Abstract

Previous research presented by the author and others into maximum-likelihood image restoration for incoherent imagery is extended to consider problems of blind deconvolution in which the impulse response of the system is assumed to be unknown. Potential applications that motivate this study are wide-field and confocal fluorescence microscopy, although applications in astronomy and infrared imaging are foreseen as well. The methodology incorporates the iterative expectation-maximization algorithm. Although the precise impulse response is assumed to be unknown, some prior knowledge about characteristics of the impulse response is used. In preliminary simulation studies that are presented, the circular symmetry and the band-limited nature of the impulse response are used as such. These simulations demonstrate the potential utility and present limitations of these methods.

© 1992 Optical Society of America

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Corrections

Timothy J. Holmes, "Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach: errata," J. Opt. Soc. Am. A 9, 2097-2097 (1992)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-9-11-2097

References

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    [CrossRef]
  3. 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]
  4. 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]
  5. 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]
  6. 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]
  7. D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imag. 10, 82–89 (1991).
    [CrossRef]
  8. 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).
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    [CrossRef]
  12. E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
    [CrossRef]
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  14. M. I. Miller, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (1985).
    [CrossRef]
  15. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  16. 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).
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    [CrossRef] [PubMed]
  21. T. J. Holmes, Y. H. Liu, D. Khosla, D. A. Agard, “Increased depth-of-field and stereo pairs of fluorescence micrographs via inverse filtering and maximum likelihood estimation,” J. Microsc. 164, 217–237 (1991).
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    [CrossRef]
  25. G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef] [PubMed]
  26. B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
    [CrossRef]
  27. B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
    [CrossRef]
  28. J. H. Seldin, J. R. Fienup, “Iterative blind deconvolution algorithm applied to phase retrieval,” J. Opt. Soc. Am. A 7, 428–433 (1990).
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  29. R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
    [CrossRef]
  30. R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
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  31. A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
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  32. T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1968).
    [CrossRef]
  33. R. W. Gerchberg, W. O. Saxton, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
    [CrossRef]
  34. A. M. Tekalp, H. Kaufman, “On statistical identification of a class of linear space-invariant blurs using nonminimum-phase ARMA models,”IEEE Trans. Acoust. Speech Signal Process. 36, 1360–1363 (1988).
    [CrossRef]
  35. R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process. 38, 1180–1191 (1990).
    [CrossRef]
  36. D. A. Agard, “Optical sectioning microscopy,” Ann. Rev. Bioeng. 13, 191–219 (1984).
    [CrossRef]
  37. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
    [CrossRef] [PubMed]
  38. M. Koshy, D. A. Agard, J. W. Sedat, “Solution of Toeplitz systems for the restoration of 3-D optical-sectioning microscopy data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1205, 64–71 (1990).
    [CrossRef]
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  42. T. Hebert, R. Leahy, “A generalized EM algorithm for Bayesian reconstruction from Poisson data using Gibbs priors,”IEEE Trans. Med. Imag. 8, 194–202 (1989).
    [CrossRef]
  43. P. J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,”IEEE Trans. Med. Imag. 9, 84–93 (1990).
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  44. S. A. Sugimoto, Y. Ichioka, “Digital composition of images with increased depth of focus considering depth information,” Appl. Opt. 24, 2076–2080 (1985).
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  45. K. Itoh, A. Hayashi, Y. Ichioka, “Digitized optical microscopy with extended depth of field,” Appl. Opt. 28, 3487–3493 (1989).
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  46. G. J. Brackenhoff, K. Visscher, H. T. M. Van Der Voort, “Size and shape of the confocal spot: control and relation to 3-D imaging and image processing,” in The Handbook of Biological Confocal Microscopy, J. Pawley, ed. (IMR, Madison, Wisc., 1989), pp. 79–82.
  47. Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system: partial confocal behavior in epifluorescence microscopy,” Biophys. J. 57, 325–333 (1990).
    [CrossRef] [PubMed]
  48. Y. Hiraoka, J. W. Sedat, D. A. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
    [CrossRef] [PubMed]
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    [CrossRef]
  51. 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]
  52. D. L. Snyder, Random Point Processes (Wiley, New York, 1975).
  53. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  54. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J.1979).
  55. N. Striebl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  56. G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).
  57. D. L. Snyder, M. M. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1990).
  58. C. R. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
    [CrossRef]
  59. B. Roysam, J. A. Shrauner, M. I. Miller, “Bayesian imaging using Good’s roughness measure implementation on a massively parallel processor,” in 1988 International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), Vol. II: Multidimensional Signal Processing, pp. 932–935.
  60. R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid state imagers for microscopy,” Methods Cell Biol. 29, 292–313 (1989).

1991 (3)

D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imag. 10, 82–89 (1991).
[CrossRef]

T. J. Holmes, Y. H. Liu, D. Khosla, D. A. Agard, “Increased depth-of-field and stereo pairs of fluorescence micrographs via inverse filtering and maximum likelihood estimation,” J. Microsc. 164, 217–237 (1991).
[CrossRef]

T. J. Holmes, Y. H. Liu, “Acceleration of maximum-likelihood image restoration for fluorescence microscopy and other noncoherent imagery,” J. Opt. Soc. Am. A 8, 893–907 (1991).
[CrossRef]

1990 (8)

J. H. Seldin, J. R. Fienup, “Iterative blind deconvolution algorithm applied to phase retrieval,” J. Opt. Soc. Am. A 7, 428–433 (1990).
[CrossRef]

R. H. T. Bates, B. K. Quek, C. R. Parker, “Some implications of zero sheets for blind deconvolution and phase retrieval,” J. Opt. Soc. Am. A 7, 468–479 (1990).
[CrossRef]

D. L. Snyder, T. J. Schulz, “High-resolution imaging at low-light levels through weak turbulence,” J. Opt. Soc. Am. A 7, 1251–1265 (1990).
[CrossRef]

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process. 38, 1180–1191 (1990).
[CrossRef]

P. J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,”IEEE Trans. Med. Imag. 9, 84–93 (1990).
[CrossRef]

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system: partial confocal behavior in epifluorescence microscopy,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
[CrossRef]

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

1989 (7)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid state imagers for microscopy,” Methods Cell Biol. 29, 292–313 (1989).

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

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

K. Itoh, A. Hayashi, Y. Ichioka, “Digitized optical microscopy with extended depth of field,” Appl. Opt. 28, 3487–3493 (1989).
[CrossRef] [PubMed]

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

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]

1988 (6)

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

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

G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
[CrossRef] [PubMed]

A. M. Tekalp, H. Kaufman, “On statistical identification of a class of linear space-invariant blurs using nonminimum-phase ARMA models,”IEEE Trans. Acoust. Speech Signal Process. 36, 1360–1363 (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]

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]

1987 (6)

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]

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]

Y. Hiraoka, J. W. Sedat, D. A. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[CrossRef] [PubMed]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (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 (4)

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, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (1985).
[CrossRef]

N. Striebl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
[CrossRef]

S. A. Sugimoto, Y. Ichioka, “Digital composition of images with increased depth of focus considering depth information,” Appl. Opt. 24, 2076–2080 (1985).
[CrossRef] [PubMed]

1984 (2)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

D. A. Agard, “Optical sectioning microscopy,” Ann. Rev. Bioeng. 13, 191–219 (1984).
[CrossRef]

1983 (2)

C. R. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
[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]

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. B 39, 1–37 (1977).

1974 (2)

R. W. Gerchberg, W. O. Saxton, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

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

1972 (1)

1968 (2)

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1968).
[CrossRef]

Agard, D. A.

T. J. Holmes, Y. H. Liu, D. Khosla, D. A. Agard, “Increased depth-of-field and stereo pairs of fluorescence micrographs via inverse filtering and maximum likelihood estimation,” J. Microsc. 164, 217–237 (1991).
[CrossRef]

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system: partial confocal behavior in epifluorescence microscopy,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid state imagers for microscopy,” Methods Cell Biol. 29, 292–313 (1989).

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Y. Hiraoka, J. W. Sedat, D. A. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[CrossRef] [PubMed]

D. A. Agard, “Optical sectioning microscopy,” Ann. Rev. Bioeng. 13, 191–219 (1984).
[CrossRef]

M. Koshy, D. A. Agard, J. W. Sedat, “Solution of Toeplitz systems for the restoration of 3-D optical-sectioning microscopy data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1205, 64–71 (1990).
[CrossRef]

Aikens, R. S.

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid state imagers for microscopy,” Methods Cell Biol. 29, 292–313 (1989).

Arfken, G.

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

Ayers, G. R.

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]

Bates, R. H. T.

Biemond, J.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process. 38, 1180–1191 (1990).
[CrossRef]

Boekee, D. E.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process. 38, 1180–1191 (1990).
[CrossRef]

Brackenhoff, G. J.

G. J. Brackenhoff, K. Visscher, H. T. M. Van Der Voort, “Size and shape of the confocal spot: control and relation to 3-D imaging and image processing,” in The Handbook of Biological Confocal Microscopy, J. Pawley, ed. (IMR, Madison, Wisc., 1989), pp. 79–82.

Cannon, T. M.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1968).
[CrossRef]

Carrington, W.

W. Carrington, “Image restoration in 3-D microscopy with limited data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1205, 72–83 (1990).
[CrossRef]

Castleman, K. R.

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J.1979).

Chen, C. J.

E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
[CrossRef]

Chen, C. T.

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

Chernoboy, E. S.

E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
[CrossRef]

Conchello, J.

J. Conchello, E. W. Hansen, “Three-dimensional reconstruction of noisy confocal scanning microscope images,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 279–285 (1989).
[CrossRef]

J. Conchello, “Three-dimensional reconstruction of noisy images from partially confocal scanning microscope,” Ph.D. dissertation (Dartmouth College, Hanover, N.H., 1990).

Dainty, J. C.

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[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. B 39, 1–37 (1977).

Fienup, J. R.

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Geman, D.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

Gerchberg, R. W.

R. W. Gerchberg, W. O. Saxton, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Green, P. J.

P. J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,”IEEE Trans. Med. Imag. 9, 84–93 (1990).
[CrossRef]

Hansen, E. W.

J. Conchello, E. W. Hansen, “Three-dimensional reconstruction of noisy confocal scanning microscope images,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 279–285 (1989).
[CrossRef]

Hayashi, A.

Hebert, T.

T. Hebert, R. Leahy, “A generalized EM algorithm for 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]

Hiraoka, Y.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system: partial confocal behavior in epifluorescence microscopy,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Y. Hiraoka, J. W. Sedat, D. A. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[CrossRef] [PubMed]

Holmes, T. J.

Ichioka, Y.

Ingebretsen, R. B.

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1968).
[CrossRef]

Itoh, K.

Kaufman, H.

A. M. Tekalp, H. Kaufman, “On statistical identification of a class of linear space-invariant blurs using nonminimum-phase ARMA models,”IEEE Trans. Acoust. Speech Signal Process. 36, 1360–1363 (1988).
[CrossRef]

Kaufman, L.

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

Khosla, D.

T. J. Holmes, Y. H. Liu, D. Khosla, D. A. Agard, “Increased depth-of-field and stereo pairs of fluorescence micrographs via inverse filtering and maximum likelihood estimation,” J. Microsc. 164, 217–237 (1991).
[CrossRef]

Koshy, M.

M. Koshy, D. A. Agard, J. W. Sedat, “Solution of Toeplitz systems for the restoration of 3-D optical-sectioning microscopy data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1205, 64–71 (1990).
[CrossRef]

Lagendijk, R. L.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process. 38, 1180–1191 (1990).
[CrossRef]

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. B 39, 1–37 (1977).

Lane, R. G.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

R. G. Lane, R. H. T. Bates, “Automatic multidimensional deconvolution,” J. Opt. Soc. Am. A 4, 180–188 (1987).
[CrossRef]

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]

Larson, K. B.

M. I. Miller, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (1985).
[CrossRef]

Leahy, R.

T. Hebert, R. Leahy, “A generalized EM algorithm for 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, L.

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.

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]

Mandel, L.

L. Mandel, “Fluctuations of light beams,” in Progress in Optics, E. Wolf, ed. (Wiley, New York, 1963), Vol. II, pp. 181–248.
[CrossRef]

McCallum, B. C.

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

Metz, C. E.

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

Miller, M. I.

E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
[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]

M. I. Miller, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (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]

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

Miller, M. M.

D. L. Snyder, M. M. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1990).

Miller, T. R.

E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
[CrossRef]

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]

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Parker, C. R.

Politte, D. G.

D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imag. 10, 82–89 (1991).
[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]

Quek, B. K.

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 1988 International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), Vol. II: Multidimensional Signal Processing, pp. 932–935.

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. B 39, 1–37 (1977).

Saffitz, J. E.

M. I. Miller, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (1985).
[CrossRef]

Saxton, W. O.

R. W. Gerchberg, W. O. Saxton, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Schafer, R. W.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

Schulz, T. J.

Sedat, J. W.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system: partial confocal behavior in epifluorescence microscopy,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid state imagers for microscopy,” Methods Cell Biol. 29, 292–313 (1989).

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Y. Hiraoka, J. W. Sedat, D. A. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[CrossRef] [PubMed]

M. Koshy, D. A. Agard, J. W. Sedat, “Solution of Toeplitz systems for the restoration of 3-D optical-sectioning microscopy data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1205, 64–71 (1990).
[CrossRef]

Seldin, J. H.

Shaw, P.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

Shepp, L. A.

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 1988 International Conference on Acoustics, Speech, and Signal Processing (Institute of Electrical and Electronics Engineers, New York, 1988), Vol. II: Multidimensional Signal Processing, pp. 932–935.

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]

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, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imag. 10, 82–89 (1991).
[CrossRef]

D. L. Snyder, T. J. Schulz, “High-resolution imaging at low-light levels through weak turbulence,” J. Opt. Soc. Am. A 7, 1251–1265 (1990).
[CrossRef]

E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
[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]

M. I. Miller, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (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]

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]

D. L. Snyder, M. M. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1990).

D. L. Snyder, Random Point Processes (Wiley, New York, 1975).

Stockham, T. G.

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1968).
[CrossRef]

Striebl, N.

Sugimoto, S. A.

Tekalp, A. M.

A. M. Tekalp, H. Kaufman, “On statistical identification of a class of linear space-invariant blurs using nonminimum-phase ARMA models,”IEEE Trans. Acoust. Speech Signal Process. 36, 1360–1363 (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]

M. I. Miller, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (1985).
[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]

Van Der Voort, H. T. M.

G. J. Brackenhoff, K. Visscher, H. T. M. Van Der Voort, “Size and shape of the confocal spot: control and relation to 3-D imaging and image processing,” in The Handbook of Biological Confocal Microscopy, J. Pawley, ed. (IMR, Madison, Wisc., 1989), pp. 79–82.

Vardi, Y.

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]

Visscher, K.

G. J. Brackenhoff, K. Visscher, H. T. M. Van Der Voort, “Size and shape of the confocal spot: control and relation to 3-D imaging and image processing,” in The Handbook of Biological Confocal Microscopy, J. Pawley, ed. (IMR, Madison, Wisc., 1989), pp. 79–82.

Wu, C. R. J.

C. R. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
[CrossRef]

Ann. Rev. Bioeng. (1)

D. A. Agard, “Optical sectioning microscopy,” Ann. Rev. Bioeng. 13, 191–219 (1984).
[CrossRef]

Ann. Stat. (1)

C. R. J. Wu, “On the convergence properties of the EM algorithm,” Ann. Stat. 11, 95–103 (1983).
[CrossRef]

Appl. Opt. (3)

Astron. J. (1)

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

Biophys. J. (1)

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system: partial confocal behavior in epifluorescence microscopy,” Biophys. J. 57, 325–333 (1990).
[CrossRef] [PubMed]

IEEE Trans. Acoust. Speech Signal Process. (2)

A. M. Tekalp, H. Kaufman, “On statistical identification of a class of linear space-invariant blurs using nonminimum-phase ARMA models,”IEEE Trans. Acoust. Speech Signal Process. 36, 1360–1363 (1988).
[CrossRef]

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identification and restoration of noisy blurred images using the expectation-maximization algorithm,”IEEE Trans. Acoust. Speech Signal Process. 38, 1180–1191 (1990).
[CrossRef]

IEEE Trans. Med. Imag. (9)

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

P. J. Green, “Bayesian reconstructions from emission tomography data using a modified EM algorithm,”IEEE Trans. Med. Imag. 9, 84–93 (1990).
[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]

E. S. Chernoboy, C. J. Chen, M. I. Miller, T. R. Miller, D. L. Snyder, “An evaluation of maximum likelihood reconstruction for SPECT,”IEEE Trans. Med. Imag. 9, 99–110 (1990).
[CrossRef]

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

D. G. Politte, D. L. Snyder, “Corrections for accidental coincidences and attenuation in maximum-likelihood image reconstruction for positron-emission tomography,”IEEE Trans. Med. Imag. 10, 82–89 (1991).
[CrossRef]

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

IEEE Trans. Nucl. Sci. (5)

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]

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]

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]

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]

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]

IEEE Trans. Pattern Anal. Mach. Intell. (1)

S. Geman, D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,”IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6, 721–741 (1984).
[CrossRef]

J. Electron Microsc. Tech. (1)

M. I. Miller, K. B. Larson, J. E. Saffitz, D. L. Snyder, L. J. Thomas, “Maximum-likelihood estimation applied to electron microscopic autoradiography,”J. Electron Microsc. Tech. 2, 611–636 (1985).
[CrossRef]

J. Microsc. (1)

T. J. Holmes, Y. H. Liu, D. Khosla, D. A. Agard, “Increased depth-of-field and stereo pairs of fluorescence micrographs via inverse filtering and maximum likelihood estimation,” J. Microsc. 164, 217–237 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (9)

J. R. Stat. Soc. B (1)

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

Methods Cell Biol. (2)

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
[CrossRef] [PubMed]

R. S. Aikens, D. A. Agard, J. W. Sedat, “Solid state imagers for microscopy,” Methods Cell Biol. 29, 292–313 (1989).

Opt. Acta (1)

R. W. Gerchberg, W. O. Saxton, “Super-resolution through error energy reduction,” Opt. Acta 21, 709–720 (1974).
[CrossRef]

Opt. Commun. (2)

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (3)

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]

A. V. Oppenheim, R. W. Schafer, T. G. Stockham, “Nonlinear filtering of multiplied and convolved signals,” Proc. IEEE 56, 1264–1291 (1968).
[CrossRef]

T. G. Stockham, T. M. Cannon, R. B. Ingebretsen, “Blind deconvolution through digital signal processing,” Proc. IEEE 63, 678–692 (1968).
[CrossRef]

Science (1)

Y. Hiraoka, J. W. Sedat, D. A. Agard, “The use of a charge-coupled device for quantitative optical microscopy of biological structures,” Science 238, 36–41 (1987).
[CrossRef] [PubMed]

Other (13)

L. Mandel, “Fluctuations of light beams,” in Progress in Optics, E. Wolf, ed. (Wiley, New York, 1963), Vol. II, pp. 181–248.
[CrossRef]

G. J. Brackenhoff, K. Visscher, H. T. M. Van Der Voort, “Size and shape of the confocal spot: control and relation to 3-D imaging and image processing,” in The Handbook of Biological Confocal Microscopy, J. Pawley, ed. (IMR, Madison, Wisc., 1989), pp. 79–82.

D. L. Snyder, Random Point Processes (Wiley, New York, 1975).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J.1979).

G. Arfken, Mathematical Methods for Physicists (Academic, New York, 1985).

D. L. Snyder, M. M. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1990).

M. Koshy, D. A. Agard, J. W. Sedat, “Solution of Toeplitz systems for the restoration of 3-D optical-sectioning microscopy data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1205, 64–71 (1990).
[CrossRef]

W. Carrington, “Image restoration in 3-D microscopy with limited data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1205, 72–83 (1990).
[CrossRef]

J. Conchello, E. W. Hansen, “Three-dimensional reconstruction of noisy confocal scanning microscope images,” in New Methods in Microscopy and Low Light Imaging, J. E. Wampler, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1161, 279–285 (1989).
[CrossRef]

J. Conchello, “Three-dimensional reconstruction of noisy images from partially confocal scanning microscope,” Ph.D. dissertation (Dartmouth College, Hanover, N.H., 1990).

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

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

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Figures (11)

Fig. 1
Fig. 1

Flowchart of the algorithm with a symmetry constraint.

Fig. 2
Fig. 2

Simulation 1: (a) Original object centered in a 64 × 64 array; minimum (min) = 0, maximum (max) = 221,405, average (avg) = 4000. (b) Degraded image with 4000 photons per pixel; min = 0, max = 98.2, avg = 3999.16. (c) Reconstruction at 50 iterations; min = 0, max = 339,516, avg = 3999.17. (d) Reconstruction at 150 iterations; min = 0, max = 902,675, avg = 3999.16.

Fig. 3
Fig. 3

Simulation 1: (a) Impulse response, chosen as an approximate circle function; min = 0, max = 1.047 × 10−2, avg = 2.441 × 10−4. (b) Reconstructed impulse response at 50 iterations; min = 0, max = 1.177 × 10−2, avg = 2.442 × 10−4.

Fig. 4
Fig. 4

Simulation 1: (a) Real part of the Fourier transform of the impulse response; min = −0.1298, max = 1.0, avg = 1.047 × 10−2. (b) Real part of the Fourier transform of Fig. 3(b); min = −0.0943, max = 1.0, avg = 7.713 × 10−3.

Fig. 5
Fig. 5

Simulation 2: (a) Degraded image with 4000 photons per pixel. The true object is shown in Fig. 2(a); min = 0, max = 100,841, avg = 4001.51. (b) Reconstruction at 50 iterations; min = 0, max = 337,164, avg = 4001.49.

Fig. 6
Fig. 6

Simulation 2: (a) Impulse response, chosen as an approximate annular function; min = 0, max = 1.212 × 10−2, avg = 2.441 × 10−4. (b) Reconstructed impulse response at 50 iterations; min = 0, max = 1.211 × 10−2, avg = 2.448 × 10−4.

Fig. 7
Fig. 7

Simulation 2: (a) Real part of the Fourier transform of the impulse response; min = −0.244, max = 1.0, avg = −1.12 × 10−9. (b) Real part of the Fourier transform of Fig. 6(b); min = −0.159, max = 1.0, avg = 1.10 × 10−3.

Fig. 8
Fig. 8

Simulation 3: (a) Degraded image with 4000 photons per pixel. The true object is shown in Fig. 2(a); min = 0, max = 255, avg = 3999.36. (b) Reconstruction at 500 iterations; min = 0, max = 260,590, avg = 3995.12.

Fig. 9
Fig. 9

Simulation 3: (a) Impulse response, chosen to be an Airy diffraction pattern; min = 0, max = 5.748 × 10−2, avg = 2.441 × 10−4. (b) Reconstructed impulse response at 500 iterations; min = 0, max = 6.372 × 10−2, avg = 2.444 × 10−4.

Fig. 10
Fig. 10

Simulation 3: (a) Real part of the Fourier transform of the impulse response; min = −3.725 × 10−7, max = 1.0, avg = 5.748 × 10−2. (b) Real part of the Fourier transform of Fig. 9(b); min = −1.539 × 10−2, max = 1.001, avg = 6.372 × 10−2.

Fig. 11
Fig. 11

Simulation 3: (a) Original object; min = 0, max = 5535.125, avg = 100. (b) Impulse response (Airy diffraction pattern) [same as Fig. 9(a)]. (c) Degraded image with 100 photons per pixel; min = 0, max = 4112, avg = 99.762. (d) Blind deconvolution reconstruction of the object at 50 iterations; min = 0, max = 6916.26, avg = 95.214. (e) Reconstruction of the impulse response at 50 iterations; min = 0, max = 6.816 × 10−2, avg = 2.455 × 10−4. (f) Blind deconvolution reconstruction of the object at 500 iterations; min = 0, max = 12,012.8, avg = 95.202. (g) Reconstruction of the impulse response at 500 iterations; min = 0, max = 5.805 × 10−2, avg = 2.46 × 10−4. (h) Nonblind reconstruction, with the impulse response assumed to be known, following 50 iterations; min = 0, max = 9000.79, avg = 9244. (i) Nonblind reconstruction, with the impulse response assumed to be known, following 500 iterations; min = 0, max = 105,837, avg = 95.244.

Tables (1)

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Table 1 Log-Likelihood Values Calculated from Simulation 3 with Λ = 16,384,000

Equations (31)

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Λ = R 2 λ ( x ) d x .
f X ( x ) = λ ( x ) / Λ ,
γ ( b ) = Λ h ( b ) .
U i = X i + B i ,
f U ( u ) = f X ( u ) * h ( u ) ,
μ ( u ) = λ ( u ) * h ( u ) .
[ u 1 , u 1 + d u 1 ) [ u 2 , u 2 + d u 2 ) ,
u = ( u 1 , u 2 ) .
U = { U 1 , U 2 , U 3 , } .
X = { X 1 , X 2 , X 3 , } ,
B = { B 1 , B 2 , B 3 , } .
l 1 ( X ; λ ) = - R 2 λ ( x ) d x + R 2 ln [ λ ( x ) ] N t ( d x ) ,
[ x 1 , x 1 + d x 1 ) [ x 2 , x 2 + d x 2 ) ,
x = ( x 1 , x 2 ) .
l 2 ( B ; h , Λ ) = - R 2 γ ( b ) d b + R 2 ln [ γ ( b ) ] N e ( d b ) ,
[ b 1 , b 1 + d b 1 ) [ b 2 , b 2 + d b 2 ) ,
b = ( b 1 , b 2 ) .
l 3 ( X , B ; λ , h , Λ ) = - R 2 λ ( x ) d x + R 2 ln [ λ ( x ) ] N t ( d x ) + i = 1 N ln [ h ( B i ) ]
l 3 ( X , B ; λ , h , Λ ) = - R 2 γ ( b ) d b + R 2 ln [ γ ( b ) ] N e ( d b ) + i = 1 N ln [ λ ( X i ) / Λ ] .
E { l 3 [ X , B ; λ ( k + 1 ) , h ( k + 1 ) , Λ ] C ( k ) } = - R 2 λ ( k + 1 ) ( x ) d x + R 2 ln [ λ ( k + 1 ) ( x ) ] E [ N t ( d x ) ] C ( k ) ] + i = 1 N E { ln [ h ( k + 1 ) ( B i ) ] C ( k ) } ,
E { l 3 [ X , B ; λ ( k + 1 ) , h ( k + 1 ) , Λ ] C ( k ) } = - R 2 Λ h ( k + 1 ) ( b ) d b + R 2 ln [ Λ h ( k + 1 ) ( b ) ] E [ N e ( d b ) ] C ( k ) ] + i = 1 N E { ln [ λ ( k + 1 ) ( X i ) / Λ ] C ( k ) } .
E [ N t ( d x ) C ( k ) ] = λ ^ ( k ) ( x ) d x R 2 h ^ ( k ) ( u - x ) h ^ ( k ) ( u - z ) λ ^ ( k ) ( z ) d z N d ( d u ) ,
E [ N e ( d b ) C ( k ) ] = h ^ ( k ) ( b ) d b R 2 λ ^ ( k ) ( u - b ) λ ^ ( k ) ( u - z ) h ^ ( k ) ( z ) d z N d ( d u ) ,
λ ^ ( k + 1 ) ( x ) = E [ N t ( d x ) C ( k ) ] / d x .
h ^ ( k + 1 ) ( x ) = E [ N e ( d x ) C ( k ) ] / ( Λ ^ d x ) ,
Λ ^ = R 2 N d ( d u ) ,
λ ^ ( k + 1 ) ( x ) = λ ^ ( k ) ( x ) R 2 h ^ ( k ) ( u - x ) h ^ ( k ) ( u - z ) λ ^ ( k ) ( z ) d z N d ( d u ) ,
h ^ ( k + 1 ) ( b ) = h ^ ( k ) ( b ) N R 2 λ ^ ( k ) ( u - b ) λ ^ ( k ) ( u - z ) h ^ ( k ) ( z ) d z N d ( d u ) ,
E { l 3 [ X , B ; λ ( k + 1 ) , h ( k + 1 ) , Λ ] C ( k ) } = - Λ all J h ( k + 1 ) ( R j ) R j d b + all J ln [ Λ h ( k + 1 ) ( R j ) ] R j E [ N e ( d b ) C ( k ) ] + i = 1 N E { ln [ λ ( k + 1 ) ( X i ) / Λ ] C ( k ) } .
h ^ ( k + 1 ) ( R i ) = R i E [ N e ( d b ) C ( k ) ] N R i d b = R i E [ N e ( d b ) C ( k ) ] N A i ,
A i = R i d b ,

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