Abstract

Maximum-likelihood image restoration for noncoherent imagery, which is based on the generic expectation maximization (EM) algorithm of Dempster et al. [ J. R. Stat. Soc. B 39, 1 ( 1977)], is an iterative method whose convergence can be slow. We discuss an accelerative version of this algorithm. The EM algorithm is interpreted as a hill-climbing technique in which each iteration takes a step up the likelihood functional. The basic principle of the acceleration technique presented is to provide larger steps in the same vector direction and to find some optimal step size. This basic line-search principle is adapted from the research of Kaufman [ IEEE Trans. Med. Imag. MI-6, 37 ( 1987)]. Modifications to her original acceleration algorithm are introduced, which involve extensions in considering truncated data and an alternative way of implementing the search for an optimal step size. Log-likelihood calculations and reconstructed images from simulations show the execution time’s being shortened from the nonaccelerated algorithm by approximately a factor of 7.

© 1991 Optical Society of America

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  1. T. J. Holmes, “Maximum-likelihood image-restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  2. T. J. Holmes, “Expectation-maximization restoration of band-limited, truncated point-process intensities with application in microscopy,” J. Opt. Soc. Am. A 6, 1006–1014 (1989).
    [CrossRef]
  3. T. J. Holmes, Y. H. Liu, “Richardson–Lucy/maximum-likelihood image restoration algorithm for fluorescence microscopy: further testing,” Appl. Opt. 28, 4930–4938 (1989).
    [CrossRef] [PubMed]
  4. T. J. Holmes, Y. H. Liu, “Simulation tests of restoring truncated fluorescence micrographs,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 197–204 (1989).
    [CrossRef]
  5. L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–122 (1982).
    [CrossRef]
  6. Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80(3), 8–19 (1985).
    [CrossRef]
  7. D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
    [CrossRef]
  8. M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 768–778 (1985).
  9. K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomography 8, 306–316 (1984).
  10. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–37 (1977).
  11. R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
    [CrossRef]
  12. C. T. Chen, C. Metz, X. Hu, “Maximum likelihood reconstruction in PET and TOFPET,” in Mathematics and Computer Science in Medical Imaging, M. A. Viergever, A. E. Todd-Pokropek, eds. (Springer-Verlag, Berlin, 1988), pp. 319–329.
    [CrossRef]
  13. C. E. Metz, C. T. Chen, “On the acceleration of maximum likelihood algorithms,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.914, 344–349 (1988).
    [CrossRef]
  14. L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imag. MI-6, 37–51 (1987).
    [CrossRef]
  15. K. Lange, M. Bahn, R. Little, “A theoretical study of some maximum likelihood algorithms for emission and transmission tomography,” IEEE Trans. Med. Imag. MI-6, 106–114 (1987).
    [CrossRef]
  16. P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).
  17. D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).
  18. D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
    [CrossRef]
  19. D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
    [CrossRef]
  20. N. Striebl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
    [CrossRef]
  21. A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).
  22. A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).
  23. E. Veklerov, J. Llacer, “MLE reconstruction of a brain phantom using a Monte Carlo transition matrix and a statistical stopping rule,”IEEE Trans. Nucl. Sci. 35, 603–607 (1988).
    [CrossRef]
  24. T. Hebert, R. Leahy, M. Singh, “Fast MLE for SPECT using an intermediate polar representation and a stopping criterion,”IEEE Trans. Nucl. Sci. 35, 615–619 (1988).
    [CrossRef]
  25. E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1972).
  26. W. H. Richardson, “Baysian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  27. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–765 (1974).
    [CrossRef]
  28. K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
    [CrossRef]
  29. M. I. Miller, D. L. Snyder, “The role of likelihood and entropy in incomplete-data problems: applications to estimating point-process intensities and Toeplitz constrained covariances,” Proc. IEEE 75, 892–907 (1987).
    [CrossRef]
  30. D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
    [CrossRef]
  31. T. Hebert, R. Leahy, “A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors,”IEEE Trans. Med. Imag. 8, 194–202 (1989).
    [CrossRef]
  32. Z. Liang, “Statistical models of a prioriinformation for image processing,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.914, 677–683 (1988).
    [CrossRef]
  33. Z. Liang, “Statistical models of a prioriinformation for image processing: neighboring correlation constraints,” J. Opt. Soc. Am. A 5, 2026–2031 (1988).
    [CrossRef]
  34. B. Roysam, J. A. Shrauner, M. I. Miller, “Bayesian imaging using Good’s roughness measure—implementation on a massively parallel processor,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), Vol. II, pp. 932–935.
    [CrossRef]
  35. D. G. Politte, D. L. Snyder, “The use of constraints to eliminate artifacts in maximum-likelihood image estimation for emission tomography,”IEEE Trans. Nucl. Sci. 35, 608–610 (1988).
    [CrossRef]

1989 (3)

1988 (6)

Z. Liang, “Statistical models of a prioriinformation for image processing: neighboring correlation constraints,” J. Opt. Soc. Am. A 5, 2026–2031 (1988).
[CrossRef]

D. G. Politte, D. L. Snyder, “The use of constraints to eliminate artifacts in maximum-likelihood image estimation for emission tomography,”IEEE Trans. Nucl. Sci. 35, 608–610 (1988).
[CrossRef]

T. J. Holmes, “Maximum-likelihood image-restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
[CrossRef]

E. Veklerov, J. Llacer, “MLE reconstruction of a brain phantom using a Monte Carlo transition matrix and a statistical stopping rule,”IEEE Trans. Nucl. Sci. 35, 603–607 (1988).
[CrossRef]

T. Hebert, R. Leahy, M. Singh, “Fast MLE for SPECT using an intermediate polar representation and a stopping criterion,”IEEE Trans. Nucl. Sci. 35, 615–619 (1988).
[CrossRef]

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

1987 (4)

M. I. Miller, D. L. Snyder, “The role of likelihood and entropy in incomplete-data problems: applications to estimating point-process intensities and Toeplitz constrained covariances,” Proc. IEEE 75, 892–907 (1987).
[CrossRef]

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imag. MI-6, 37–51 (1987).
[CrossRef]

K. Lange, M. Bahn, R. Little, “A theoretical study of some maximum likelihood algorithms for emission and transmission tomography,” IEEE Trans. Med. Imag. MI-6, 106–114 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

1986 (1)

R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
[CrossRef]

1985 (3)

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 768–778 (1985).

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80(3), 8–19 (1985).
[CrossRef]

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

1984 (2)

N. Striebl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomography 8, 306–316 (1984).

1983 (1)

D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

1982 (1)

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–122 (1982).
[CrossRef]

1981 (1)

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–37 (1977).

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–765 (1974).
[CrossRef]

1972 (1)

Bahn, M.

K. Lange, M. Bahn, R. Little, “A theoretical study of some maximum likelihood algorithms for emission and transmission tomography,” IEEE Trans. Med. Imag. MI-6, 106–114 (1987).
[CrossRef]

Capps, R. W.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Carson, R.

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomography 8, 306–316 (1984).

Chen, C. T.

C. T. Chen, C. Metz, X. Hu, “Maximum likelihood reconstruction in PET and TOFPET,” in Mathematics and Computer Science in Medical Imaging, M. A. Viergever, A. E. Todd-Pokropek, eds. (Springer-Verlag, Berlin, 1988), pp. 319–329.
[CrossRef]

C. E. Metz, C. T. Chen, “On the acceleration of maximum likelihood algorithms,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.914, 344–349 (1988).
[CrossRef]

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–37 (1977).

Gill, P. E.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

Grasdalen, G. L.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Hebert, T.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors,”IEEE Trans. Med. Imag. 8, 194–202 (1989).
[CrossRef]

T. Hebert, R. Leahy, M. Singh, “Fast MLE for SPECT using an intermediate polar representation and a stopping criterion,”IEEE Trans. Nucl. Sci. 35, 615–619 (1988).
[CrossRef]

Hodapp, K.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Holmes, T. J.

Hu, X.

C. T. Chen, C. Metz, X. Hu, “Maximum likelihood reconstruction in PET and TOFPET,” in Mathematics and Computer Science in Medical Imaging, M. A. Viergever, A. E. Todd-Pokropek, eds. (Springer-Verlag, Berlin, 1988), pp. 319–329.
[CrossRef]

Kak, A. C.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Kaufman, L.

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imag. MI-6, 37–51 (1987).
[CrossRef]

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80(3), 8–19 (1985).
[CrossRef]

Kreyszig, E.

E. Kreyszig, Advanced Engineering Mathematics (Wiley, New York, 1972).

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–37 (1977).

Lange, K.

K. Lange, M. Bahn, R. Little, “A theoretical study of some maximum likelihood algorithms for emission and transmission tomography,” IEEE Trans. Med. Imag. MI-6, 106–114 (1987).
[CrossRef]

K. Lange, R. Carson, “EM reconstruction algorithms for emission and transmission tomography,” J. Comput. Assisted Tomography 8, 306–316 (1984).

Leahy, R.

T. Hebert, R. Leahy, “A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors,”IEEE Trans. Med. Imag. 8, 194–202 (1989).
[CrossRef]

T. Hebert, R. Leahy, M. Singh, “Fast MLE for SPECT using an intermediate polar representation and a stopping criterion,”IEEE Trans. Nucl. Sci. 35, 615–619 (1988).
[CrossRef]

Lewitt, R. M.

R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
[CrossRef]

Liang, Z.

Z. Liang, “Statistical models of a prioriinformation for image processing: neighboring correlation constraints,” J. Opt. Soc. Am. A 5, 2026–2031 (1988).
[CrossRef]

Z. Liang, “Statistical models of a prioriinformation for image processing,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.914, 677–683 (1988).
[CrossRef]

Little, R.

K. Lange, M. Bahn, R. Little, “A theoretical study of some maximum likelihood algorithms for emission and transmission tomography,” IEEE Trans. Med. Imag. MI-6, 106–114 (1987).
[CrossRef]

Liu, Y. H.

T. J. Holmes, Y. H. Liu, “Richardson–Lucy/maximum-likelihood image restoration algorithm for fluorescence microscopy: further testing,” Appl. Opt. 28, 4930–4938 (1989).
[CrossRef] [PubMed]

T. J. Holmes, Y. H. Liu, “Simulation tests of restoring truncated fluorescence micrographs,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 197–204 (1989).
[CrossRef]

Llacer, J.

E. Veklerov, J. Llacer, “MLE reconstruction of a brain phantom using a Monte Carlo transition matrix and a statistical stopping rule,”IEEE Trans. Nucl. Sci. 35, 603–607 (1988).
[CrossRef]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–765 (1974).
[CrossRef]

Luenberger, D. G.

D. G. Luenberger, Linear and Nonlinear Programming (Addison-Wesley, Reading, Mass., 1984).

Metz, C.

C. T. Chen, C. Metz, X. Hu, “Maximum likelihood reconstruction in PET and TOFPET,” in Mathematics and Computer Science in Medical Imaging, M. A. Viergever, A. E. Todd-Pokropek, eds. (Springer-Verlag, Berlin, 1988), pp. 319–329.
[CrossRef]

Metz, C. E.

C. E. Metz, C. T. Chen, “On the acceleration of maximum likelihood algorithms,” in Medical Imaging II, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.914, 344–349 (1988).
[CrossRef]

Miller, M. I.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

M. I. Miller, D. L. Snyder, “The role of likelihood and entropy in incomplete-data problems: applications to estimating point-process intensities and Toeplitz constrained covariances,” Proc. IEEE 75, 892–907 (1987).
[CrossRef]

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 768–778 (1985).

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

B. Roysam, J. A. Shrauner, M. I. Miller, “Bayesian imaging using Good’s roughness measure—implementation on a massively parallel processor,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), Vol. II, pp. 932–935.
[CrossRef]

Miller, T. R.

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 768–778 (1985).

Muehllehner, G.

R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
[CrossRef]

Murray, W.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Politte, D. G.

D. G. Politte, D. L. Snyder, “The use of constraints to eliminate artifacts in maximum-likelihood image estimation for emission tomography,”IEEE Trans. Nucl. Sci. 35, 608–610 (1988).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

Richardson, W. H.

Roysam, B.

B. Roysam, J. A. Shrauner, M. I. Miller, “Bayesian imaging using Good’s roughness measure—implementation on a massively parallel processor,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), Vol. II, pp. 932–935.
[CrossRef]

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,”J. R. Stat. Soc. Ser. B 39, 1–37 (1977).

Salas, L.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975).

Shepp, L. A.

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80(3), 8–19 (1985).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–122 (1982).
[CrossRef]

Shrauner, J. A.

B. Roysam, J. A. Shrauner, M. I. Miller, “Bayesian imaging using Good’s roughness measure—implementation on a massively parallel processor,” in Proceedings of the International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), Vol. II, pp. 932–935.
[CrossRef]

Singh, M.

T. Hebert, R. Leahy, M. Singh, “Fast MLE for SPECT using an intermediate polar representation and a stopping criterion,”IEEE Trans. Nucl. Sci. 35, 615–619 (1988).
[CrossRef]

Slaney, M.

A. C. Kak, M. Slaney, Principles of Computerized Tomographic Imaging (Institute of Electrical and Electronics Engineers, New York, 1988).

Snyder, D.

D. Snyder, D. G. Politte, “Image reconstruction from list-mode data in an emission tomography system having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-30, 1843–1849 (1983).
[CrossRef]

Snyder, D. L.

D. G. Politte, D. L. Snyder, “The use of constraints to eliminate artifacts in maximum-likelihood image estimation for emission tomography,”IEEE Trans. Nucl. Sci. 35, 608–610 (1988).
[CrossRef]

M. I. Miller, D. L. Snyder, “The role of likelihood and entropy in incomplete-data problems: applications to estimating point-process intensities and Toeplitz constrained covariances,” Proc. IEEE 75, 892–907 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

M. I. Miller, D. L. Snyder, T. R. Miller, “Maximum-likelihood reconstruction for single-photon emission computed tomography,” IEEE Trans. Nucl. Sci. NS-32, 768–778 (1985).

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

Striebl, N.

N. Striebl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
[CrossRef]

Strom, S. E.

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

Ter-Pogossian, M. M.

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

Thomas, L. J.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

D. L. Snyder, L. J. Thomas, M. M. Ter-Pogossian, “A mathematical model for positron-emission tomography systems having time-of-flight measurements,” IEEE Trans. Nucl. Sci. NS-28, 3575–3583 (1981).
[CrossRef]

Vardi, Y.

Y. Vardi, L. A. Shepp, L. Kaufman, “A statistical model for positron emission tomography,”J. Am. Stat. Assoc. 80(3), 8–19 (1985).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–122 (1982).
[CrossRef]

Veklerov, E.

E. Veklerov, J. Llacer, “MLE reconstruction of a brain phantom using a Monte Carlo transition matrix and a statistical stopping rule,”IEEE Trans. Nucl. Sci. 35, 603–607 (1988).
[CrossRef]

Wright, M. H.

P. E. Gill, W. Murray, M. H. Wright, Practical Optimization (Academic, New York, 1981).

Appl. Opt. (1)

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–765 (1974).
[CrossRef]

Astrophys. J. (1)

K. Hodapp, R. W. Capps, S. E. Strom, L. Salas, G. L. Grasdalen, “Near-infrared imaging of Lynds 1551 IRS 5,” Astrophys. J. 335, 814–819 (1988).
[CrossRef]

IEEE Trans. Med. Imag. (6)

T. Hebert, R. Leahy, “A generalized EM algorithm for 3D Bayesian reconstruction from Poisson data using Gibbs priors,”IEEE Trans. Med. Imag. 8, 194–202 (1989).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imag. MI-6, 228–238 (1987).
[CrossRef]

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,” IEEE Trans. Med. Imag. MI-1, 113–122 (1982).
[CrossRef]

R. M. Lewitt, G. Muehllehner, “Accelerated iterative reconstruction for positron emission tomography based on the EM algorithm for maximum likelihood estimation,” IEEE Trans. Med. Imag. MI-5, 16–22 (1986).
[CrossRef]

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Figures (11)

Fig. 1
Fig. 1

Flowchart of the EM-based algorithm without acceleration.

Fig. 2
Fig. 2

Superresolution simulation with a binary object: (a) original object to be restored, (b) diffraction-limited, quantum-limited image with an average of 160 photons per pixel over the 64 × 64 image, (c) restored image after 50,000 iterations without acceleration, (d) truncated version of (b).

Fig. 3
Fig. 3

(a) Image restored from Fig. 2(d) after 1000 iterations without acceleration. (b) Image restored from Fig. 2(d) after 1000 iterations with acceleration.

Fig. 4
Fig. 4

Flowchart of the accelerative version of the algorithm.

Fig. 5
Fig. 5

Images obtained in the superresolution simulation with a nonbinary object: (a) the original object to be restored (the dimensions of the object are described in Subsection 3.B), (b) diffraction-limited image with an average of 4000 photons per pixel, (c) restored image without acceleration after 10,000 iterations and with regularization having σr = 0.04 μ m, (d) restored image with acceleration after 10,000 iterations and with regularization having σr = 0.04 μ m, (e) restored image without acceleration and without regularization after 10,000 iterations, (f) restored image with acceleration and without regularization after 10,000 iterations.

Fig. 6
Fig. 6

Images obtained in the optical-sectioning simulation: (a) The original object to be restored. The entire object by simulation is 84 pixels wide by 100 pixels high, with sampling of 0.1 μm per pixel. (b) Image degraded with an average of 64,000 photons per pixel to simulate the optical-sectioning problem with truncated data. The observation window W used for the truncation is 108 pixels wide by 124 pixels high. (c), (e), (g) Restored images without acceleration after 100, 1000, and 10,000 iterations, respectively. (d), (f), (h) Restored images with acceleration after 100, 1000, and 10,000 iterations, respectively.

Fig. 7
Fig. 7

Log-likelihood plot for the superresolution simulation of Subsection 3.A.

Fig. 8
Fig. 8

Log-likelihood plot for the superresolution simulation of Subsection 3.B.

Fig. 9
Fig. 9

Log-likelihood plot for the optical-sectioning simulation of Subsection 3.C.

Fig. 10
Fig. 10

Log-likelihood plots for four different cases of arithmetic precision. See the text in Appendix B for an explanation of these different cases.

Fig. 11
Fig. 11

Log-likelihood plots for different cases of arithmetic precision with finer sampling of the iterations between 1000 and 10,000.

Tables (8)

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Table 1 Log-Likelihood Calculations for the Superresolution Simulation of Subsection 3.A

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Table 2 Log-Likelihood Calculations for the Superresolution Simulation of Subsection 3.B

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Table 3 Log-Likelihood Calculations for the Optical-Sectioning Simulation of Subsection 3.C

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Table 4 Execution Times of Individual Steps in Fig. 4

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Table 5 Average α, Maximum α, and Average log2α Tabulated in the Superresolution Simulation of Subsection 3.B

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Table 6 Average α, Maximum α, and Average log2α Tabulated in the Optical-Sectioning Simulation of Subsection 3.C

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Table 7 Log-Likelihood Calculations for the Superresolution Simulation of Subsection 3.B and for the Cases Described in Appendix B from Iterations of 1–10,000

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Table 8 Log-Likelihood Calculations for the Superresolution Simulation of Subsection 3.B and for the Cases Described in Appendix B from Iterations of 1000–10,000

Equations (36)

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L ( λ ) = j [ μ ( u j ) ] N d j exp [ - μ ( u j ) ] / N d j ! ,
μ ( u j ) = i p ( u j - x i ) λ ( x i ) Δ x Δ u
λ = [ λ ( x 1 ) , λ ( x 2 ) , ] T ,
μ = [ μ ( u 1 ) , μ ( u 2 ) , ] T .
L ( λ ) = j u j W [ μ ( u j ) ] N d j exp [ - μ ( u j ) ] / N d j ! .
l ( λ ) = ln [ L ( λ ) ] .
Δ λ ^ = λ ^ - λ ^
μ ^ ( u j ) = i p ( u j - x i ) λ ^ ( x i ) Δ x Δ u ,
μ ^ ( u j ) = i p ( u j - x i ) λ ^ ( x i ) Δ x Δ u ,
Δ μ ^ ( u j ) = i p ( u j - x i ) Δ λ ^ ( x i ) Δ x Δ u ,
μ ^ = ( μ 1 , μ 2 , ) T ,
μ ^ = ( μ 1 , μ 2 , ) T ,
Δ μ ^ = ( Δ μ 1 , Δ μ 2 , ) T .
Δ l = l ( λ ^ ) - l ( λ ) = j u j W [ N d j ln μ ^ ( u j ) μ ^ ( u j ) - μ ^ ( u j ) + μ ^ ( u j ) ] ,
Δ l = l ( λ ^ ) - l ( λ ) = j u j W [ N d j ln μ ^ ( u j ) + Δ μ ^ ( u j ) μ ^ ( u j ) - Δ μ ^ ( u j ) ] .
α > 1 ,
λ ^ a = λ ^ + α Δ λ ^ ,
λ ^ a = λ ^ + ( α - 1 ) Δ λ ^ ,
λ ^ a = [ λ ^ a ( x 1 ) , λ ^ a ( x 2 ) , ] T ,
Δ l a = l [ λ ^ a ] - l [ λ ^ ] ,
Δ l a = j u j W { N d j ln μ ^ ( u j ) + α Δ μ ^ ( u j ) μ ^ ( u j ) - α Δ μ ^ ( u j ) } ,
Δ l a = j u j W { N d j ln μ ^ ( u j ) + ( α - 1 ) Δ μ ^ ( u j ) μ ^ ( u j ) - α Δ μ ^ ( u j ) } .
α ^ = min all x i [ α i = - λ ( x i ) / Δ λ ( x i ) ] ,             Δ λ ( x i ) < 0 ,
α = α ^ .
N fft = M fft log 2 ( M fft ) ,
log 2 α + 2 ,
log 2 α ¯ ,
T = N its ( T fft + T r + T α ) ,
T α = ( log 2 α ¯ + 2 ) T 10 ,
T fft = k fft M fft log 2 ( M fft ) ,
T r = k r M fft ,
T α = k α M w ( log 2 α ¯ + 2 ) ,
d ( x ) = Y r ( x - y ) λ ( y ) d y ,
S d ^ = { d ^ : d ^ ( x ) = Y s ( x - y ) ξ ( y ) d y } ,
S ξ = { ξ ( y ) : Y ξ ( y ) = E [ N ] } ,
σ r = σ s .

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