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References
D.L. Anderson and A.M. Dziewonski, Seismic tomography, Scientific American 251 (1984), pp. 58–66.
M. Avriel, Nonlinear Programming, Analysis and Methods, Prentice-Hall, New Jersey, 1976.
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.
J.P. Burg, Maximum entropy spectral analysis, in: Proceedings of the 37th Annual Meeting of the Society of Exploration Geophysicists, Oklahoma City, Oklahoma, 1967.
Y. Censor, Row-action methods for huge and sparse systems and their applications, SIAM Review 23 (1981), pp. 444–466.
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.
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.
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.
Y. Censor, A.R. de Pierro and A.N. Iusem, Optimization of Burg's entropy over linear constraints, Applied Numerical Mathematics, to appear.
Y. Censor and G.T. Herman, On some optimization techniques in image reconstruction from projections, Applied Numerical Mathematics 3 (1987), pp. 365–391.
Y. Censor and A. Lent, An iterative row-action method for interval convex programming, J. Optim. Theory Appl. 34 (1981), pp. 321–353.
Y. Censor and A. Lent, Optimization of “log x”-entropy over linear equality constraints, SIAM J. Control Optim. 25 (1987), pp. 921–933.
Y. Censor and J. Segman, On the block-iterative entropy maximization, Journal of Information & Optimization Sciences, vol. 8 (1987), 3, pp. 275–291.
M.T. Chahine, Determination of the temperature profile in an atmosphere from its outgoing radiance, J. Opt. Soc. Amer. 58 (1968) 1634–1637.
M.T. Chahine, Remote sounding of cloud parameters, J. Atmos. Sci., 38, 1, (1982) pp. 159–170.
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.
I. Csiszár and G. Tusnády, Information geometry and alternating minimization procedures, in: Statistics & Decisions, Suppl. No. 1 (1984), pp. 205–237.
J.N. Darroch and D. Ratcliff, Generalized iterative scaling for log-linear models. The Annals of Mathematical Statistics, Vol. 43 (1972), pp. 1470–1480.
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.
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.
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.
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.
A.R. De Pierro, On some nonlinear iterative relaxation methods in remote sensing, Matemática Aplicada e Computacional, vol. 8 (1989), pp. 153–166.
A.R. De Pierro, Nonlinear relaxation methods for solving symmetric linar complementarity problems, J. Optim. Theory Appl. 64 (1990), pp. 87–99.
A.R. De Pierro, Parallel Bregman methods for convex programming and entropy maximization, forthcoming paper.
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.
N.J. Dusaussoy and I.E. Abdou, Some new multiplicative algorithms for image reconstruction from projections, Linear Algebra Appl., 130 (1990), pp. 111–132.
P.P.B. Eggermont, Multiplicative iterative algorithms for convex programming, Linear algebra Appl., 130 (1990), pp. 25–42.
H.E. Fleming, Satellite remote sensing by the technique of computed tomography, J. Appl., Metereology 21 (1982), pp. 1538–1549.
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.
P.J. Green, Penalized likelihood reconstructions from emission tomography data using a modified EM algorithm, to appear.
P.J. Green, On the use of the EM algorithm for penalized likelihood estimation, to appear.
G.T. Herman, Image Reconstruction from Projections: The Fundamentals of Computerized Tomography, Academic Press, New York, 1980.
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.
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.
A.N. Iusem, Convergence analysis for a multiplicatively relaxed EM algorithm, with applications in Position Emission Tomography, to be published.
A.N. Iusem and A.R. De Pierro, Convergence results for an accelerated nonlinear Cimmino algorithm, Numer. Math. 49 (1986), pp. 367–378.
E.T. Jaynes, On the rationale of maximum-entropy methods, Proc. IEEE 70 (1982), pp. 939–952.
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.
J.N. Kapur, Twenty-five years of maximum-entropy principle, J. Math. Phys. Sci. 17 (1983), pp. 102–156.
S. Kullback, Information Theory and Statistics, Wiley, New York, 1959.
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.
K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Comput. Assist. Tomog., vol. 8 (1984), pp. 306–316.
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.
R.D. Levine and M. Tribus, Eds., The Maximum Entropy Formalism, MIT Press, Cambridge, MA, 1978.
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.
R.M. Lewitt, Reconstruction algorithms: transform methods, Proc. IEEE, 71 (1983), pp. 390–408.
L.B. Lucy, An iterative technique for the rectification of observed distributions, Astron. J. 79 (1974), pp. 745–754.
E.S. Meinel, Origins of linear and nonlinear recursive restoration algorithms, J. Opt. Soc. Amer. A, vol. 36 (1986), pp. 787–799.
F. Natterer, The Mathematics of Computerized Tomography, J. Wiley & Sons, New York, 1986.
D.N. Nychka, Some properties of an EM algorithm that includes a smoothing step, to be published.
W.H. Richardson, Bayesian-based iterative method of image restoration, J. Opt. Soc. Am. 62 (1972), pp. 55–59.
L.A. Shepp and Y. Vardi, Maximum likelihood reconstruction in position emission tomography, IEEE Trans. Med. Imaging, MT-1 (1982), pp. 113–122.
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.
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.
C.R. Smith and W.T. Grandy, Jr., Eds., Maximum-Entropy and Bayesian Methods in Inverse Problems, Reidel, Dordrecht, 1985.
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.
M. Teboulle, On ø-divergence and its applications, Proceedings of the Conference in Honor of A. Charnes 70th. Birthday, Austin, Texas, to appear.
M.M. Ter-Pogossian, M. Raichle and B.E. Sobel, Positron Emission Tomography, Scientific American, 243 (4) (1980), pp. 170–181.
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.
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.
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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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