Skip to main content
Log in

Solving emission tomography problems on vector machines

  • Published:
Annals of Operations Research Aims and scope Submit manuscript

Abstract

Shepp and Vardi's maximum likelihood reconstruction algorithm for emission tomography has mostly been ignored by the medical community because of its perceived high computational costs. However, the algorithm is very suited to parallel and vector machines, and on these machine can be made economically feasible even on medically reasonable problems of 16000 variables. It is also a good test problem for parallel optimization algorithms for large simply bounded nonlinear problems.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others

References

  1. D.P. Bertsekas, Projected Newton methods for optimization problems with simple constraints, SIAM J. on Control and Optimization 20(1982)221–246.

    Article  Google Scholar 

  2. P.H. Calamai and J.J. More, Projected gradient methods for linearly constrained problems, Math. Progr. 39(1987)93–116.

    Google Scholar 

  3. A.P. Dempster, N.M. Laird and D.B. Rubin, Maximum likelihood from incomplete data via the EM algorithm, J. Roy. Stat. Soc., Ser. B, 39(1977)1–38.

    Google Scholar 

  4. J. Dongarra, J. Du Croz, S. Hammarling and R. Hanson, An extended set of FORTRAN basic linear algebra subprograms, ACM Trans. on Mathematical Software (March 1988) 1–17.

  5. J. Dongarra and S.C. Eisenstat, Squeezing the most out of an algorithm in CRAY FORTRAN, ACM Trans. on Mathematical Software (Sept. 1984)219–230.

  6. J. Dongarra, L. Kaufman and S. Hammarling, Squeezing the most out of eigenvalue solvers on high-performance computers, Linear Algebra and its Applications 77(1986)113–136.

    Article  MathSciNet  Google Scholar 

  7. P.E. Gill, W. Murray and M.H. Wright,Practical Optimization (Academic Press, 1981).

  8. N. Karmarkar, A new polynomial-time algorithm for linear programming, Combinatorica 4(1984)373–395.

    Google Scholar 

  9. L. Kaufman, Implementing and accelerating the EM algorithm for positron emission tomography, IEEE Trans. on Medical Imaging MI-6(1987)37–51.

    Google Scholar 

  10. L. Kaufman, S. Morgenthaler and Y. Vardi, Maximum likelihood reconstruction in emission tomography with time-of-flight information: A limited study, in:Automated Image Analysis: Theory and Experiments, ed. Cooper, Laurner and McClure (Academic Press, 1984).

  11. K. Lange and R. Carson, EM reconstruction algorithms for emission and transmission tomography, J. Computer Assisted Tomography 8(1984)302–316.

    Google Scholar 

  12. C. Lawson, R. Hanson, D. Kincaid and F. Krogh, Basic linear algebra subproblems for FORTRAN usage, ACM Trans. on Mathematical Software 5(1979)308–371.

    Article  Google Scholar 

  13. R.M. Lewitt and G. Muehllehner, Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximim likelihood estimation, IEEE Trans. on Medical Imaging MI-5(1986)12–22.

    Google Scholar 

  14. G.P. McCormick and R.A. Tapia, The gradient projection method under mild differentiability conditions, SIAM J. on Control 10(1972)93–98.

    Article  Google Scholar 

  15. C.C. Paige and M.A. Saunders, LSQR: An algorithm for computing the singular value decomposition, ACM Trans. on Mathematical Software (March 1982)43–71.

  16. L.A. Shepp and Y. Vardi, Maximum likelihood reconstruction in positron emission tomography, IEEE Trans. on Medical Imaging (1982) 113–122.

  17. D.L. Snyder and M.I. Miller, The use of sieves to stabilize images produced with the EM algorithm for emission tomography, IEEE Trans. on Nucl. Sci. (October 1985)3864–3777.

  18. D.L. Snyder and D.G. Politte, Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements, IEEE Trans. on Nucl. Sci. NS-30(1983)1843–1849.

    Article  Google Scholar 

  19. M.M. Ter-Pogossian, N.A. Mulani, D.C. Ficke, J. Markham and D.L. Snyder, Photon time-of-flight positron emission tomography, J. Computer Assisted Tomography 5(1981)227–239.

    Article  Google Scholar 

  20. M.M. Ter-Pogossian, M.E. Raichle and B.E. Sobel, Positron emission tomography, Sci. Amer. 243(1980)170–181.

    Article  PubMed  Google Scholar 

  21. Y. Vardi, L.A. Shepp and L. Kaufman, A statistical model for positron emission tomography, J. Amer. Stat. Assoc. 80, 389(1985)8–20; 34–37.

    Google Scholar 

  22. E. Verklerov and J. Llacer, Stopping rules for the MLE algorithm based on statistical hypothesis testing, IEEE Trans. on Medical Imaging MI-7(1987)313–319.

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Cite this article

Kaufman, L. Solving emission tomography problems on vector machines. Ann Oper Res 22, 325–353 (1990). https://doi.org/10.1007/BF02023059

Download citation

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02023059

Keywords

Navigation