Abstract

We present efficient algorithms for image restoration by using the maximum a posteriori (MAP) method. Assuming Gaussian or Poisson statistics for the noise and either a Gaussian or an entropy prior distribution for the image, corresponding functionals are formulated and minimized to produce MAP estimations. Efficient algorithms are presented for finding the minimum of these functionals in the presence of nonnegativity and support constraints. Performance was tested by using simulated three-dimensional (3-D) imaging with a fluorescence confocal laser scanning microscope. Results are compared with those from two existing algorithms for superresolution in fluorescence imaging. An example is given of the restoration of a 3-D confocal image of a biological specimen.

© 1997 Optical Society of America

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  1. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” in Methods in Cell Biology, D. L. Taylor, Y. Wang, eds. (Academic, San Diego, Calif., 1989), Vol. 3, pp. 353–377.
  2. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).
  3. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  4. T. J. Holmes, Y. Liu, “Acceleration of maximum-likelihood image restoration for fluorescence microscopy and other noncoherent imagery,” J. Opt. Soc. Am. A 8, 893–907 (1991).
    [CrossRef]
  5. W. 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]
  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. R. L. Lagendijk, “Iterative identification and restoration of images,” Ph.D. dissetation (Delft Technical University, Delft, The Netherlands, 1990).
  8. H. T. M. van der Voort, K. C. Strasters, “Restoration of confocal images for quantitative image analysis,” J. Microsc. 178, 165–181 (1995).
    [CrossRef]
  9. B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
    [CrossRef]
  10. B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
  11. J. Skilling, “Classic maximum entropy,” in Maximum Entropy and Bayesian Methods, J. Skilling, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 45–52.
  12. T. M. Jovin, D. J. Arndt-Jovin, “Luminescence digital imaging microscopy,” Annu. Rev. Biophys. Chem. 18, 271–308 (1989).
    [PubMed]
  13. M. Bertero, P. Boccacci, “Regularization methods in im- age restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
  14. A. N. Tikhonov, V. Y. Arsenin, Solutions of Ill-Posed Problems (Wiley, New York, 1977).
  15. K. Miller, “Least-squares method for ill-posed problems with prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
    [CrossRef]
  16. H.-M. Adorf, R. N. Hook, L. B. Lucy, “HST image restoration developments at the ST-ETF,” Int. J. Imaging Syst. Technol. 6, 339–349 (1995).
  17. D. L. Snyder, J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
    [CrossRef]
  18. I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2033–2066 (1991).
  19. H. J. Trussell, B. R. Hunt, “Improved methods of maximum a posteriori restoration,” IEEE Trans. Comput. C-27, 57–62 (1979).
    [CrossRef]
  20. K. M. Hanson, “Bayesian and related methods in image reconstruction from incomplete data,” in Image Recovery: Theory and Application, H. Stark, ed. (Academic, San Diego, Calif., 1987), pp. 79–125.
  21. 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]
  22. S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
    [CrossRef]
  23. A. M. Thompson, J. C. Brown, J. W. Kay, D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 326–339 (1991).
    [CrossRef]
  24. A. K. Katsaggelos, J. Biemond, R. W. Schafer, R. M. Mersereau, “A regularized iterative image restoration algorithm,” IEEE Trans. Acoust. Speech Signal Process. 39, 913–929 (1991).
  25. 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]
  26. G. M. P. van Kempen, H. T. M. van der Voort, J. G. J. Bauman, K. C. Strasters, “Comparing maximum likelihood estimation and constrained Tikhonov–Miller restoration,” IEEE Eng. Med. Biol. Mag. 15, 76–83 (1996).
    [CrossRef]
  27. P. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).
  28. C. J. R. Sheppard, X. Gan, M. Gu, M. Roy, “Signal-to-noise in confocal microscopes,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 363–371.
  29. 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]
  30. S. I. Olsen, “Estimation of noise in images: an evaluation,” Comput. Vis. Graph. Image Process. 55, 319–323 (1993).
  31. S. F. Gull, “Developments in maximum entropy data analysis,” in Maximum Entropy and Bayesian Methods, J. Skilling, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 53–71.

1997 (1)

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]

1996 (2)

G. M. P. van Kempen, H. T. M. van der Voort, J. G. J. Bauman, K. C. Strasters, “Comparing maximum likelihood estimation and constrained Tikhonov–Miller restoration,” IEEE Eng. Med. Biol. Mag. 15, 76–83 (1996).
[CrossRef]

P. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).

1995 (5)

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]

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).

M. Bertero, P. Boccacci, “Regularization methods in im- age restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).

H.-M. Adorf, R. N. Hook, L. B. Lucy, “HST image restoration developments at the ST-ETF,” Int. J. Imaging Syst. Technol. 6, 339–349 (1995).

1994 (1)

B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
[CrossRef]

1993 (1)

S. I. Olsen, “Estimation of noise in images: an evaluation,” Comput. Vis. Graph. Image Process. 55, 319–323 (1993).

1992 (2)

D. L. Snyder, J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (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]

1991 (4)

A. M. Thompson, J. C. Brown, J. W. Kay, D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 326–339 (1991).
[CrossRef]

A. K. Katsaggelos, J. Biemond, R. W. Schafer, R. M. Mersereau, “A regularized iterative image restoration algorithm,” IEEE Trans. Acoust. Speech Signal Process. 39, 913–929 (1991).

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2033–2066 (1991).

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

1990 (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]

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[CrossRef]

1989 (1)

T. M. Jovin, D. J. Arndt-Jovin, “Luminescence digital imaging microscopy,” Annu. Rev. Biophys. Chem. 18, 271–308 (1989).
[PubMed]

1988 (1)

1979 (1)

H. J. Trussell, B. R. Hunt, “Improved methods of maximum a posteriori restoration,” IEEE Trans. Comput. C-27, 57–62 (1979).
[CrossRef]

1977 (1)

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

1970 (1)

K. Miller, “Least-squares method for ill-posed problems with prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Adorf, H.-M.

H.-M. Adorf, R. N. Hook, L. B. Lucy, “HST image restoration developments at the ST-ETF,” Int. J. Imaging Syst. Technol. 6, 339–349 (1995).

Agard, D. A.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” in Methods in Cell Biology, D. L. Taylor, Y. Wang, eds. (Academic, San Diego, Calif., 1989), Vol. 3, pp. 353–377.

Arndt-Jovin, D. J.

T. M. Jovin, D. J. Arndt-Jovin, “Luminescence digital imaging microscopy,” Annu. Rev. Biophys. Chem. 18, 271–308 (1989).
[PubMed]

Arsenin, V. Y.

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

Bastiaens, P. I. H.

P. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).

Bauman, J. G. J.

G. M. P. van Kempen, H. T. M. van der Voort, J. G. J. Bauman, K. C. Strasters, “Comparing maximum likelihood estimation and constrained Tikhonov–Miller restoration,” IEEE Eng. Med. Biol. Mag. 15, 76–83 (1996).
[CrossRef]

Bertero, M.

M. Bertero, P. Boccacci, “Regularization methods in im- age restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).

Biemond, J.

A. K. Katsaggelos, J. Biemond, R. W. Schafer, R. M. Mersereau, “A regularized iterative image restoration algorithm,” IEEE Trans. Acoust. Speech Signal Process. 39, 913–929 (1991).

Boccacci, P.

M. Bertero, P. Boccacci, “Regularization methods in im- age restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).

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]

Brown, J. C.

A. M. Thompson, J. C. Brown, J. W. Kay, D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 326–339 (1991).
[CrossRef]

Carrington, W.

W. 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]

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]

Csiszár, I.

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2033–2066 (1991).

Dempster, A. P.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

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]

Gan, X.

C. J. R. Sheppard, X. Gan, M. Gu, M. Roy, “Signal-to-noise in confocal microscopes,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 363–371.

Gu, M.

C. J. R. Sheppard, X. Gan, M. Gu, M. Roy, “Signal-to-noise in confocal microscopes,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 363–371.

Gull, S. F.

S. F. Gull, “Developments in maximum entropy data analysis,” in Maximum Entropy and Bayesian Methods, J. Skilling, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 53–71.

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, San Diego, Calif., 1987), pp. 79–125.

Hiraoka, Y.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” in Methods in Cell Biology, D. L. Taylor, Y. Wang, eds. (Academic, San Diego, Calif., 1989), Vol. 3, pp. 353–377.

Holmes, T. J.

Hook, R. N.

H.-M. Adorf, R. N. Hook, L. B. Lucy, “HST image restoration developments at the ST-ETF,” Int. J. Imaging Syst. Technol. 6, 339–349 (1995).

Hunt, B. R.

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).

B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
[CrossRef]

H. J. Trussell, B. R. Hunt, “Improved methods of maximum a posteriori restoration,” IEEE Trans. Comput. C-27, 57–62 (1979).
[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]

Jovin, T. M.

P. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).

T. M. Jovin, D. J. Arndt-Jovin, “Luminescence digital imaging microscopy,” Annu. Rev. Biophys. Chem. 18, 271–308 (1989).
[PubMed]

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]

A. K. Katsaggelos, J. Biemond, R. W. Schafer, R. M. Mersereau, “A regularized iterative image restoration algorithm,” IEEE Trans. Acoust. Speech Signal Process. 39, 913–929 (1991).

Kay, J. W.

A. M. Thompson, J. C. Brown, J. W. Kay, D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 326–339 (1991).
[CrossRef]

Lagendijk, R. L.

R. L. Lagendijk, “Iterative identification and restoration of images,” Ph.D. dissetation (Delft Technical University, Delft, The Netherlands, 1990).

Laird, N. M.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

Liu, Y.

Lucy, L. B.

H.-M. Adorf, R. N. Hook, L. B. Lucy, “HST image restoration developments at the ST-ETF,” Int. J. Imaging Syst. Technol. 6, 339–349 (1995).

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]

Majoul, I. V.

P. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).

Mersereau, R. M.

A. K. Katsaggelos, J. Biemond, R. W. Schafer, R. M. Mersereau, “A regularized iterative image restoration algorithm,” IEEE Trans. Acoust. Speech Signal Process. 39, 913–929 (1991).

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[CrossRef]

Miller, K.

K. Miller, “Least-squares method for ill-posed problems with prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

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, J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Olsen, S. I.

S. I. Olsen, “Estimation of noise in images: an evaluation,” Comput. Vis. Graph. Image Process. 55, 319–323 (1993).

Reeves, S. J.

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[CrossRef]

Roy, M.

C. J. R. Sheppard, X. Gan, M. Gu, M. Roy, “Signal-to-noise in confocal microscopes,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 363–371.

Rubin, D. B.

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

Schafer, R. W.

A. K. Katsaggelos, J. Biemond, R. W. Schafer, R. M. Mersereau, “A regularized iterative image restoration algorithm,” IEEE Trans. Acoust. Speech Signal Process. 39, 913–929 (1991).

Schulz, J.

D. L. Snyder, 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,” in Methods in Cell Biology, D. L. Taylor, Y. Wang, eds. (Academic, San Diego, Calif., 1989), Vol. 3, pp. 353–377.

Shaw, P.

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” in Methods in Cell Biology, D. L. Taylor, Y. Wang, eds. (Academic, San Diego, Calif., 1989), Vol. 3, pp. 353–377.

Sheppard, C. J. R.

C. J. R. Sheppard, X. Gan, M. Gu, M. Roy, “Signal-to-noise in confocal microscopes,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 363–371.

Skilling, J.

J. Skilling, “Classic maximum entropy,” in Maximum Entropy and Bayesian Methods, J. Skilling, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 45–52.

Snyder, D. L.

D. L. Snyder, J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Söling, H.-D.

P. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).

Strasters, K. C.

G. M. P. van Kempen, H. T. M. van der Voort, J. G. J. Bauman, K. C. Strasters, “Comparing maximum likelihood estimation and constrained Tikhonov–Miller restoration,” IEEE Eng. Med. Biol. Mag. 15, 76–83 (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]

Thompson, A. M.

A. M. Thompson, J. C. Brown, J. W. Kay, D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 326–339 (1991).
[CrossRef]

Tikhonov, A. N.

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

Titterington, D. M.

A. M. Thompson, J. C. Brown, J. W. Kay, D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 326–339 (1991).
[CrossRef]

Trussell, H. J.

H. J. Trussell, B. R. Hunt, “Improved methods of maximum a posteriori restoration,” IEEE Trans. Comput. C-27, 57–62 (1979).
[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]

G. M. P. van Kempen, H. T. M. van der Voort, J. G. J. Bauman, K. C. Strasters, “Comparing maximum likelihood estimation and constrained Tikhonov–Miller restoration,” IEEE Eng. Med. Biol. Mag. 15, 76–83 (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]

G. M. P. van Kempen, H. T. M. van der Voort, J. G. J. Bauman, K. C. Strasters, “Comparing maximum likelihood estimation and constrained Tikhonov–Miller restoration,” IEEE Eng. Med. Biol. Mag. 15, 76–83 (1996).
[CrossRef]

van Vliet, L. J.

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]

Verveer, P. J.

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. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).

Ann. Stat. (1)

I. Csiszár, “Why least squares and maximum entropy? An axiomatic approach to inference for linear inverse problems,” Ann. Stat. 19, 2033–2066 (1991).

Annu. Rev. Biophys. Chem. (1)

T. M. Jovin, D. J. Arndt-Jovin, “Luminescence digital imaging microscopy,” Annu. Rev. Biophys. Chem. 18, 271–308 (1989).
[PubMed]

Comput. Vis. Graph. Image Process. (1)

S. I. Olsen, “Estimation of noise in images: an evaluation,” Comput. Vis. Graph. Image Process. 55, 319–323 (1993).

EMBO J. (1)

P. I. H. Bastiaens, I. V. Majoul, P. J. Verveer, H.-D. Söling, T. M. Jovin, “Imaging the intracellular trafficking and state of the AB5 quaternary structure of cholera toxin,” EMBO J. 15, 4246–4253 (1996).

IEEE Eng. Med. Biol. Mag. (1)

G. M. P. van Kempen, H. T. M. van der Voort, J. G. J. Bauman, K. C. Strasters, “Comparing maximum likelihood estimation and constrained Tikhonov–Miller restoration,” IEEE Eng. Med. Biol. Mag. 15, 76–83 (1996).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

A. K. Katsaggelos, J. Biemond, R. W. Schafer, R. M. Mersereau, “A regularized iterative image restoration algorithm,” IEEE Trans. Acoust. Speech Signal Process. 39, 913–929 (1991).

IEEE Trans. Comput. (1)

H. J. Trussell, B. R. Hunt, “Improved methods of maximum a posteriori restoration,” IEEE Trans. Comput. C-27, 57–62 (1979).
[CrossRef]

IEEE Trans. Image Process. (1)

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]

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

A. M. Thompson, J. C. Brown, J. W. Kay, D. M. Titterington, “A study of methods of choosing the smoothing parameter in image restoration by regularization,” IEEE Trans. Pattern Anal. Mach. Intell. 13, 326–339 (1991).
[CrossRef]

IEEE Trans. Signal Process. (1)

D. L. Snyder, J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Int. J. Imaging Syst. Technol. (3)

H.-M. Adorf, R. N. Hook, L. B. Lucy, “HST image restoration developments at the ST-ETF,” Int. J. Imaging Syst. Technol. 6, 339–349 (1995).

M. Bertero, P. Boccacci, “Regularization methods in im- age restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).

B. R. Hunt, “Super-resolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).

Int. J. Mod. Phys. C (1)

B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
[CrossRef]

J. Microsc. (3)

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]

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]

J. Opt. Soc. Am. A (2)

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

A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).

Opt. Eng. (1)

S. J. Reeves, R. M. Mersereau, “Optimal estimation of the regularization parameter and stabilizing functional for regularized image restoration,” Opt. Eng. 29, 446–454 (1990).
[CrossRef]

Science (1)

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]

SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. (1)

K. Miller, “Least-squares method for ill-posed problems with prescribed bound,” SIAM (Soc. Ind. Appl. Math.) J. Math. Anal. 1, 52–74 (1970).
[CrossRef]

Other (8)

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

J. Skilling, “Classic maximum entropy,” in Maximum Entropy and Bayesian Methods, J. Skilling, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 45–52.

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

R. L. Lagendijk, “Iterative identification and restoration of images,” Ph.D. dissetation (Delft Technical University, Delft, The Netherlands, 1990).

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” in Methods in Cell Biology, D. L. Taylor, Y. Wang, eds. (Academic, San Diego, Calif., 1989), Vol. 3, pp. 353–377.

W. 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]

C. J. R. Sheppard, X. Gan, M. Gu, M. Roy, “Signal-to-noise in confocal microscopes,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 363–371.

S. F. Gull, “Developments in maximum entropy data analysis,” in Maximum Entropy and Bayesian Methods, J. Skilling, ed. (Kluwer Academic, Dordrecht, The Netherlands, 1989), pp. 53–71.

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Figures (8)

Fig. 1
Fig. 1

Objects that are used in the simulations: (a) test object; (b) test object, blurred by simulated confocal PSF; (c) blurred object distorted with Gaussian noise with SNR=45; (d) blurred object distorted with Poisson noise with β=10. A single slice from the center of each stack is shown.

Fig. 2
Fig. 2

MSE as a function of κ for a simulated confocal image. Poisson noise with β=10 has been added for the MAPPG and MAPPE algorithms. Gaussian noise with SNR=45 has been added for the MAPGG and MAPGE algorithms. For all four algorithms 500 iterations were used to find the MSE for given κ.

Fig. 3
Fig. 3

Results for the MAP algorithms for the simulated images with use of the optimal κ found from Fig. 2: (a) MAPGG result with κ=3.7-3; (b) MAPGE result with κ=2.9-2; (c) MAPPG result with κ=2.5-3; (d) MAPPE result with κ=5.6-2. Results (a) and (b) are for the image with Gaussian noise, and results (c) and (d) are for the image with Poisson noise. In all four cases 500 iterations were used. A single slice from the center of each stack is shown.

Fig. 4
Fig. 4

MSE as a function of the number of iterations for the MAPGG, MAPGE, MLG, and Carrington algorithms. A simulated confocal image was used with Gaussian noise with SNR =45. The same values for κ were used as those given in Fig. 3 for the MAP algorithms. For Carrington’s algorithm the same regularization parameters were used as those for the MAPGG algorithm.

Fig. 5
Fig. 5

MSE as a function of the number of iterations for the MAPPG, MAPPE, and MLP algorithms and the accelerated EM algorithm of Holmes. A simulated confocal image was used with Poisson noise with β=10. The same values for κ were used as those given in Fig. 3 for the MAP algorithms.

Fig. 6
Fig. 6

MSE as a function of the SNR for the MAPGG and MAPGE algorithms. A simulated confocal image was used with Gaussian noise with varying SNR. The values for κ were the same as those given in Fig. 3. The number of iterations was 500.

Fig. 7
Fig. 7

MSE as a function of β, the reciprocal of the photon-conversion factor of the Poisson noise for the MAPPG and MAPPE algorithms. A simulated confocal image was used with Poisson noise with varying β. The values for κ were the same as those given in Fig. 3. The number of iterations was 500.

Fig. 8
Fig. 8

Confocal image of a Vero cell, treated with cholera toxin and incubated with Cy5-labeled antibody against the A subunit of the toxin, and restoration results. (a) original; (b) MAPPG restoration with κ=1; (c) MAPPE restoration with κ=5. In both (a) and (b), 25 iterations were used. Part of an optical section from the center of each stack is shown.

Tables (1)

Tables Icon

Table 1 Average Execution Times per Iteration on a Transtec Workstation with a 167-MHz UltraSparc Processor, Running the Solaris 2.5 Operating System

Equations (43)

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g(x, y, z)=Ni,j,kh(x, y, z, i, j, k)f(i, j, k)+b(x, y, z),
g=N(Hf+b),
p(f|g)=p(g|f)p(f)/p(g).
p(f)=C exp-12τ2 C(f-m)2,
p(f)=C exp[ρS(|Cf|, |Cm|)],
S(x, y)=i=1Mxi-yi-xi lnxiyi.
p(g|f)=C exp-12σ2 Hf+b-g2,
p(g|f)=i=1M μiNi exp(-μi)Ni!,
ΨG,G=Hf-g2+γC(f-m)2;
ΨP,G=Hf-gT ln(Hf+b)+γC(f-m)2,
ΨG,E=Hf-g2+γ|Cf|T ln|Cf|e|Cm|,
ΨP,E=Hf-gT ln(Hf+b)+γ|Cf|T ln|Cf|e|Cm|.
γG,G=σ2/τ2,
γP,G=1/(2βτ2),
γG,E=2σ2ρ,
γP,E=ρ/β.
Ψ=Hf-gT ln(Hf)-fT ln m+ ln(f!).
ln(fi!)-fi+fi ln fi+12ln(2πfi).
ΨG,G=4X[HT(Hx2-g)+γCTC(x2-m)],
ΨP,G=2XHT1-gHx2+b+2γCTC(x2-m),
ΨG,E=4XHT(Hx2-g)+12 γCTSln|Cx2|e|Cm|+1,
ΨP,E=2XHT1-gHx2+b+γCTSln|Cx2|e|Cm|+1,
γG,G=κ Mσ2C(fˆ-m)2,
γP,G=κ M2βC(fˆ-m)2,
γG,E=2κMσ2|Cfˆ|T ln|Cfˆ|e|Cm|+|Cm|-1,
γP,E=κ Mβ |Cfˆ|T ln|Cfˆ|e|Cm|+|Cm|-1,
SNR=maxi(Hf)i-mini(Hf)iσ,
MSE(fˆ,f)=1M i=1M(f^i-fi)2.
xk+1=xk+αkdk,
dk=βkdk-1-Ψ(xk)
βk=Ψ(xk)2/Ψ(xk-1)2.
Ψ(x+αd)=pα4+qα3+rα2+sα+t,
p=(d2)TAd2,
q=4(d2)TAXd,
r=4dTXAXd+2(d2)T(Ax2-q),
s=4dTX(Ax2-q),
t=(x2)TAx2-2(x2)Tq+c.
dΨP,Gdα=2(Xd+αd2)THT1-gH(x+αd)2+b+γ(4pα3+3qα2+2rα+s),
d2ΨP,Gdα2=2(d2)THT1-gH(x+αd)2+b+4gTH(Xd+αd2)H(x+αd)2+b2+γ(12pα2+6qα+2r),
dΨG,Edα=4pα3+3qα2+2rα+s+2γ(Xd+αd2)TCTS1+ln|C(x+αd)2|e|Cm|,
d2ΨG,Edα2=12pα2+6qα+2r+2γ(d2)TCTS×1+ln|C(x+αd)2|e|Cm|+4γ(Xd+αd2)TCTS C(Xd+αd2)C(x+αd)2,
dΨP,Edα=2(Xd+αd2)THT1-gH(x+αd)2+b+2γ(Xd+αd2)TCTS×1+ln|C(x+αd)2|e|Cm|,
d2ΨP,Edα2=2(d2)THT1-gH(x+αd)2+b+4gTH(Xd+αd2)H(x+αd)2+b2+2γ(d2)TCTS×1+ln|C(x+αd)2|e|Cm|+4γ(Xd+αd2)TCTS C(Xd+αd2)C(x+αd)2.

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