Abstract

We present an approach for the combined restoration of multiple different images of a single object. A linear Tikhonov filter adapted for this purpose is derived in detail. Nonlinear constrained algorithms can also be adapted, and we illustrate this possibility for an iterative constrained Tikhonov algorithm. Both the linear and the iterative constrained Tikhonov algorithms were used to analyze performance in fluorescence confocal imaging by use of simulated and experimental data. One can improve the quality of restored confocal images significantly if the signal that normally is rejected by the detection pinhole of a confocal laser scanning microscope is also recorded on a separate detector such that the two recorded signals are used together for image restoration according to the proposed algorithms.

© 1998 Optical Society of America

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  1. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).
  2. C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik (Stuttgart) 74, 128–129 (1986).
  3. C. J. R. Sheppard, “Axial resolution of confocal fluorescence microscopy,” J. Microsc. 154, 237–241 (1989).
    [CrossRef]
  4. C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
    [CrossRef]
  5. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” Meth. Cell Biol. 30, 353–377 (1989).
    [CrossRef]
  6. W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
    [CrossRef] [PubMed]
  7. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  8. T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
    [CrossRef]
  9. 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).
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  11. J.-A. Conchello, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope images. 1: deterministic and maximum likelihood reconstructions,” Appl. Opt. 29, 3795–3804 (1990).
    [CrossRef] [PubMed]
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    [CrossRef]
  13. M. Schrader, S. W. Hell, H. T. M. van der Voort, “Potential of confocal microscopes to resolve in the 50–100-nm range,” Appl. Phys. Lett. 69, 3644–3646 (1996).
    [CrossRef]
  14. G. M. P. van Kempen, L. J. van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. 185, 354–365 (1997).
    [CrossRef]
  15. P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. van Vliet, T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. 189, 192–198 (1998).
    [CrossRef]
  16. Q. S. Hanley, P. J. Verveer, T. M. Jovin, “Optical sectioning fluorescence spectroscopy in a programmable array microscope (PAM),” Appl. Spectrosc. 52, 783–789 (1998).
    [CrossRef]
  17. P. J. Verveer, T. M. Jovin, “Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy,” J. Opt. Soc. Am. A 14, 1696–1706 (1997).
    [CrossRef]
  18. P. J. Verveer, T. M. Jovin, “Efficient image restoration based on Good’s roughness penalty with application to fluorescence microscopy,” J. Opt. Soc. Am. A 15, 1077–1083 (1998).
    [CrossRef]
  19. D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
    [CrossRef]
  20. I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Statist. 19, 2032–2066 (1991).
    [CrossRef]
  21. P. J. Verveer, G. M. P. van Kempen, T. M. Jovin, “Super-resolution MAP algorithms applied to fluorescence imaging,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 125–135 (1997).
    [CrossRef]
  22. P. J. Verveer, T. M. Jovin, “Acceleration of the ICTM image restoration algorithm,” J. Microsc. 188, 191–195 (1997).
    [CrossRef]
  23. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  24. W. A. Carrington, “Image restoration in 3D microscopy with limited data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. SPIE1205, 72–83 (1990).
    [CrossRef]
  25. G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
    [CrossRef]
  26. N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 1, 322–336 (1992).
    [CrossRef] [PubMed]
  27. W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  28. H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in a high-aperture fluorescence confocal microscope: a numerical analysis,” J. Microsc. 158, 43–54 (1990).
    [CrossRef]
  29. A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1989).
  30. P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).
    [CrossRef]

1998 (3)

1997 (3)

P. J. Verveer, T. M. Jovin, “Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy,” J. Opt. Soc. Am. A 14, 1696–1706 (1997).
[CrossRef]

G. M. P. van Kempen, L. J. van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. 185, 354–365 (1997).
[CrossRef]

P. J. Verveer, T. M. Jovin, “Acceleration of the ICTM image restoration algorithm,” J. Microsc. 188, 191–195 (1997).
[CrossRef]

1996 (1)

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Potential of confocal microscopes to resolve in the 50–100-nm range,” Appl. Phys. Lett. 69, 3644–3646 (1996).
[CrossRef]

1995 (2)

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

H. T. M. van der Voort, K. C. Strasters, “Restoration of confocal images for quantitative image analysis,” J. Microsc. 178, 165–181 (1995).
[CrossRef]

1994 (1)

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).
[CrossRef]

1993 (1)

1992 (3)

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 1, 322–336 (1992).
[CrossRef] [PubMed]

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

1991 (1)

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Statist. 19, 2032–2066 (1991).
[CrossRef]

1990 (2)

H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in a high-aperture fluorescence confocal microscope: a numerical analysis,” J. Microsc. 158, 43–54 (1990).
[CrossRef]

J.-A. Conchello, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope images. 1: deterministic and maximum likelihood reconstructions,” Appl. Opt. 29, 3795–3804 (1990).
[CrossRef] [PubMed]

1989 (3)

T. Wilson, “Optical sectioning in confocal fluorescent microscopes,” J. Microsc. 154, 143–156 (1989).
[CrossRef]

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

C. J. R. Sheppard, “Axial resolution of confocal fluorescence microscopy,” J. Microsc. 154, 237–241 (1989).
[CrossRef]

1988 (1)

1986 (2)

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik (Stuttgart) 74, 128–129 (1986).

1979 (1)

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Agard, D. A.

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

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Benedetti, P. A.

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).
[CrossRef]

Bhattacharyya, S.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Brakenhoff, G. J.

H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in a high-aperture fluorescence confocal microscope: a numerical analysis,” J. Microsc. 158, 43–54 (1990).
[CrossRef]

Carrington, W. A.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

W. A. Carrington, “Image restoration in 3D microscopy with limited data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. SPIE1205, 72–83 (1990).
[CrossRef]

Conchello, J.-A.

Cooper, J. A.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Csiszár, I.

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Statist. 19, 2032–2066 (1991).
[CrossRef]

Evangelista, V.

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).
[CrossRef]

Fay, F. S.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Fogarty, K. E.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Galatsanos, N. P.

N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 1, 322–336 (1992).
[CrossRef] [PubMed]

Golub, G. H.

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Gu, M.

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

Guidarini, D.

P. A. Benedetti, V. Evangelista, D. Guidarini, S. Vestri, “Achieving confocal-point performance in confocal-line microscopy,” Bioimaging 2, 122–130 (1994).
[CrossRef]

Hanley, Q. S.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. van Vliet, T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. 189, 192–198 (1998).
[CrossRef]

Q. S. Hanley, P. J. Verveer, T. M. Jovin, “Optical sectioning fluorescence spectroscopy in a programmable array microscope (PAM),” Appl. Spectrosc. 52, 783–789 (1998).
[CrossRef]

Hansen, E. W.

Hanzel, D.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Heath, M.

G. H. Golub, M. Heath, G. Wahba, “Generalized cross-validation as a method for choosing a good ridge parameter,” Technometrics 21, 215–223 (1979).
[CrossRef]

Hell, S. W.

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Potential of confocal microscopes to resolve in the 50–100-nm range,” Appl. Phys. Lett. 69, 3644–3646 (1996).
[CrossRef]

Hiraoka, Y.

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

Holmes, T. J.

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

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Isenberg, G.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall, Englewood Cliffs, N.J., 1989).

Joshi, S.

Jovin, T. M.

Q. S. Hanley, P. J. Verveer, T. M. Jovin, “Optical sectioning fluorescence spectroscopy in a programmable array microscope (PAM),” Appl. Spectrosc. 52, 783–789 (1998).
[CrossRef]

P. J. Verveer, T. M. Jovin, “Efficient image restoration based on Good’s roughness penalty with application to fluorescence microscopy,” J. Opt. Soc. Am. A 15, 1077–1083 (1998).
[CrossRef]

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. van Vliet, T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. 189, 192–198 (1998).
[CrossRef]

P. J. Verveer, T. M. Jovin, “Efficient superresolution restoration algorithms using maximum a posteriori estimations with application to fluorescence microscopy,” J. Opt. Soc. Am. A 14, 1696–1706 (1997).
[CrossRef]

P. J. Verveer, T. M. Jovin, “Acceleration of the ICTM image restoration algorithm,” J. Microsc. 188, 191–195 (1997).
[CrossRef]

P. J. Verveer, G. M. P. van Kempen, T. M. Jovin, “Super-resolution MAP algorithms applied to fluorescence imaging,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 125–135 (1997).
[CrossRef]

Katsaggelos, A. K.

N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 1, 322–336 (1992).
[CrossRef] [PubMed]

Krishnamurthi, V.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Lin, W.-C.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Lynch, R. M.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

Miller, M. I.

Moore, E. D. W.

W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
[CrossRef] [PubMed]

O’Sullivan, J. A.

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Roysam, B.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Schrader, M.

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Potential of confocal microscopes to resolve in the 50–100-nm range,” Appl. Phys. Lett. 69, 3644–3646 (1996).
[CrossRef]

Schulz, T. J.

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Sedat, J. W.

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

Shaw, P.

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

Sheppard, C. J. R.

C. J. R. Sheppard, M. Gu, “The significance of 3-D transfer functions in confocal scanning microscopy,” J. Microsc. 165, 377–390 (1992).
[CrossRef]

C. J. R. Sheppard, “Axial resolution of confocal fluorescence microscopy,” J. Microsc. 154, 237–241 (1989).
[CrossRef]

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging,” Optik (Stuttgart) 72, 131–133 (1986).

C. J. R. Sheppard, “The spatial frequency cut-off in three-dimensional imaging II,” Optik (Stuttgart) 74, 128–129 (1986).

Snyder, D. L.

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Strasters, K. C.

H. T. M. van der Voort, K. C. Strasters, “Restoration of confocal images for quantitative image analysis,” J. Microsc. 178, 165–181 (1995).
[CrossRef]

Szarowski, D. H.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Tikhonov, A. N.

A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).

Turner, J. N.

T. J. Holmes, S. Bhattacharyya, J. A. Cooper, D. Hanzel, V. Krishnamurthi, W.-C. Lin, B. Roysam, D. H. Szarowski, J. N. Turner, “Light microscopic images reconstructed by maximum likelihood deconvolution,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), Chap. 24, pp. 389–402.
[CrossRef]

van der Voort, H. T. M.

G. M. P. van Kempen, L. J. van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. 185, 354–365 (1997).
[CrossRef]

M. Schrader, S. W. Hell, H. T. M. van der Voort, “Potential of confocal microscopes to resolve in the 50–100-nm range,” Appl. Phys. Lett. 69, 3644–3646 (1996).
[CrossRef]

H. T. M. van der Voort, K. C. Strasters, “Restoration of confocal images for quantitative image analysis,” J. Microsc. 178, 165–181 (1995).
[CrossRef]

H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in a high-aperture fluorescence confocal microscope: a numerical analysis,” J. Microsc. 158, 43–54 (1990).
[CrossRef]

van Kempen, G. M. P.

G. M. P. van Kempen, L. J. van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. 185, 354–365 (1997).
[CrossRef]

P. J. Verveer, G. M. P. van Kempen, T. M. Jovin, “Super-resolution MAP algorithms applied to fluorescence imaging,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 125–135 (1997).
[CrossRef]

van Vliet, L. J.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. van Vliet, T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. 189, 192–198 (1998).
[CrossRef]

G. M. P. van Kempen, L. J. van Vliet, P. J. Verveer, H. T. M. van der Voort, “A quantitative comparison of image restoration methods for confocal microscopy,” J. Microsc. 185, 354–365 (1997).
[CrossRef]

Verbeek, P. W.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. van Vliet, T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. 189, 192–198 (1998).
[CrossRef]

Verveer, P. J.

P. J. Verveer, Q. S. Hanley, P. W. Verbeek, L. J. van Vliet, T. M. Jovin, “Theory of confocal fluorescence imaging in the programmable array microscope (PAM),” J. Microsc. 189, 192–198 (1998).
[CrossRef]

P. J. Verveer, T. M. Jovin, “Efficient image restoration based on Good’s roughness penalty with application to fluorescence microscopy,” J. Opt. Soc. Am. A 15, 1077–1083 (1998).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

MSE as a function of the relative total intensity of the confocal signal compared with the nonconfocal image. The errors are shown for the linear Tikhonov restorations of the confocal image, the nonconfocal image, and the combined linear Tikhonov restoration of the confocal and difference images.

Fig. 2
Fig. 2

MSE as a function of the relative total intensity of the confocal signal compared with the nonconfocal image. The errors are shown for the iterative constrained Tikhonov restorations of the confocal image, the nonconfocal image, and the combined iterative constrained Tikhonov restoration of the confocal and difference images.

Fig. 3
Fig. 3

Restorations of a confocal and a nonconfocal image of nuclear envelopes of a Drosophila melanogaster embryo: (a) nonconfocal data, (b) confocal data, (c) MAPGG restoration of the nonconfocal data, (d) MAPGG restoration of the confocal data, and (e) combined MAPGG restoration.

Equations (27)

Equations on this page are rendered with MathJax. Learn more.

g i x ,   y ,   z = N i - +   h i x ,   y ,   z ,   u ,   v ,   w f u ,   v ,   w d u d v d w , i = 1 , ,   N ,
g i = H i f + n i ,     i = 1 , ,   N ,
g = Hf + n ,
g = c 1 g 1   c 2 g 2     c N g N T .
H = c 1 H 1   c 2 H 2     c N H N T ,
n = c 1 n 1   c 2 n 2     c N n N T ,
ϕ = L Hf ,   g + γ P f ,
ϕ = H i f - g i 2 + γ i Cf 2 ,
f ˆ i = H i T g i H i T H i + γ i C T C = A i γ i g i .
GCV γ i = I - H i A i γ i g i 2 trace I - H i A i γ i 2 .
ϕ = Hf - g 2 + γ Cf 2 = i = 1 N   c i 2 H i f - g i 2 + γ Cf 2 .
f ˆ γ = i = 1 N   c i 2 H i T g i j = 1 N   c j 2 H j T H j + γ C T C = B γ g ,
B γ = B 1 γ B 2 γ     B N γ ,
B i γ = c i H i T j = 1 N   c j 2 H j T H j + γ C T C .
GCV γ = i = 1 N   c i 2 g i - H i f ˆ γ 2 i = 1 N trace I - c i H i B i γ 2 .
ϕ = Hx 2 - g 2 + γ Cx 2 2 .
ϕ = i = 0 N   c i 2 H i x 2 - g i 2 + γ Cx 2 2 .
ϕ x = 4 X i = 0 N   c i 2 H i T H i x 2 - H i T g i + γ C T Cx 2 ,
x k + 1 = x k + α k d k ,
ϕ x k + α d k = p α 4 + q α 3 + r α 2 + s α + t ,
p = d 2 T Ud 2 ,
q = 4 d 2 T UXd ,
r = 4 d T XUXd + 2 d 2 T Ux 2 - v ,
s = 4 d T X Ux 2 - v ,
t = x 2 T Ux 2 - 2 x 2 T v + w ,
c i     1 σ i .
σ i 2 = I - H i A i γ i g i 2 trace I - H i A i γ i .

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