics and Computation, 25 (1988), pp. 57–87.

    Article  Google Scholar 

  8. Y. Censor, A.R. De Pierro, T. Elfving, G.T. Herman and A.N. Iusen, On iterative methods for linearly constrained entropy maximization, in: A. Wakulicz, ed., Numerical Analysis and Mathematical Modelling, Banach Center Publications, Vol. XXIV, Stefan Banach International Mathematical Center (1989), pp. 147–165.

    Google Scholar 

  9. Y. Censor, A.R. de Pierro and A.N. Iusem, Optimization of Burg's entropy over linear constraints, Applied Numerical Mathematics, to appear.

    Google Scholar 

  10. Y. Censor and G.T. Herman, On some optimization techniques in image reconstruction from projections, Applied Numerical Mathematics 3 (1987), pp. 365–391.

    Article  Google Scholar 

  11. Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981), pp. 321–353.

    Article  Google Scholar 

  12. Y. Censor and A. Lent, Optimization of “log x”-entropy over linear equality constraints, SIAM J. Control Optim. 25 (1987), pp. 921–933.

    Article  Google Scholar 

  13. Y. Censor and J. Segman, On the block-iterative entropy maximization, Journal of Information & Optimization Sciences, vol. 8 (1987), 3, pp. 275–291.

    Article  Google Scholar 

  14. M.T. Chahine, Determination of the temperature profile in an atmosphere from its outgoing radiance, J. Opt. Soc. Amer. 58 (1968) 1634–1637.

    Article  Google Scholar 

  15. M.T. Chahine, Remote sounding of cloud parameters, J. Atmos. Sci., 38, 1, (1982) pp. 159–170.

    Article  Google Scholar 

  16. M.T. Chahine, Generalization of the relaxation method for the inverse solution of nonlinear and linear transfer equations, in Inversion Methods in Atmospheric Remote Sounding. A. Deepak ed., Academic Press, New York, 1977.

    Google Scholar 

  17. I. Csiszár and G. Tusnády, Information geometry and alternating minimization procedures, in: Statistics & Decisions, Suppl. No. 1 (1984), pp. 205–237.

    Google Scholar 

  18. J.N. Darroch and D. Ratcliff, Generalized iterative scaling for log-linear models. The Annals of Mathematical Statistics, Vol. 43 (1972), pp. 1470–1480.

    Article  Google Scholar 

  19. M.E. Daube-Witherspoon and G. Muehllehner, An iterative image space reconstruction algorithm suitable for volume ECT, IEEE Trans. Med. Imaging, vol. MI-5 (1986), pp. 61–66.

    Article  Google Scholar 

  20. A.P. Dempster, N.M. Laird and D.B. Rubin, Maximum likelihood for incomplete data via the EM algorithm, J.R. Stat. Soc. Series B, 39 (1977), pp. 1–38.

    Google Scholar 

  21. A.R. De Pierro, On the convergence of the iterative image space reconstruction algorithm for volume ECT, IEEE Trans. Med. Imaging, vol. MI-6 (1987), pp. pp. 174–175.

    Google Scholar 

  22. A.R. De Pierro, A generalization of the EM algorithm for maximum likelihood estimates from incomplete data, Tech. Rept. MIPG 119, Medical Image Processing Group, Department of Radiology, Hospital of the University of Pennsylvania, PA, 1987.

    Google Scholar 

  23. A.R. De Pierro, On some nonlinear iterative relaxation methods in remote sensing, Matemática Aplicada e Computacional, vol. 8 (1989), pp. 153–166.

    Google Scholar 

  24. A.R. De Pierro, Nonlinear relaxation methods for solving symmetric linar complementarity problems, J. Optim. Theory Appl. 64 (1990), pp. 87–99.

    Article  Google Scholar 

  25. A.R. De Pierro, Parallel Bregman methods for convex programming and entropy maximization, forthcoming paper.

    Google Scholar 

  26. A.R. De Pierro and A.N. Iusem, A relaxed version of Bregman's method for convex programming, J. Optim. Theory Appl., 51 (1986), pp. 421–440.

    Article  Google Scholar 

  27. N.J. Dusaussoy and I.E. Abdou, Some new multiplicative algorithms for image reconstruction from projections, Linear Algebra Appl., 130 (1990), pp. 111–132.

    Article  Google Scholar 

  28. P.P.B. Eggermont, Multiplicative iterative algorithms for convex programming, Linear algebra Appl., 130 (1990), pp. 25–42.

    Article  Google Scholar 

  29. H.E. Fleming, Satellite remote sensing by the technique of computed tomography, J. Appl., Metereology 21 (1982), pp. 1538–1549.

    Article  Google Scholar 

  30. 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 (1970), pp. 471–481.

    Article  CAS  Google Scholar 

  31. P.J. Green, Penalized likelihood reconstructions from emission tomography data using a modified EM algorithm, to appear.

    Google Scholar 

  32. P.J. Green, On the use of the EM algorithm for penalized likelihood estimation, to appear.

    Google Scholar 

  33. G.T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.

    Google Scholar 

  34. G.T. Herman, Application of maximum entropy and Bayesian optimization methods to image reconstruction from projections, in: C.R. Smith and W.T. Grandy, Jr., Eds., Maximum-Entropy and Bayesian Methods in Inverse Problems, Reidel, Dordrecht (1985), pp. 319–338.

    Chapter  Google Scholar 

  35. G.T. Herman, D. Odhner, K.D. Toennies and S.A. Zenios, A parallelized algorithm for image reconstruction from noisy projections, Tech. Rept. MIPG 155, Medical Image Processing Group, Department of Radiology, Hospital of the University of Pennsylvania, PA, 1989.

    Google Scholar 

  36. A.N. Iusem, Convergence analysis for a multiplicatively relaxed EM algorithm, with applications in Position Emission Tomography, to be published.

    Google Scholar 

  37. A.N. Iusem and A.R. De Pierro, Convergence results for an accelerated nonlinear Cimmino algorithm, Numer. Math. 49 (1986), pp. 367–378.

    Article  Google Scholar 

  38. E.T. Jaynes, On the rationale of maximum-entropy methods, Proc. IEEE 70 (1982), pp. 939–952.

    Article  Google Scholar 

  39. R. Johnson and J.E. Shore, Which is the better entropy expression for speech processing: -S log S or log S?, IEEE Trans. Acoust. Speech Signal Process, 32 (1984), pp. 129–136.

    Article  Google Scholar 

  40. J.N. Kapur, Twenty-five years of maximum-entropy principle, J. Math. Phys. Sci. 17 (1983), pp. 102–156.

    Google Scholar 

  41. S. Kullback, Information Theory and Statistics, Wiley, New York, 1959.

    Google Scholar 

  42. K. Lange, M. Bahn and R. Little, A theoretical study of some maximum likelihood algorithms for emission and transmission tomography, IEEE Trans. Med. Imaging., vol. MI-6 (1987), pp. 106–114.

    Article  Google Scholar 

  43. K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomog., vol. 8 (1984), pp. 306–316.

    CAS  Google Scholar 

  44. A. Lent, A convergent algorithm for maximum entropy image restoration with a medical X-ray application, in: R. Shaw, Ed., Image Analysis and Evaluation, Society of Photographic Scientists and Engineers, Washington, DC (1977), pp. 249–247.

    Google Scholar 

  45. R.D. Levine and M. Tribus, Eds., The Maximum Entropy Formalism, MIT Press, Cambridge, MA, 1978.

    Google Scholar 

  46. E. Levitan and G.T. Herman, A maximum a posteriori probability expectation maximization algorithm for image reconstruction in emission tomography, IEEE Trans. Med. Imaging, MI-6 (1987), pp. 185–192.

    Article  Google Scholar 

  47. R.M. Lewitt, Reconstruction algorithms: transform methods, Proc. IEEE, 71 (1983), pp. 390–408.

    Article  Google Scholar 

  48. L.B. Lucy, An iterative technique for the rectification of observed distributions, Astron. J. 79 (1974), pp. 745–754.

    Article  Google Scholar 

  49. E.S. Meinel, Origins of linear and nonlinear recursive restoration algorithms, J. Opt. Soc. Amer. A, vol. 36 (1986), pp. 787–799.

    Article  Google Scholar 

  50. F. Natterer, The Mathematics of Computerized Tomography, J. Wiley & Sons, New York, 1986.

    Google Scholar 

  51. D.N. Nychka, Some properties of an EM algorithm that includes a smoothing step, to be published.

    Google Scholar 

  52. W.H. Richardson, Bayesian-based iterative method of image restoration, J. Opt. Soc. Am. 62 (1972), pp. 55–59.

    Article  Google Scholar 

  53. L.A. Shepp and Y. Vardi, Maximum likelihood reconstruction in position emission tomography, IEEE Trans. Med. Imaging, MT-1 (1982), pp. 113–122.

    Article  Google Scholar 

  54. L.A. Shepp, Y. Vardi and L. Kaufman, A statistical model for positron emission tomography, J. of the Am. Stat. Assoc., vol. 80, No. 389 (1985), pp. 8–37.

    Article  Google Scholar 

  55. B.W. Silverman, M.C. Jones, J.D. Wilson and D.W. Nychka, A smoothed EM approach to a class of problems in image analysis and integral equations, to be published.

    Google Scholar 

  56. C.R. Smith and W.T. Grandy, Jr., Eds., Maximum-Entropy and Bayesian Methods in Inverse Problems, Reidel, Dordrecht, 1985.

    Google Scholar 

  57. E. Tanaka, A fast reconstruction algorithm for stationary positron emission tomography based on a modified EM algorithm, IEEE Trans. Med. Imaging, MI-6,2 (1987), pp. 98–105.

    Article  Google Scholar 

  58. M. Teboulle, On ø-divergence and its applications, Proceedings of the Conference in Honor of A. Charnes 70th. Birthday, Austin, Texas, to appear.

    Google Scholar 

  59. M.M. Ter-Pogossian, M. Raichle and B.E. Sobel, Positron Emission Tomography, Scientific American, 243 (4) (1980), pp. 170–181.

    Article  CAS  PubMed  Google Scholar 

  60. S. Twomey, Comparison of constrained linear inversion and an iterative nonlinear algorithm applied to the indirect estimation of particle size distributions, J. Comput. Phys., 18 (1975), pp. 188–200.

    Article  Google Scholar 

  61. S.A. Zenios and Y. Censor, Parallel computing with block iterative image reconstruction algorithms, Tech. Rept. MIPG 134, Medical Image Processing Group, Department of Radiology, University of Pennsylvania, PA, 1990.

    Google Scholar 

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De Pierro, A.R. (1991). Multiplicative iterative methods in computed tomography. In: Herman, G.T., Louis, A.K., Natterer, F. (eds) Mathematical Methods in Tomography. Lecture Notes in Mathematics, vol 1497. Springer, Berlin, Heidelberg. https://doi.org/10.1007/BFb0084517

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  • DOI: https://doi.org/10.1007/BFb0084517

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