Abstract

The three-dimensional image-reconstruction problem solved here for optical-sectioning microscopy is to estimate the fluorescence intensity λ(x), where x3, given a series of Poisson counting process measurements {Mj(dx)}j=1J, each with intensity sj(y) ∫3pj(y|x)λ(x)dx, with pj(y|x) being the point spread of the optics focused to the jth plane and sj(y) the detection probability for detector pointy at focal depth j. A maximum a posteriori reconstruction generated by inducing a prior distribution on the space of images via Good’s three-dimensional rotationally invariant roughness penalty ∫3 [|Δλ(x)|2/λ(x)]dx. It is proven that the sequence of iterates that is generated by using the expectation maximization algorithm is monotonically increasing in posterior probability, with stable points of the iteration satisfying the necessary maximizer conditions of the maximum a posteriori solution. The algorithms were implemented on the DECmpp-SX, a 64 × 64 parallel processor, running at <2 s/(643, 3-D iteration). Results are demonstrated from simulated as well as amoebae and volvox data. We study performance comparisons of the algorithms for the missing-data problems corresponding to fast data collection for rapid motion studies in which every other focal plane is removed and for imaging with limited detector areas and efficiency.

© 1993 Optical Society of America

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  1. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N. J., 1979).
  2. D. A. Agard, “Optical sectioning microscopy,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
    [CrossRef]
  3. C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for 3-D fluorescence microscopy,” Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 158–169 (1992).
    [CrossRef]
  4. C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
    [CrossRef] [PubMed]
  5. F. S. Gibson, F. Lanni, “Diffraction by a circular aperture as a model for three-dimensional optical microscopy,” J. Opt. Soc. Am. A 6, 1357–1367 (1989).
    [CrossRef] [PubMed]
  6. 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]
  7. L. S. Joyce, W. L. Root, “Precision bounds in superresolution processing,” J. Opt. Soc. Am. A 1, 149–168 (1984).
    [CrossRef]
  8. A. D. 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).
  9. D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
    [CrossRef] [PubMed]
  10. L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
    [CrossRef]
  11. E. Veklerov, J. Llacer, “Stopping rule for the mle algorithm based on statistical hypothesis testing,”IEEE Trans. Med. Imag. MI-6, 313–319 (1987).
    [CrossRef]
  12. D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-6, 228–237 (1987).
    [CrossRef]
  13. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
    [CrossRef] [PubMed]
  14. W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.
  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).
    [CrossRef]
  17. M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography: implementation of the EM algorithm and Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. 88, 3223–3227 (1991).
    [CrossRef]
  18. A. W. McCarthy, M. I. Miller, “Maximum likelihood SPECT in clinical computation times using mesh-connected parallel computers,”IEEE Trans. Med. Imag. 10, 426–436 (1991).
    [CrossRef]
  19. C. S. Butler, M. I. Miller, “Maximum a posterioriestimation for SPECT using regularization techniques on massively-parallel computers,”IEEE Trans. Med. Imag. (to be published).
  20. M. I. Miller, C. S. Butler, “3-D maximum a posterioriestimation for SPECT on massively-parallel computers.” IEEE Trans. Med. Imag. (to be published).
  21. I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
    [CrossRef]
  22. I. J. Good, “A nonparametric roughness penalty for probability densities,” Nature (London) 229, 29–30 (1971).
  23. I. J. Good, “Roughness penalties, invariant under rotation for multidimensional probability density estimation,”J. Stat. Comput. Simul. 12, 142–144 (1981); J. Stat. Comput. Simul. 13, 63 (1981).
    [CrossRef]
  24. The system’s principal components are an inverted Olympus IMT-2 microscope and a cooled −45°C CCD camera made by Photometrics, Ltd., equipped with a Kodak KAF1400 CCD chip and a Stardent Titan computer (Kubota Pacific Computer, Inc.).
  25. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Phys. 24, 194–200 (1985).
  26. N. Streibl, “Depth transfer by an imaging system,” Opt. Acta 31, 1233–1241 (1984).
    [CrossRef]
  27. R. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” Methods Cell Biol. 29, 291–313 (1989).
    [CrossRef] [PubMed]
  28. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
    [CrossRef]
  29. U. Genander, Abstract Inference (Wiley, New York, 1981).
  30. D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithms for emission tomography,”IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
    [CrossRef]
  31. R. A. Tapia, J. R. Thompson, Nonparametric Probability Density Estimation (Johns Hopkins U. Press, Baltimore, Md., 1978).
  32. D. L. Kirk, M. R. Kaufman, R. M. Keeling, K. A. Stammer, “Genetic and cytological control of the asymmetric divisions that pattern the Volvox embryo,” Dev. Biol. Suppl. 1, 67–82 (1991).
  33. K. J. Green, D. L. Kirk, “Cleavage patterns, cell lineages, and development of a cytoplasmic bridge system in Volvox embryos,”J. Cell Biol. 91, 743–755 (1981).
    [CrossRef] [PubMed]
  34. D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

1993 (1)

1992 (1)

1991 (3)

M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography: implementation of the EM algorithm and Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. 88, 3223–3227 (1991).
[CrossRef]

A. W. McCarthy, M. I. Miller, “Maximum likelihood SPECT in clinical computation times using mesh-connected parallel computers,”IEEE Trans. Med. Imag. 10, 426–436 (1991).
[CrossRef]

D. L. Kirk, M. R. Kaufman, R. M. Keeling, K. A. Stammer, “Genetic and cytological control of the asymmetric divisions that pattern the Volvox embryo,” Dev. Biol. Suppl. 1, 67–82 (1991).

1989 (4)

1988 (1)

1987 (3)

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]

E. Veklerov, J. Llacer, “Stopping rule for the mle algorithm based on statistical hypothesis testing,”IEEE Trans. Med. Imag. MI-6, 313–319 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-6, 228–237 (1987).
[CrossRef]

1985 (2)

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Phys. 24, 194–200 (1985).

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

1984 (3)

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

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

L. S. Joyce, W. L. Root, “Precision bounds in superresolution processing,” J. Opt. Soc. Am. A 1, 149–168 (1984).
[CrossRef]

1982 (1)

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

1981 (2)

K. J. Green, D. L. Kirk, “Cleavage patterns, cell lineages, and development of a cytoplasmic bridge system in Volvox embryos,”J. Cell Biol. 91, 743–755 (1981).
[CrossRef] [PubMed]

I. J. Good, “Roughness penalties, invariant under rotation for multidimensional probability density estimation,”J. Stat. Comput. Simul. 12, 142–144 (1981); J. Stat. Comput. Simul. 13, 63 (1981).
[CrossRef]

1977 (1)

A. D. 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).

1971 (2)

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

I. J. Good, “A nonparametric roughness penalty for probability densities,” Nature (London) 229, 29–30 (1971).

Agard, D. A.

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. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” Methods Cell Biol. 29, 291–313 (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. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef]

Aikens, R.

R. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” Methods Cell Biol. 29, 291–313 (1989).
[CrossRef] [PubMed]

Bille, J.

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Phys. 24, 194–200 (1985).

Butler, C. S.

C. S. Butler, M. I. Miller, “Maximum a posterioriestimation for SPECT using regularization techniques on massively-parallel computers,”IEEE Trans. Med. Imag. (to be published).

M. I. Miller, C. S. Butler, “3-D maximum a posterioriestimation for SPECT on massively-parallel computers.” IEEE Trans. Med. Imag. (to be published).

Carrington, W.

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.

Castleman, K. R.

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

Dempster, A. D.

A. D. 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).

Erhardt, A.

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Phys. 24, 194–200 (1985).

Fogarty, K.

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.

Gaskins, R. A.

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

Genander, U.

U. Genander, Abstract Inference (Wiley, New York, 1981).

Gibson, F. S.

Good, I. J.

I. J. Good, “Roughness penalties, invariant under rotation for multidimensional probability density estimation,”J. Stat. Comput. Simul. 12, 142–144 (1981); J. Stat. Comput. Simul. 13, 63 (1981).
[CrossRef]

I. J. Good, “A nonparametric roughness penalty for probability densities,” Nature (London) 229, 29–30 (1971).

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

Green, K. J.

K. J. Green, D. L. Kirk, “Cleavage patterns, cell lineages, and development of a cytoplasmic bridge system in Volvox embryos,”J. Cell Biol. 91, 743–755 (1981).
[CrossRef] [PubMed]

Hammoud, A. M.

Hiraoka, Y.

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.

Joyce, L. S.

Kaufman, M. R.

D. L. Kirk, M. R. Kaufman, R. M. Keeling, K. A. Stammer, “Genetic and cytological control of the asymmetric divisions that pattern the Volvox embryo,” Dev. Biol. Suppl. 1, 67–82 (1991).

Keeling, R. M.

D. L. Kirk, M. R. Kaufman, R. M. Keeling, K. A. Stammer, “Genetic and cytological control of the asymmetric divisions that pattern the Volvox embryo,” Dev. Biol. Suppl. 1, 67–82 (1991).

Kirk, D. L.

D. L. Kirk, M. R. Kaufman, R. M. Keeling, K. A. Stammer, “Genetic and cytological control of the asymmetric divisions that pattern the Volvox embryo,” Dev. Biol. Suppl. 1, 67–82 (1991).

K. J. Green, D. L. Kirk, “Cleavage patterns, cell lineages, and development of a cytoplasmic bridge system in Volvox embryos,”J. Cell Biol. 91, 743–755 (1981).
[CrossRef] [PubMed]

Komitowski, D.

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Phys. 24, 194–200 (1985).

Laird, N. M.

A. D. 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).

Lanni, F.

Llacer, J.

E. Veklerov, J. Llacer, “Stopping rule for the mle algorithm based on statistical hypothesis testing,”IEEE Trans. Med. Imag. MI-6, 313–319 (1987).
[CrossRef]

Luenberger, D. G.

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

McCarthy, A. W.

A. W. McCarthy, M. I. Miller, “Maximum likelihood SPECT in clinical computation times using mesh-connected parallel computers,”IEEE Trans. Med. Imag. 10, 426–436 (1991).
[CrossRef]

McNally, J. G.

C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
[CrossRef] [PubMed]

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for 3-D fluorescence microscopy,” Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 158–169 (1992).
[CrossRef]

Miller, M. I.

C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
[CrossRef] [PubMed]

A. W. McCarthy, M. I. Miller, “Maximum likelihood SPECT in clinical computation times using mesh-connected parallel computers,”IEEE Trans. Med. Imag. 10, 426–436 (1991).
[CrossRef]

M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography: implementation of the EM algorithm and Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. 88, 3223–3227 (1991).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-6, 228–237 (1987).
[CrossRef]

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

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

C. S. Butler, M. I. Miller, “Maximum a posterioriestimation for SPECT using regularization techniques on massively-parallel computers,”IEEE Trans. Med. Imag. (to be published).

M. I. Miller, C. S. Butler, “3-D maximum a posterioriestimation for SPECT on massively-parallel computers.” IEEE Trans. Med. Imag. (to be published).

Ollinger, J. M.

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for 3-D fluorescence microscopy,” Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 158–169 (1992).
[CrossRef]

Politte, D. G.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-6, 228–237 (1987).
[CrossRef]

Preza, C.

C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
[CrossRef] [PubMed]

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for 3-D fluorescence microscopy,” Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 158–169 (1992).
[CrossRef]

Root, W. L.

Roysam, B.

M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography: implementation of the EM algorithm and Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. 88, 3223–3227 (1991).
[CrossRef]

Rubin, D. B.

A. D. 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).

Sedat, J. W.

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. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” Methods Cell Biol. 29, 291–313 (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]

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, 113–121 (1982).
[CrossRef]

Snyder, D. L.

D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
[CrossRef] [PubMed]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-6, 228–237 (1987).
[CrossRef]

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

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

Stammer, K. A.

D. L. Kirk, M. R. Kaufman, R. M. Keeling, K. A. Stammer, “Genetic and cytological control of the asymmetric divisions that pattern the Volvox embryo,” Dev. Biol. Suppl. 1, 67–82 (1991).

Streibl, N.

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

Tapia, R. A.

R. A. Tapia, J. R. Thompson, Nonparametric Probability Density Estimation (Johns Hopkins U. Press, Baltimore, Md., 1978).

Thomas, L. J.

C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized linear method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
[CrossRef] [PubMed]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-6, 228–237 (1987).
[CrossRef]

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for 3-D fluorescence microscopy,” Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 158–169 (1992).
[CrossRef]

Thompson, J. R.

R. A. Tapia, J. R. Thompson, Nonparametric Probability Density Estimation (Johns Hopkins U. Press, Baltimore, Md., 1978).

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

Veklerov, E.

E. Veklerov, J. Llacer, “Stopping rule for the mle algorithm based on statistical hypothesis testing,”IEEE Trans. Med. Imag. MI-6, 313–319 (1987).
[CrossRef]

White, R. L.

Zinser, G.

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Phys. 24, 194–200 (1985).

Ann. Rev. Biophys. Bioeng. (1)

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

Appl. Phys. (1)

A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Phys. 24, 194–200 (1985).

Biometrika (1)

I. J. Good, R. A. Gaskins, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
[CrossRef]

Dev. Biol. Suppl. (1)

D. L. Kirk, M. R. Kaufman, R. M. Keeling, K. A. Stammer, “Genetic and cytological control of the asymmetric divisions that pattern the Volvox embryo,” Dev. Biol. Suppl. 1, 67–82 (1991).

IEEE Trans. Med. Imag. (4)

L. A. Shepp, Y. Vardi, “Maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-1, 113–121 (1982).
[CrossRef]

E. Veklerov, J. Llacer, “Stopping rule for the mle algorithm based on statistical hypothesis testing,”IEEE Trans. Med. Imag. MI-6, 313–319 (1987).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstruction for emission tomography,”IEEE Trans. Med. Imag. MI-6, 228–237 (1987).
[CrossRef]

A. W. McCarthy, M. I. Miller, “Maximum likelihood SPECT in clinical computation times using mesh-connected parallel computers,”IEEE Trans. Med. Imag. 10, 426–436 (1991).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

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

J. Cell Biol. (1)

K. J. Green, D. L. Kirk, “Cleavage patterns, cell lineages, and development of a cytoplasmic bridge system in Volvox embryos,”J. Cell Biol. 91, 743–755 (1981).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (6)

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

A. D. 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).

J. Stat. Comput. Simul. (1)

I. J. Good, “Roughness penalties, invariant under rotation for multidimensional probability density estimation,”J. Stat. Comput. Simul. 12, 142–144 (1981); J. Stat. Comput. Simul. 13, 63 (1981).
[CrossRef]

Methods Cell Biol. (2)

R. Aikens, D. A. Agard, J. W. Sedat, “Solid-state imagers for microscopy,” Methods Cell Biol. 29, 291–313 (1989).
[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]

Nature (London) (1)

I. J. Good, “A nonparametric roughness penalty for probability densities,” Nature (London) 229, 29–30 (1971).

Opt. Acta (1)

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

Proc. Natl. Acad. Sci. (1)

M. I. Miller, B. Roysam, “Bayesian image reconstruction for emission tomography: implementation of the EM algorithm and Good’s roughness prior on massively parallel processors,” Proc. Natl. Acad. Sci. 88, 3223–3227 (1991).
[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 (10)

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for 3-D fluorescence microscopy,” Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 158–169 (1992).
[CrossRef]

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

C. S. Butler, M. I. Miller, “Maximum a posterioriestimation for SPECT using regularization techniques on massively-parallel computers,”IEEE Trans. Med. Imag. (to be published).

M. I. Miller, C. S. Butler, “3-D maximum a posterioriestimation for SPECT on massively-parallel computers.” IEEE Trans. Med. Imag. (to be published).

W. Carrington, K. Fogarty, “3-D molecular distribution in living cells by deconvolution of optical sections using light microscopy,” in Proceedings of the Thirteenth Annual Northeast Bioengineering Conference, K. R. Foster, ed. (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 108–111.

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
[CrossRef]

U. Genander, Abstract Inference (Wiley, New York, 1981).

The system’s principal components are an inverted Olympus IMT-2 microscope and a cooled −45°C CCD camera made by Photometrics, Ltd., equipped with a Kodak KAF1400 CCD chip and a Stardent Titan computer (Kubota Pacific Computer, Inc.).

D. G. Luenberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

R. A. Tapia, J. R. Thompson, Nonparametric Probability Density Estimation (Johns Hopkins U. Press, Baltimore, Md., 1978).

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Figures (9)

Fig. 1
Fig. 1

(a) xy and (b) xz cuts through the experimentally measured point-spread function of the microscope for a 100× magnification and a 1.3-N.A. lens.

Fig. 2
Fig. 2

Surface rendering of the phantom.

Fig. 3
Fig. 3

xz cuts (a) through the phantom, (b) through the simulated data at high SNR, and (c) through the low SNR simulated data.

Fig. 4
Fig. 4

Top row shows xz cuts through the MAP (left) and ML (right) reconstructions at high SNR. The bottom row shows the same at low SNR.

Fig. 5
Fig. 5

MAP reconstructions corresponding to truncated or windowed detection of the data. (a) The MAP reconstruction corresponding to Eq. (3) without accommodation of sj(y) = 0 for deleted detector points. (b) The MAP reconstruction of Eq. (4) accounting for the deletion of detector points (see text).

Fig. 6
Fig. 6

xz slice (a) through the data associated with the deletion of every other focal plane and (b) through the MAP reconstruction.

Fig. 7
Fig. 7

The left column shows xy (top) and xz (bottom) cuts through the low-resolution 3-D amoebae data set collected at 20× magnification. The right column shows the slices through the MAP reconstruction with α = 0.001.

Fig. 8
Fig. 8

The left column shows two xy slices through the high-resolution 3-D amoebae data set collected at 100× magnification. The right column shows the slices through the MAP reconstructions at the same level.

Plate I
Plate I

The left-hand column shows two xy slices through the 3-D volvox embryo data set collected at 100× magnification. The right-hand column shows the MAP reconstructions for α = 0.001.

Tables (2)

Tables Icon

Table 1 Timing for 32-Bit Complex 643 and 1283 FFT’S on the 64 × 64 DECmppa

Tables Icon

Table 2 Breakdown of Times for the 643 and 1283 MAP Image Reconstruction on the 64 × 64 DECmppa

Equations (20)

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M j ( d y ) = M j obj ( d y ) + M j back ( d y ) + G ( d y ) ,
μ j ( d y ) = R 3 p j ( y x ) λ ( x ) d x ,
L ( λ ) = - j = 1 J D R 3 p j ( y x ) λ ( x ) d x + j = 1 J D log [ R 3 p j ( y x ) λ ( x ) d x ] M j ( d y ) .
L ( λ ) = - j = 1 J D s j ( y ) R 3 p j ( y x ) λ ( x ) d x d y + j = 1 J D log [ s j ( y ) R 3 p j ( y x ) λ ( x ) d x ] M j ( d y ) .
L [ r N ( d x ) ] = - R 3 λ [ x + r e ^ ( θ , φ ) ] d x + R 3 log { λ [ x + r e ^ ( θ , φ ) ] } N ( d x ) ,
E ( λ ) = R 3 λ ( x ) 2 λ ( x ) d x = R 3 1 λ ( x ) [ | λ ( x ) x | 2 + | λ ( x ) y | 2 + | λ ( x ) z | 2 ] d x d y d z .
E ( λ ) = i , j , k 1 λ ( i , j , k ) [ λ ( i + 1 , j , k ) - λ ( i , j , k ) 2 + λ ( i , j + 1 , k ) - λ ( i , j , k ) 2 + λ ( i , j , k + 1 ) - λ ( i , j , k ) 2 ] .
H ( λ ) = - j = 1 J D s j ( y ) R 3 p j ( y x ) λ ( x ) d x d y - α R 3 λ ( x ) 2 λ ( x ) d x + j = 1 J D log [ s j ( y ) R 3 p j ( y x ) λ ( x ) d x ] M j ( d y ) ,
λ ( k + 1 ) = arg max λ = γ 2 0 - R 3 λ ( x ) d x - α R 3 λ ( x ) 2 λ ( x ) d x + R 3 λ k ( x ) j = 1 J D { [ 1 - s j ( y ) ] p j ( y x ) d y + p j ( y x ) R 3 p j ( y x ) λ k ( x ) d x M j ( d y ) } log [ λ ( x ) ] d x ,
L c d ( λ ) = - R 3 λ ( x ) d x + R 3 log [ λ ( x ) ] N ( d x ) - α R 3 λ ( x ) 2 λ ( x ) d x .
Q [ λ λ ( k ) ] = - R 3 λ ( x ) d x + R 3 log λ ( x ) E { N ( d x ) M , λ ( k ) } × α R 3 λ ( x ) 2 λ ( x ) d x ,
E { N ( d x ) M , λ ( k ) } = λ ( k ) ( x ) d x j = 1 J D { [ 1 - s j ( y ) ] × p j ( y x ) d y + p j ( y x ) R 3 p j ( y x ) λ ( k ) ( x ) d x M j ( d y ) } ,
Q [ λ λ ( k ) ] = - R 3 λ ( x ) d x - α R 3 λ ( x ) 2 λ ( x ) d x + R 3 log [ λ ( x ) ] λ k ( x ) j = 1 J D { [ 1 - s j ( y ) ] × p j ( y x ) d y + p j ( y x ) R 3 p j ( y x ) λ k ( x ) d x M j ( d y ) } d x .
λ ( k + 1 ) = arg max λ = γ 2 0 ( - R 3 λ ( x ) d x - α R 3 λ ( x ) 2 λ ( x ) d x + R 3 log [ λ ( x ) ] λ k ( x ) j = 1 J D { [ 1 - s j ( y ) ] p j ( y x ) d y + p j ( y x ) R 3 p j ( y x ) λ k ( x ) d x M j ( d y ) } d x ) ,
δ T ( λ ; η ) = lim 0 T ( λ + η ) - T ( λ ) | = 0 .
δ Q [ λ ; η λ ( k ) ] = lim 0 Q [ ( γ + η ) 2 λ ( k ) ] - Q [ γ 2 λ ( k ) ] | = 0 .
δ Q [ λ ( k + 1 ) λ ( k ) ] = - 2 R 3 λ ( k + 1 ) ( x ) η ( x ) d x - α δ E ( λ ; η )
+ R 3 λ ( k ) ( x ) j = 1 J D { [ 1 - s j ( y ) ] p j ( y x ) d y + p j ( y x ) R 3 p j ( y x ) λ ( k ) ( x ) d x M j ( d y ) } × 2 η ( x ) λ ( k + 1 ) ( x ) d x .
0 = - 2 R 3 λ ¯ ( x ) η ( x ) d x - α δ E ( λ ¯ ; η ) + R 3 λ ¯ ( x ) 2 η ( x ) j = 1 J D { [ 1 - s j ( y ) ] p j ( y x ) d y + p j ( y x ) R 3 p j ( y x ) λ ¯ ( x ) d x M j ( d y ) } d x
= - 2 R 3 λ ¯ ( x ) η ( x ) j = 1 J D s j ( y ) p j ( y x ) d y d x - α δ E ( λ ¯ ; η ) + 2 R 3 λ ¯ ( x ) η ( x ) × j = 1 J D p j ( y x ) R 3 p j ( y x ) λ ¯ ( x ) d x M j ( d y ) d x .

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