Abstract

Significant improvement in the longitudinal (optical axis) resolution of a microscope has previously been obtained either by posterior digital processing of optical sections collected with a conventional microscope, or by collecting sections with a confocal scanning microscope. In this paper we report the feasibility of obtaining longitudinal resolution comparable to the lateral diffraction limit by posterior processing of confocal sections. A confocal through-focus image series was simulated numerically and restored by constrained iterative deconvolution and by maximum likelihood. Several typical imaging situations were simulated. The results support the possibility of achieving equal longitudinal and lateral resolution.

© 1990 Optical Society of America

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References

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  1. F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of a-Actinin in Single Isolated Smooth Muscle Cells,” J. Cell Biol. 96, 783–795 (1983).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  3. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D Light-Microscopy Images by Digital Image Processing,” Appl. Opt. 24, 194–200 (1985).
    [CrossRef] [PubMed]
  4. R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
    [CrossRef]
  5. K. Carlsson, P. E. Danielsson, R. Lenz, A. Liljeborg, L. Majlof, N. Åslund, “Three-Dimensional Microscopy Using a Confocal Laser Scanning Microscope,” Opt. Lett. 10, 53–55 (1985).
    [CrossRef] [PubMed]
  6. G. J. Brakenhoff, P. Blom, P. Barends, “Confocal Scanning Light Microscopy with High Aperture Immersion Lenses,” J. Microsc. 117, 219–232 (1979).
    [CrossRef]
  7. T. Wilson, “Imaging Properties and Applications of Scanning Optical Microscopes,” Appl. Phys. 22, 119–128 (1980).
    [CrossRef]
  8. T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).
  9. J. G. White, A. B. Amos, M. Fordham, “An Evaluation of Confocal Versus Conventional Imaging of Biological Structures by Fluorescence Light Microscopy,” J. Cell Biol. 105, 41–48 (1987).
    [CrossRef] [PubMed]
  10. J. A. Conchello, E. W. Hansen, “Three-Dimensional Reconstruction of Noisy Confocal Scanning Microscope Images,” Proc. Soc. Photo-Opt. Instrum. Eng. 1161, 279–285 (1989).
  11. W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional Imaging on Confocal and Wide-Field Microscopes,” in The Handbook of Biological Confocal Microscopy, J. Pawley, Ed. (IMR Press, Madison, WI, 1989), pp. 137–146.
  12. B. R. Frieden, “Optical Transfer of the Three-Dimensional Object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  13. W. B. Amos, J. G. White, M. Fordham, “Use of Confocal Imaging in the Study of Biological Structures,” Appl. Opt. 26, 3239–3243 (1987).
    [CrossRef] [PubMed]
  14. K. Carlsson, N. Åslund, “Confocal Imaging for 3–D Digital Microscopy,” Appl. Opt. 26, 3232–3238 (1987).
    [CrossRef] [PubMed]
  15. B. Bertero, C. D. Mol, E. R. Pike, “Analytic Inversion Formula for Confocal Scanning Microscopy,” J. Opt. Soc. Am. A 4, 1748–1750 (1987).
    [CrossRef]
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    [CrossRef]
  17. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).
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    [CrossRef]
  19. D. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
    [CrossRef]
  20. B. R. Frieden, “Image Enhancement and Restoration,” in Picture Processing and Digital Filtering, T. S. Huang, Ed. (Springer-Verlag, New York, 1975), pp. 177–248.
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  22. T. J. Holmes, “Expectation-Maximization Restoration of Band-Limited, Truncated Point-Process Intensities with Application to Microscopy,” J. Opt. Soc. Am. A 6, 1006–1014 (1989).
    [CrossRef]
  23. B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).
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    [CrossRef]
  25. E. W. Hansen, J. P. Zelten, B. A. Wiseman, “Laser Scanning Fluorescence Microscope,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 304–311 (1988).
  26. C. J. R. Sheppard, “The Spatial Frequency Cut-Off in Three-Dimensional Imaging II,” Optik 74, 128–129 (1986).
  27. J. A. Conchello, E. W. Hansen, “Resolution and Signal-to-Noise Tradeoffs in Confocal Microscopy,” in OSA Annual Meeting, Technical Digest Series, Vol. 18 (Optical Society of America, Washington, DC, 1989), paper FI2.
  28. K. M. Hanson, “Bayesian and Related Methods in Image Reconstruction From Incomplete Data,” in Image Recovery: Theory and Application, H. Stark, Ed. (Academic Press, New York, 1987), pp. 79–125.

1989 (2)

T. J. Holmes, “Expectation-Maximization Restoration of Band-Limited, Truncated Point-Process Intensities with Application to Microscopy,” J. Opt. Soc. Am. A 6, 1006–1014 (1989).
[CrossRef]

J. A. Conchello, E. W. Hansen, “Three-Dimensional Reconstruction of Noisy Confocal Scanning Microscope Images,” Proc. Soc. Photo-Opt. Instrum. Eng. 1161, 279–285 (1989).

1988 (2)

Z. Liang, “Statistical Models of a priori Information for Image Processing: Neighboring Correlation Constraints,” J. Opt. Soc. Am. A 5, 2026–2031 (1988).
[CrossRef]

E. W. Hansen, J. P. Zelten, B. A. Wiseman, “Laser Scanning Fluorescence Microscope,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 304–311 (1988).

1987 (5)

W. B. Amos, J. G. White, M. Fordham, “Use of Confocal Imaging in the Study of Biological Structures,” Appl. Opt. 26, 3239–3243 (1987).
[CrossRef] [PubMed]

K. Carlsson, N. Åslund, “Confocal Imaging for 3–D Digital Microscopy,” Appl. Opt. 26, 3232–3238 (1987).
[CrossRef] [PubMed]

B. Bertero, C. D. Mol, E. R. Pike, “Analytic Inversion Formula for Confocal Scanning Microscopy,” J. Opt. Soc. Am. A 4, 1748–1750 (1987).
[CrossRef]

M. Bertero, P. Brianzi, E. R. Pike, “Super-Resolution in Confocal Scanning Microscopy,” Inverse Probl. 3, 195–212 (1987).
[CrossRef]

J. G. White, A. B. Amos, M. Fordham, “An Evaluation of Confocal Versus Conventional Imaging of Biological Structures by Fluorescence Light Microscopy,” J. Cell Biol. 105, 41–48 (1987).
[CrossRef] [PubMed]

1986 (1)

C. J. R. Sheppard, “The Spatial Frequency Cut-Off in Three-Dimensional Imaging II,” Optik 74, 128–129 (1986).

1985 (3)

1984 (1)

D. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef]

1983 (2)

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of a-Actinin in Single Isolated Smooth Muscle Cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

D. A. Agard, J. W. Sedat, “Three-Dimensional Structure of a Polytene Nucleus,” Nature London 302, 676–681 (1983).
[CrossRef] [PubMed]

1980 (1)

T. Wilson, “Imaging Properties and Applications of Scanning Optical Microscopes,” Appl. Phys. 22, 119–128 (1980).
[CrossRef]

1979 (1)

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal Scanning Light Microscopy with High Aperture Immersion Lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

1967 (1)

1955 (1)

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Agard, D.

D. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef]

Agard, D. A.

D. A. Agard, J. W. Sedat, “Three-Dimensional Structure of a Polytene Nucleus,” Nature London 302, 676–681 (1983).
[CrossRef] [PubMed]

Amos, A. B.

J. G. White, A. B. Amos, M. Fordham, “An Evaluation of Confocal Versus Conventional Imaging of Biological Structures by Fluorescence Light Microscopy,” J. Cell Biol. 105, 41–48 (1987).
[CrossRef] [PubMed]

Amos, W. B.

Åslund, N.

Barends, P.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal Scanning Light Microscopy with High Aperture Immersion Lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Bertero, B.

Bertero, M.

M. Bertero, P. Brianzi, E. R. Pike, “Super-Resolution in Confocal Scanning Microscopy,” Inverse Probl. 3, 195–212 (1987).
[CrossRef]

Bille, J.

Blom, P.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal Scanning Light Microscopy with High Aperture Immersion Lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

Brakenhoff, G. J.

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal Scanning Light Microscopy with High Aperture Immersion Lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

Brianzi, P.

M. Bertero, P. Brianzi, E. R. Pike, “Super-Resolution in Confocal Scanning Microscopy,” Inverse Probl. 3, 195–212 (1987).
[CrossRef]

Carlsson, K.

Carnahan, B.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Carrington, W. A.

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional Imaging on Confocal and Wide-Field Microscopes,” in The Handbook of Biological Confocal Microscopy, J. Pawley, Ed. (IMR Press, Madison, WI, 1989), pp. 137–146.

Conchello, J. A.

J. A. Conchello, E. W. Hansen, “Three-Dimensional Reconstruction of Noisy Confocal Scanning Microscope Images,” Proc. Soc. Photo-Opt. Instrum. Eng. 1161, 279–285 (1989).

J. A. Conchello, E. W. Hansen, “Resolution and Signal-to-Noise Tradeoffs in Confocal Microscopy,” in OSA Annual Meeting, Technical Digest Series, Vol. 18 (Optical Society of America, Washington, DC, 1989), paper FI2.

Danielsson, P. E.

Davoust, J.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

Dudgeon, D. E.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Erhardt, A.

Fay, F. S.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of a-Actinin in Single Isolated Smooth Muscle Cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional Imaging on Confocal and Wide-Field Microscopes,” in The Handbook of Biological Confocal Microscopy, J. Pawley, Ed. (IMR Press, Madison, WI, 1989), pp. 137–146.

Fogarty, K. E.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of a-Actinin in Single Isolated Smooth Muscle Cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional Imaging on Confocal and Wide-Field Microscopes,” in The Handbook of Biological Confocal Microscopy, J. Pawley, Ed. (IMR Press, Madison, WI, 1989), pp. 137–146.

Fordham, M.

J. G. White, A. B. Amos, M. Fordham, “An Evaluation of Confocal Versus Conventional Imaging of Biological Structures by Fluorescence Light Microscopy,” J. Cell Biol. 105, 41–48 (1987).
[CrossRef] [PubMed]

W. B. Amos, J. G. White, M. Fordham, “Use of Confocal Imaging in the Study of Biological Structures,” Appl. Opt. 26, 3239–3243 (1987).
[CrossRef] [PubMed]

Frieden, B. R.

B. R. Frieden, “Optical Transfer of the Three-Dimensional Object,” J. Opt. Soc. Am. 57, 56–66 (1967).
[CrossRef]

B. R. Frieden, “Image Enhancement and Restoration,” in Picture Processing and Digital Filtering, T. S. Huang, Ed. (Springer-Verlag, New York, 1975), pp. 177–248.

Fujiwara, K.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of a-Actinin in Single Isolated Smooth Muscle Cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

Hansen, E. W.

J. A. Conchello, E. W. Hansen, “Three-Dimensional Reconstruction of Noisy Confocal Scanning Microscope Images,” Proc. Soc. Photo-Opt. Instrum. Eng. 1161, 279–285 (1989).

E. W. Hansen, J. P. Zelten, B. A. Wiseman, “Laser Scanning Fluorescence Microscope,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 304–311 (1988).

J. A. Conchello, E. W. Hansen, “Resolution and Signal-to-Noise Tradeoffs in Confocal Microscopy,” in OSA Annual Meeting, Technical Digest Series, Vol. 18 (Optical Society of America, Washington, DC, 1989), paper FI2.

Hanson, K. M.

K. M. Hanson, “Bayesian and Related Methods in Image Reconstruction From Incomplete Data,” in Image Recovery: Theory and Application, H. Stark, Ed. (Academic Press, New York, 1987), pp. 79–125.

Holmes, T. J.

Hopkins, H. H.

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Kaplan, R.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

Komitowski, D.

Lenz, R.

Liang, Z.

Lifschitz, L.

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional Imaging on Confocal and Wide-Field Microscopes,” in The Handbook of Biological Confocal Microscopy, J. Pawley, Ed. (IMR Press, Madison, WI, 1989), pp. 137–146.

Liljeborg, A.

Luther, H. A.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Majlof, L.

Marsman, H. K. B.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

Mersereau, R. M.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

Mol, C. D.

Pike, E. R.

B. Bertero, C. D. Mol, E. R. Pike, “Analytic Inversion Formula for Confocal Scanning Microscopy,” J. Opt. Soc. Am. A 4, 1748–1750 (1987).
[CrossRef]

M. Bertero, P. Brianzi, E. R. Pike, “Super-Resolution in Confocal Scanning Microscopy,” Inverse Probl. 3, 195–212 (1987).
[CrossRef]

Rees, D. D.

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of a-Actinin in Single Isolated Smooth Muscle Cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

Sedat, J. W.

D. A. Agard, J. W. Sedat, “Three-Dimensional Structure of a Polytene Nucleus,” Nature London 302, 676–681 (1983).
[CrossRef] [PubMed]

Sheppard, C. J. R.

C. J. R. Sheppard, “The Spatial Frequency Cut-Off in Three-Dimensional Imaging II,” Optik 74, 128–129 (1986).

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Steltzer, E. H. K.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

Striker, R.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

White, J. G.

J. G. White, A. B. Amos, M. Fordham, “An Evaluation of Confocal Versus Conventional Imaging of Biological Structures by Fluorescence Light Microscopy,” J. Cell Biol. 105, 41–48 (1987).
[CrossRef] [PubMed]

W. B. Amos, J. G. White, M. Fordham, “Use of Confocal Imaging in the Study of Biological Structures,” Appl. Opt. 26, 3239–3243 (1987).
[CrossRef] [PubMed]

Wijnaendts van Resandt, R. W.

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

Wilkes, J. O.

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

Wilson, T.

T. Wilson, “Imaging Properties and Applications of Scanning Optical Microscopes,” Appl. Phys. 22, 119–128 (1980).
[CrossRef]

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

Wiseman, B. A.

E. W. Hansen, J. P. Zelten, B. A. Wiseman, “Laser Scanning Fluorescence Microscope,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 304–311 (1988).

Zelten, J. P.

E. W. Hansen, J. P. Zelten, B. A. Wiseman, “Laser Scanning Fluorescence Microscope,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 304–311 (1988).

Zinser, G.

Ann. Rev. Biophys. Bioeng. (1)

D. Agard, “Optical Sectioning Microscopy: Cellular Architecture in Three Dimensions,” Ann. Rev. Biophys. Bioeng. 13, 191–219 (1984).
[CrossRef]

Appl. Opt. (3)

Appl. Phys. (1)

T. Wilson, “Imaging Properties and Applications of Scanning Optical Microscopes,” Appl. Phys. 22, 119–128 (1980).
[CrossRef]

Inverse Probl. (1)

M. Bertero, P. Brianzi, E. R. Pike, “Super-Resolution in Confocal Scanning Microscopy,” Inverse Probl. 3, 195–212 (1987).
[CrossRef]

J. Cell Biol. (2)

J. G. White, A. B. Amos, M. Fordham, “An Evaluation of Confocal Versus Conventional Imaging of Biological Structures by Fluorescence Light Microscopy,” J. Cell Biol. 105, 41–48 (1987).
[CrossRef] [PubMed]

F. S. Fay, K. Fujiwara, D. D. Rees, K. E. Fogarty, “Distribution of a-Actinin in Single Isolated Smooth Muscle Cells,” J. Cell Biol. 96, 783–795 (1983).
[CrossRef] [PubMed]

J. Microsc. (2)

R. W. Wijnaendts van Resandt, H. K. B. Marsman, R. Kaplan, J. Davoust, E. H. K. Steltzer, R. Striker, “Optical Fluorescence Microscopy in Three Dimensions: Microtomoscopy,” J. Microsc. 138, 29–34 (1985).
[CrossRef]

G. J. Brakenhoff, P. Blom, P. Barends, “Confocal Scanning Light Microscopy with High Aperture Immersion Lenses,” J. Microsc. 117, 219–232 (1979).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

Nature London (1)

D. A. Agard, J. W. Sedat, “Three-Dimensional Structure of a Polytene Nucleus,” Nature London 302, 676–681 (1983).
[CrossRef] [PubMed]

Opt. Lett. (1)

Optik (1)

C. J. R. Sheppard, “The Spatial Frequency Cut-Off in Three-Dimensional Imaging II,” Optik 74, 128–129 (1986).

Proc. R. Soc. London Ser. A (1)

H. H. Hopkins, “The Frequency Response of a Defocused Optical System,” Proc. R. Soc. London Ser. A 231, 91–103 (1955).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (2)

E. W. Hansen, J. P. Zelten, B. A. Wiseman, “Laser Scanning Fluorescence Microscope,” Proc. Soc. Photo-Opt. Instrum. Eng. 909, 304–311 (1988).

J. A. Conchello, E. W. Hansen, “Three-Dimensional Reconstruction of Noisy Confocal Scanning Microscope Images,” Proc. Soc. Photo-Opt. Instrum. Eng. 1161, 279–285 (1989).

Other (8)

W. A. Carrington, K. E. Fogarty, L. Lifschitz, F. S. Fay, “Three-dimensional Imaging on Confocal and Wide-Field Microscopes,” in The Handbook of Biological Confocal Microscopy, J. Pawley, Ed. (IMR Press, Madison, WI, 1989), pp. 137–146.

T. Wilson, C. J. R. Sheppard, Theory and Practice of Scanning Optical Microscopy (Academic, New York, 1984).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978).

B. R. Frieden, “Image Enhancement and Restoration,” in Picture Processing and Digital Filtering, T. S. Huang, Ed. (Springer-Verlag, New York, 1975), pp. 177–248.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1984).

B. Carnahan, H. A. Luther, J. O. Wilkes, Applied Numerical Methods (Wiley, New York, 1969).

J. A. Conchello, E. W. Hansen, “Resolution and Signal-to-Noise Tradeoffs in Confocal Microscopy,” in OSA Annual Meeting, Technical Digest Series, Vol. 18 (Optical Society of America, Washington, DC, 1989), paper FI2.

K. M. Hanson, “Bayesian and Related Methods in Image Reconstruction From Incomplete Data,” in Image Recovery: Theory and Application, H. Stark, Ed. (Academic Press, New York, 1987), pp. 79–125.

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Figures (11)

Fig. 1
Fig. 1

Geometry of the confocal scanning microscope. The specimen is illuminated by a focused beam; light emitted by points outside the plane of focus (dashed rays) is rejected by the aperture in front of the detector.

Fig. 2
Fig. 2

Jansson-van Cittert reconstruction of simulated cross and circle of equal intensity 200 nm apart. From top to bottom each column has the optical slice immediately above the cross, the optical slices where the cross (marked X) and the circle (marked ○) are in focus, and one slice below the circle. Columns from left to right: recorded (unprocessed) image, reconstructions after 10, 20, and 40 iterations, respectively. Bar = 1 μm.

Fig. 3
Fig. 3

Jansson-van Cittert reconstruction of simulated cross and circle of equal intensity, 50 nm apart. From top to bottom each column displays two optical slices above the circle, the plane immediately above the circle, the plane where the circle is located (marked ○), and one plane below the circle. The cross is 50 nm above the circle, i.e., at 1/4 of the distance between the third and the second slices. Columns from left to right: recorded (unprocessed) image, reconstructions after 10, 20, and 40 iterations. Bar = 1 μm.

Fig. 4
Fig. 4

Jansson-van Cittert reconstruction of simulated cross and circle 200 nm apart, appearing in focus in adjacent optical slices. The intensity of the cross is 1/10 the intensity of the circle. Columns from left to right: recorded (unprocessed) image, reconstructions after 10, 20, and 40 iterations, respectively. Each of the sixteen panels was normalized independently to have maximum intensity equal to white. Bar = 1 μm.

Fig. 5
Fig. 5

Jansson-van Cittert reconstruction of noisy image. The data used in Fig. 2 were corrupted by Poisson noise with maximum SNR = 3.16. Columns from left to right: recorded (unprocessed) noisy image, reconstructions after 10, 20, and 40 iterations, respectively. Cross and circle are in focus at rows marked X and ○, respectively. Bar = 1 μm.

Fig. 6
Fig. 6

Jansson-van Cittert reconstruction of noisy image data. The data used in Fig. 3 were corrupted by Poisson noise with maximum SNR = 3.16. Columns from left to right: recorded (unprocessed) noisy image, reconstructions after 10, 20, and 40 iterations, respectively. Circle is in focus at row marked ○. Bar = 1 μm.

Fig. 7
Fig. 7

Maximum likelihood reconstructions of simulated cross and circle of equal brightness, 200 nm apart and appearing in focus in adjacent optical slices. Columns from left to right: recorded (unprocessed) image, reconstructions after 5, 10, and 20 iterations, respectively. The cross and the circle are in focus at rows labeled X and ○, respectively. Bar = 1 μm.

Fig. 8
Fig. 8

Maximum likelihood reconstructions of image data used in Fig. 7, corrupted by Poisson noise with maximum SNR = 3.16. Columns from left to right: recorded (unprocessed) noisy image, reconstructions after 5, 10, and 20 iterations, respectively. The cross and the circle are in focus at rows labeled X and ○, respectively. Bar = 1 μm.

Fig. 9
Fig. 9

Maximum likelihood reconstruction of image data used in Fig. 7, corrupted by Poisson noise with maximum SNR = 1. Columns from left to right: recorded (unprocessed) noisy image, reconstructions after 5, 10, and 20 iterations, respectively. Bar = 1 μm.

Fig. 10
Fig. 10

Maximum likelihood reconstruction of simulated cross and circle 200 nm apart appearing in focus in adjacent optical slices. The intensity of the cross is 1/10 the intensity of the circle. Columns from left to right: recorded (unprocessed) image, reconstructions after 5, 10, and 20 iterations, respectively. Bar = 1 μm.

Fig. 11
Fig. 11

Maximum likelihood reconstructions of image data in Fig. 10, corrupted by Poisson noise with maximum SNR = 3.16. Each panel was normalized to bring out detail. Columns from left to right: recorded (unprocessed) noisy image, reconstructions after 5, 10, and 20 iterations, respectively. Bar = 1 μm.

Equations (11)

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I ( x , z ) = d x d P 3 ( x d ) 2 d x s d z s p 2 ( x s - x d , z s ) × p 1 ( x s , z s ) 2 t ( x - x s , z - z s )
h ( x , z ) = p 1 ( x , z ) 2 p 2 ( x , z ) 2 .
h ( x , z ) = F 2 - 1 [ P ( η , ξ ; z ) ] 4 ,
H ( ν ) = F 3 { h ( x , z ) } h ( x , z ) d x d z ,
P ( ρ ; z ) = exp { j 2 π z z 0 ( ρ ρ 0 ) 2 } × circ ( ρ ρ 0 ) ,
circ ( ρ ) = { 1 if ρ 1 0 otherwise .
s ^ ( k ) ( x ) { 0 , if s ^ ( k ) ( x ) < 0 2 A , if s ^ ( k ) ( x ) > 2 A s ^ ( k ) ( x ) , otherwise ,
f ( s g ) f ( g s ) f ( s ) .
f ( g s ) = i = 1 N ( Hs ) i g i g i ! exp [ - ( Hs ) i ] ,
s ^ i ( k + 1 ) = s ^ i ( k ) [ H d ( k ) ] i ( j ( H ) i j ) + ξ ( k ) z i ( k ) ,
d i ( k ) = g i [ H s ^ ( k ) ] i , z i ( k ) = [ - s i ln f ( s ) ] s = s ^ ( k ) ,

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