Abstract

We tested the most complete optical model available for computational optical-sectioning microscopy and obtained four main results. First, we observed good agreement between experimental and theoretical point-spread functions (PSF∙s) under a variety of imaging conditions. Second, using these PSF’s, we found that a linear restoration method yielded reconstructed images of a well-defined phantom object (a 10-μm-diameter fluorescent bead) that closely resembled the theoretically determined, best-possible linear reconstruction of the object. Third, this best linear reconstruction suffered from a (to our knowledge) previously undescribed artifactual axial elongation whose principal cause was not increased axial blur but rather the conical shape of the null space intrinsic to nonconfocal three-dimensional (3D) microscopy. Fourth, when 10-μm phantom beads were embedded at different depths in a transparent medium, reconstructed bead images were progressively degraded with depth unless they were reconstructed with use of a PSF determined at the bead’s depth. We conclude that (1) the optical model for optical sectioning is reasonably accurate; (2) if PSF shift variance cannot be avoided by adjustment of the optics, then reconstruction methods must be modified to account for this effect; and (3) alternative microscopical or nonlinear algorithmic approaches are required for overcoming artifacts imposed by the missing cone of frequencies that is intrinsic to nonconfocal 3D microscopy.

© 1994 Optical Society of America

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  1. D. A. Agard, “Optical sectioning microscopy,” Ann. Rev. Bio-phys. Bioeng. 13, 191–219 (1984).
    [CrossRef]
  2. M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
    [CrossRef]
  3. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
    [CrossRef] [PubMed]
  4. T. J. Holmes, “Maximum-likelihood image restoration for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  5. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Methods Cell Biol. 30, 353–377 (1989).
    [CrossRef] [PubMed]
  6. F. S. Fay, W. Carrington, K. E. Fogarty, “Three-dimensional molecular distribution in single cells analysed using the digital imaging microscope,” J. Microsc. 153, 133–149 (1989).
    [CrossRef] [PubMed]
  7. W. A. Carrington, “Image-restoration in 3D 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]
  8. A. Diaspro, M. Sartore, C. Nicolini, “3D representation of biostructures imaged with an optical microscope,” Image Vis. Computing 8, 130–141 (1990).
    [CrossRef]
  9. 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]
  10. 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]
  11. S. Joshi, M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for three-dimensional optical-sectioning microscopy,” J. Opt. Soc. Am. A 10, 1078–1085 (1993).
    [CrossRef] [PubMed]
  12. K. R. Castleman, Digital Image Processing (Prentice-Hall, Englewood Cliffs, N.J., 1979).
  13. I. T. Young, “Image fidelity: characterizing the imaging transfer function,” Methods Cell Biol. 30, 1–44 (1989).
    [CrossRef] [PubMed]
  14. F. S. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
    [CrossRef]
  15. C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for computational optical-sectioning microscopy,” Micron Microsc. Acta 23, 501–513 (1992).
    [CrossRef]
  16. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).
  17. 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]
  18. B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [CrossRef]
  19. N. Streibl, “Three-dimensional imaging by a microscope,” J. Opt. Soc. Am. A 2, 121–127 (1985).
    [CrossRef]
  20. J. A. Conchello, “Three-dimensional reconstruction of noisy images from partially confocal scanning microscope,” Ph.D. dissertation (Dartmouth College, Hanover, N.H., 1990)
  21. The error in our measurement of distances by determining half-maximal intensities at an edge is affected predominantly by the pixel size. The Rayleigh diffraction limit would contribute to this error only if the width of the bead’s fluorescent shell were comparable with the size of the Airy disk. However, the shell’s thickness is approximately ten times the radius of the Airy disk (0.35 μm), and so the bead’s fluorescent layer can be considered a step. The incoherent step response preserves the location of the edge as measured by the half-intensity point.33
  22. Exact matches of phantom-bead depth and PSF depth were not always possible for experimental PSF’s, because the latter were measured from microspheres embedded at random depths in optical cement. From many such measurements we selected experimental PSF’s at depths as close as possible to those of the phantom beads. Subsequently, for comparing reconstructions having these experimental PSF’s with those having theoretical PSF’s, we computed each theoretical PSF at the same depth as that of the experimental PSF, so as not to give the theoretical PSF’s an advantage.
  23. F. Lanni, G. J. Baxter, “Sampling theorem for square-pixel image data,” in Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 140–147 (1992).
    [CrossRef]
  24. Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” Biophys. J. 57, 325–333 (1990).
    [CrossRef] [PubMed]
  25. F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
    [CrossRef]
  26. K. M. Hanson, “Bayesian and related methods in image reconstruction from incomplete data,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 79–125.
  27. D. A. Agard, R. M. Stroud, “Linking regions between helices in bacteriorhodopsin revealed,” Biophys. J. 37, 589–602 (1981).
  28. 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]
  29. J.-M. Carazo, “The fidelity of 3D reconstructions from incomplete data and the use of restoration methods,” in Electron Tomography: Three-Dimensional Imaging with the Transmission Electron Microscope (J. Frank, ed. (Plenum, New York, 1992), pp. 117–164.
  30. J. J. Lemasters, E. Chacon, G. Zahrebelski, J. M. Reece, A.-L. Nieminen, “Laser scanning confocal microscopy of living cells,” in Optical Microscopy: Emerging Methods and Applications, B. Herman, J. J. Lemasters, eds. (Academic, San Diego, Calif., 1993), pp. 339–354.
  31. J.-A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstructions from confocal scanning microscope images. 2. Depth discrimination versus signal-to-noise ratio in partially confocal images,” Appl. Opt. (to be published).
  32. J.-A. Conchello, J. W. Lichtman, “Theoretical analysis of a rotating-disk partially confocal scanning microscope,” Appl. Opt. 33, 585–596 (1994).
    [CrossRef] [PubMed]
  33. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 132.

1994 (1)

1993 (1)

1992 (2)

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 computational optical-sectioning microscopy,” Micron Microsc. Acta 23, 501–513 (1992).
[CrossRef]

1991 (1)

1990 (2)

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

A. Diaspro, M. Sartore, C. Nicolini, “3D representation of biostructures imaged with an optical microscope,” Image Vis. Computing 8, 130–141 (1990).
[CrossRef]

1989 (5)

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

F. S. Fay, W. Carrington, K. E. Fogarty, “Three-dimensional molecular distribution in single cells analysed using the digital imaging microscope,” J. Microsc. 153, 133–149 (1989).
[CrossRef] [PubMed]

F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
[CrossRef]

I. T. Young, “Image fidelity: characterizing the imaging transfer function,” Methods Cell Biol. 30, 1–44 (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]

1988 (1)

1987 (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]

1985 (2)

1984 (1)

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

1981 (1)

D. A. Agard, R. M. Stroud, “Linking regions between helices in bacteriorhodopsin revealed,” Biophys. J. 37, 589–602 (1981).

1967 (1)

Agard, D. A.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” 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]

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

D. A. Agard, R. M. Stroud, “Linking regions between helices in bacteriorhodopsin revealed,” Biophys. J. 37, 589–602 (1981).

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]

Aggarwal, J. K.

F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
[CrossRef]

Aggarwal, S. J.

F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
[CrossRef]

Baxter, G. J.

F. Lanni, G. J. Baxter, “Sampling theorem for square-pixel image data,” in Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 140–147 (1992).
[CrossRef]

Bille, J.

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Bovik, A. C.

F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
[CrossRef]

Carazo, J.-M.

J.-M. Carazo, “The fidelity of 3D reconstructions from incomplete data and the use of restoration methods,” in Electron Tomography: Three-Dimensional Imaging with the Transmission Electron Microscope (J. Frank, ed. (Plenum, New York, 1992), pp. 117–164.

Carrington, W.

F. S. Fay, W. Carrington, K. E. Fogarty, “Three-dimensional molecular distribution in single cells analysed using the digital imaging microscope,” J. Microsc. 153, 133–149 (1989).
[CrossRef] [PubMed]

Carrington, W. A.

W. A. Carrington, “Image-restoration in 3D 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.

M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
[CrossRef]

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

Chacon, E.

J. J. Lemasters, E. Chacon, G. Zahrebelski, J. M. Reece, A.-L. Nieminen, “Laser scanning confocal microscopy of living cells,” in Optical Microscopy: Emerging Methods and Applications, B. Herman, J. J. Lemasters, eds. (Academic, San Diego, Calif., 1993), pp. 339–354.

Conchello, J. A.

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

Conchello, J.-A.

J.-A. Conchello, J. W. Lichtman, “Theoretical analysis of a rotating-disk partially confocal scanning microscope,” Appl. Opt. 33, 585–596 (1994).
[CrossRef] [PubMed]

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstructions from confocal scanning microscope images. 2. Depth discrimination versus signal-to-noise ratio in partially confocal images,” Appl. Opt. (to be published).

Diaspro, A.

A. Diaspro, M. Sartore, C. Nicolini, “3D representation of biostructures imaged with an optical microscope,” Image Vis. Computing 8, 130–141 (1990).
[CrossRef]

Diller, K. R.

F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
[CrossRef]

Erhardt, A.

Fay, F. S.

F. S. Fay, W. Carrington, K. E. Fogarty, “Three-dimensional molecular distribution in single cells analysed using the digital imaging microscope,” J. Microsc. 153, 133–149 (1989).
[CrossRef] [PubMed]

Fogarty, K. E.

F. S. Fay, W. Carrington, K. E. Fogarty, “Three-dimensional molecular distribution in single cells analysed using the digital imaging microscope,” J. Microsc. 153, 133–149 (1989).
[CrossRef] [PubMed]

Frieden, B. R.

Gibson, F. S.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 132.

Hansen, E. W.

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstructions from confocal scanning microscope images. 2. Depth discrimination versus signal-to-noise ratio in partially confocal images,” Appl. Opt. (to be published).

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, New York, 1987), pp. 79–125.

Hiraoka, Y.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” 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.

Joshi, S.

Kim, J. J.

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstructions from confocal scanning microscope images. 2. Depth discrimination versus signal-to-noise ratio in partially confocal images,” Appl. Opt. (to be published).

Komitowski, D.

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]

Lanni, F.

F. S. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
[CrossRef]

F. Lanni, G. J. Baxter, “Sampling theorem for square-pixel image data,” in Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 140–147 (1992).
[CrossRef]

Lemasters, J. J.

J. J. Lemasters, E. Chacon, G. Zahrebelski, J. M. Reece, A.-L. Nieminen, “Laser scanning confocal microscopy of living cells,” in Optical Microscopy: Emerging Methods and Applications, B. Herman, J. J. Lemasters, eds. (Academic, San Diego, Calif., 1993), pp. 339–354.

Lichtman, J. W.

Liu, Y. H.

Macias-Garza, F.

F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
[CrossRef]

McNally, J. G.

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for computational optical-sectioning microscopy,” Micron Microsc. Acta 23, 501–513 (1992).
[CrossRef]

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]

Miller, M. I.

Nicolini, C.

A. Diaspro, M. Sartore, C. Nicolini, “3D representation of biostructures imaged with an optical microscope,” Image Vis. Computing 8, 130–141 (1990).
[CrossRef]

Nieminen, A.-L.

J. J. Lemasters, E. Chacon, G. Zahrebelski, J. M. Reece, A.-L. Nieminen, “Laser scanning confocal microscopy of living cells,” in Optical Microscopy: Emerging Methods and Applications, B. Herman, J. J. Lemasters, eds. (Academic, San Diego, Calif., 1993), pp. 339–354.

Ollinger, J. M.

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for computational optical-sectioning microscopy,” Micron Microsc. Acta 23, 501–513 (1992).
[CrossRef]

Preza, C.

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for computational optical-sectioning microscopy,” Micron Microsc. Acta 23, 501–513 (1992).
[CrossRef]

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]

Reece, J. M.

J. J. Lemasters, E. Chacon, G. Zahrebelski, J. M. Reece, A.-L. Nieminen, “Laser scanning confocal microscopy of living cells,” in Optical Microscopy: Emerging Methods and Applications, B. Herman, J. J. Lemasters, eds. (Academic, San Diego, Calif., 1993), pp. 339–354.

Sartore, M.

A. Diaspro, M. Sartore, C. Nicolini, “3D representation of biostructures imaged with an optical microscope,” Image Vis. Computing 8, 130–141 (1990).
[CrossRef]

Sedat, J. W.

Y. Hiraoka, J. W. Sedat, D. A. Agard, “Determination of three-dimensional imaging properties of a light microscope system,” 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]

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]

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]

Streibl, N.

Stroud, R. M.

D. A. Agard, R. M. Stroud, “Linking regions between helices in bacteriorhodopsin revealed,” Biophys. J. 37, 589–602 (1981).

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]

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for computational optical-sectioning microscopy,” Micron Microsc. Acta 23, 501–513 (1992).
[CrossRef]

Weinstein, M.

M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
[CrossRef]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

Young, I. T.

I. T. Young, “Image fidelity: characterizing the imaging transfer function,” Methods Cell Biol. 30, 1–44 (1989).
[CrossRef] [PubMed]

Zahrebelski, G.

J. J. Lemasters, E. Chacon, G. Zahrebelski, J. M. Reece, A.-L. Nieminen, “Laser scanning confocal microscopy of living cells,” in Optical Microscopy: Emerging Methods and Applications, B. Herman, J. J. Lemasters, eds. (Academic, San Diego, Calif., 1993), pp. 339–354.

Zinser, G.

Ann. Rev. Bio-phys. Bioeng. (1)

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

Appl. Opt. (3)

Biophys. J. (2)

D. A. Agard, R. M. Stroud, “Linking regions between helices in bacteriorhodopsin revealed,” Biophys. J. 37, 589–602 (1981).

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

Image Vis. Computing (1)

A. Diaspro, M. Sartore, C. Nicolini, “3D representation of biostructures imaged with an optical microscope,” Image Vis. Computing 8, 130–141 (1990).
[CrossRef]

J. Microsc. (2)

F. S. Fay, W. Carrington, K. E. Fogarty, “Three-dimensional molecular distribution in single cells analysed using the digital imaging microscope,” J. Microsc. 153, 133–149 (1989).
[CrossRef] [PubMed]

F. Macias-Garza, K. R. Diller, A. C. Bovik, S. J. Aggarwal, J. K. Aggarwal, “Improvement in the resolution of three-dimensional data sets collected using optical serial sectioning,” J. Microsc. 153, 205–221 (1989).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

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]

I. T. Young, “Image fidelity: characterizing the imaging transfer function,” Methods Cell Biol. 30, 1–44 (1989).
[CrossRef] [PubMed]

Micron Microsc. Acta (1)

C. Preza, J. M. Ollinger, J. G. McNally, L. J. Thomas, “Point-spread sensitivity analysis for computational optical-sectioning microscopy,” Micron Microsc. Acta 23, 501–513 (1992).
[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 (14)

J.-M. Carazo, “The fidelity of 3D reconstructions from incomplete data and the use of restoration methods,” in Electron Tomography: Three-Dimensional Imaging with the Transmission Electron Microscope (J. Frank, ed. (Plenum, New York, 1992), pp. 117–164.

J. J. Lemasters, E. Chacon, G. Zahrebelski, J. M. Reece, A.-L. Nieminen, “Laser scanning confocal microscopy of living cells,” in Optical Microscopy: Emerging Methods and Applications, B. Herman, J. J. Lemasters, eds. (Academic, San Diego, Calif., 1993), pp. 339–354.

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstructions from confocal scanning microscope images. 2. Depth discrimination versus signal-to-noise ratio in partially confocal images,” Appl. Opt. (to be published).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), p. 132.

K. M. Hanson, “Bayesian and related methods in image reconstruction from incomplete data,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 79–125.

M. Weinstein, K. R. Castleman, “Reconstructing 3-D specimens from 2-D section images,” in Quantitative Imagery in the Biomedical Sciences I, R. E. Herron, ed., Proc. Soc. Photo-Opt. Instrum. Eng.26, 131–138 (1971).
[CrossRef]

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

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

The error in our measurement of distances by determining half-maximal intensities at an edge is affected predominantly by the pixel size. The Rayleigh diffraction limit would contribute to this error only if the width of the bead’s fluorescent shell were comparable with the size of the Airy disk. However, the shell’s thickness is approximately ten times the radius of the Airy disk (0.35 μm), and so the bead’s fluorescent layer can be considered a step. The incoherent step response preserves the location of the edge as measured by the half-intensity point.33

Exact matches of phantom-bead depth and PSF depth were not always possible for experimental PSF’s, because the latter were measured from microspheres embedded at random depths in optical cement. From many such measurements we selected experimental PSF’s at depths as close as possible to those of the phantom beads. Subsequently, for comparing reconstructions having these experimental PSF’s with those having theoretical PSF’s, we computed each theoretical PSF at the same depth as that of the experimental PSF, so as not to give the theoretical PSF’s an advantage.

F. Lanni, G. J. Baxter, “Sampling theorem for square-pixel image data,” in Biomedical Image Processing and Three-Dimensional Microscopy, R. S. Acharya, C. J. Cogswell, D. B. Goldgof, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1660, 140–147 (1992).
[CrossRef]

W. A. Carrington, “Image-restoration in 3D 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]

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]

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1964).

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Figures (12)

Fig. 1
Fig. 1

Schematic of the bead phantom imaged with an inverted microscope. The dashed boxes represent the two 128 × 128 × 128 data sets examined and also indicate the orientation used for all subsequent xz sections: the top row of pixels is farthest from the objective and deepest in the phantom.

Fig. 2
Fig. 2

xz medial sections of (a,c) experimental and (b,d) theoretical PSF’s for (a,b) a 20×/0.7-N.A. dry lens and (c,d) a 40×/1.0-N.A. oil lens measured under design conditions. A logarithmic intensity scale (see Section 2) was used to enhance small values on the PSF tails. Scale bar, 3 μm.

Fig. 3
Fig. 3

The profiles are taken from the images of Fig. 2. Intensities of theoretical and experimental PSF’s are indicated by dotted and solid curves, respectively. The axial profiles intersect the PSF point-source location, whereas the transverse profiles lie 6.5 and 3.5 μm above focus for the 20× and the 40× lenses, respectively. The curves agree quantitatively, except that high-frequency oscillations in the theoretical PSF cannot be detected in the experimental PSF because of both the coarse spatial sampling and the noise present in the experimental data. Spikes in the tails of experimental PSF’s indicate the height of the noise floor.

Fig. 4
Fig. 4

xz medial sections of (a–e) experimental and (f–j) theoretical PSF’s for a 20×/0.7-N.A. dry lens focused at the indicated depths in a medium of n = 1.56. For both experiment and theory increasing depth increases the width of the central peak axially and also generates an increasingly distinct inverted Y profile‥ For theoretical PSF’s the actual location of the point source corresponds to the center of each image, thereby demonstrating the apparent axial shift introduced by increasing spherical aberration (f–j). Scale bar, 8 μm.

Fig. 5
Fig. 5

xz medial sections of (a–c) experimental and (d–f) theoretical PSF’s for cover slips at the indicated nondesign thicknesses. For both experiment and theory an upright Y profile is transformed to an inverted Y profile as cover slips change from thinner to thicker than the design thickness of 0.17 mm (Figs. 4a and 4f show design PSF’s for this lens). Scale bar, 8 μm.

Fig. 6
Fig. 6

Bead images obtained by (a,b) confocal microscopy and (c) physical sectioning, in all cases with use of a 1.0-N.A. oil-immersion lens. For the confocal image, (a) xy and (b) xz medial sections are shown. Scale bar, 3 μm.

Fig. 7
Fig. 7

(a–d) xy and (e–h) xz medial sections of bead images obtained by conventional fluorescence microscopy (a, e) and then reconstructed by the regularized linear least-squares method with (b, f) an experimental PSF and (c, g) a theoretical PSF. A 40×/1.0-N.A. oil-immersion objective was used. The analytically determined (see text) best-possible linear reconstruction of a hollow fluorescent shell is quite similar (d, h). The reconstructed images presented in this and all subsequent figures contain negative intensities (see Fig. 8 below), because the linear least-squares estimation procedure does not impose a positivity constraint on the estimated intensities. Negative intensities have been retained in the displayed images in order to preserve as much information as possible about the restoration procedure. In practice, negative values can be set to zero for improved contrast in the images. Scale bar, 3 μm.

Fig. 8
Fig. 8

Representative intensity profiles through selected portions of the images in Fig. 7g. a, Axial profiles were generated by plotting intensities from a column of pixels in Fig. 7g that passed through the bead center (solid curve) or at 1.7 μm left of center (dotted curve), b, Transverse profiles were generated by plotting intensities from a row of pixels in Fig. 7g that passed through the bead center (solid curve) or at 4 μm up from center (dotted curve). The plots show the extent of negative intensities generated by the estimation procedure. The largest negative undershoots occur at the poles and the equator (solid curves). Elsewhere the undershoots are significantly smaller (dotted curves). In all the reconstructed images displayed, negative numbers are included in the linear gray scale.

Fig. 9
Fig. 9

(a–d) xy and (e–h) xz medial sections of reconstructed images of beads located in the phantom at the indicated depths. Images were obtained with a 20×/0.7-N.A. dry lens and reconstructed with an experimental PSF measured under design conditions (see Fig. 4a). Note that the upper half of the bead’s xz profile becomes progressively degraded with depth, whereas little change is observed in the medial xy profile. Scale bar, 8 μm.

Fig. 10
Fig. 10

xz medial sections of reconstructed images of phantom beads at the indicated depths with experimental PSF’s at the indicated depths. The 20× /0.7-N.A. dry lens was used. The best reconstructions for each bead occur along the diagonal (a, e, i), where the bead’s depth is closest to the PSF’s depth. Above this diagonal (b,c, f), where bead images were reconstructed with deeper PSF’s, the lower half of the bead image is degraded. The opposite holds below the diagonal (d, g, h). Scale bar, 8 μm.

Fig. 11
Fig. 11

xz medial sections of reconstructed images of beads at the indicated depths with theoretical PSF’s at the indicated depths. Imaging conditions as in Fig. 10. The reconstructions are similar to those of Fig. 10, except that with theoretical PSF’s the signal-to-noise ratio is improved, and sensitivity to mismatch of object and PSF depth is reduced. Scale bar, 8 μm.

Fig. 12
Fig. 12

xz medial sections of (a) a simulated torus and (d) polar caps, (b, e) their simulated images obtained by convolution with a theoretical PSF for a 40×/1.0-N.A. lens, and (c, f) the best-possible linear reconstructions. The simulated objects (a,c) sum to yield a simulated 10-μm bead, namely, a uniformly intense shell of fluorescence with an outer diameter of 10 μm. and an inner diameter of 8 μm, and zero fluorescence elsewhere. Both the simulated images (b, e) and their reconstructions (c, f) reveal artifacts. Most notably, in the polar-cap images it is difficult to discern the location of the caps (e, f), whereas in the torus images (b, c) caps appear to be present. These results demonstrate that, in the reconstructions of the actual or the simulated bead (see Fig. 7), the apparent axial elongation arises not from blurring at the poles but rather primarily from residual out-of-focus light originally emanating from more-equatorial regions. Scale bar, 4 μm.

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