Abstract

We have evaluated three constrained, iterative restoration algorithms to find a fast, reliable algorithm for maximum-likelihood estimation of fluorescence microscopic images. Two algorithms used a Gaussian approximation to Poisson statistics, with variances computed assuming Poisson noise for the images. The third method used Csiszár’s information-divergence (I-divergence) discrepancy measure. Each method included a nonnegativity constraint and a penalty term for regularization; optimization was performed with a conjugate gradient method. Performance of the methods was analyzed with simulated as well as biological images and the results compared with those obtained with the expectation-maximization–maximum-likelihood (EM-ML) algorithm. The I-divergence-based algorithm converged fastest and produced images similar to those restored by EM-ML as measured by several metrics. For a noiseless simulated specimen, the number of iterations required for the EM-ML method to reach a given log-likelihood value was approximately the square of the number required for the I-divergence-based method to reach the same value.

© 2001 Optical Society of America

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2000 (1)

1999 (1)

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. 193, 50–61 (1999).
[CrossRef]

1998 (1)

1997 (3)

1996 (2)

M. Al-Baali, R. Fletcher, “On the order of convergence of preconditioned nonlinear conjugate gradient methods,” SIAM J. Sci. Comput. 17, 658–665 (1996).
[CrossRef]

H. Kano, H. T. M. van der Voort, M. Schrader, G. M. P. van Kempen, S. W. Hell, “Avalanche photodiode detection with object scanning and image restoration provides 2–4 fold resolution increase in two-photon fluorescence microscopy,” Bioimaging 4, 187–197 (1996).
[CrossRef]

1995 (2)

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

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]

1994 (2)

1993 (2)

N. H. Clinthorne, T.-S. Pan, P.-C. Chiao, W. L. Rogers, “Preconditioning methods for improved convergence rates in iterative reconstructions,” IEEE Trans. Med. Imaging 12, 78–83 (1993).
[CrossRef] [PubMed]

S. Joshi, M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for optical sectioning microscopy,” J. Opt. Soc. Am. A 10, 1078–1085 (1993).
[CrossRef] [PubMed]

1992 (2)

1991 (3)

1990 (1)

1989 (2)

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

T. J. Holmes, “Expectation-maximization restoration of band limited, truncated point-process intensities with application in microscopy,” J. Opt. Soc. Am. A 6, 1006–1014 (1989).
[CrossRef]

1988 (1)

1987 (1)

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

1985 (1)

1984 (1)

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

1981 (1)

J. P. Butler, J. A. Reeds, S. V. Dawson, “Estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing,” SIAM J. Numer. Anal. 18, 381–397 (1981).
[CrossRef]

1977 (1)

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

1971 (1)

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

1967 (1)

1964 (1)

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Agard, D. A.

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

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

Al-Baali, M.

M. Al-Baali, R. Fletcher, “On the order of convergence of preconditioned nonlinear conjugate gradient methods,” SIAM J. Sci. Comput. 17, 658–665 (1996).
[CrossRef]

Andrews, M.

Arsenin, V. Y.

A. N. Tikhonov, V. Y. Arsenin, Solutions to Ill-Posed Problems (Wiley, New York, 1977), p. 70.

Biggs, D. S. C.

Bille, J.

Butler, J. P.

J. P. Butler, J. A. Reeds, S. V. Dawson, “Estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing,” SIAM J. Numer. Anal. 18, 381–397 (1981).
[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. Smith, ed., Proc. SPIE1205, 72–83 (1990).
[CrossRef]

W. A. Carrington, F. E. Fogarty, F. S. Fay, “3D fluorescence imaging of single cells using image restoration,” in Noninvasive Techniques in Cell Biology, J. K. Fosket, S. Grinstein, eds. (Wiley–Liss, New York, 1990), pp. 53–72.

Chiao, P.-C.

N. H. Clinthorne, T.-S. Pan, P.-C. Chiao, W. L. Rogers, “Preconditioning methods for improved convergence rates in iterative reconstructions,” IEEE Trans. Med. Imaging 12, 78–83 (1993).
[CrossRef] [PubMed]

Clinthorne, N. H.

N. H. Clinthorne, T.-S. Pan, P.-C. Chiao, W. L. Rogers, “Preconditioning methods for improved convergence rates in iterative reconstructions,” IEEE Trans. Med. Imaging 12, 78–83 (1993).
[CrossRef] [PubMed]

Conchello, J. A.

J. A. Conchello, “Superresolution and convergence properties of the expectation-maximization algorithm for maximum-likelihood deconvolution of incoherent images,” J. Opt. Soc. Am. A 15, 2609–2619 (1998).
[CrossRef]

J. A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope images. II. Depth discrimination vs. signal-to-noise ratio in partially confocal images,” Appl. Opt. 33, 3740–3750 (1994).
[CrossRef] [PubMed]

J. G. McNally, C. Preza, J. A. Conchello, L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11, 1056–1067 (1994).
[CrossRef]

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

J. A. Conchello, “Super-resolution and point spread function sensitivity analysis of the expectation-maximization algorithm for computational optical sectioning microscopy,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 369–378 (1994).
[CrossRef]

J. A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).

J. Markham, J. A. Conchello, “Tradeoffs in regularized maximum-likelihood image restoration,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. Cogswell, J. A. Conchello, T. Wilson, eds., Proc. SPIE2984, 136–145 (1997).

Conchello, J.-A.

C. Preza, M. I. Miller, J.-A. Conchello, “Image reconstruction for 3-D light microscopy with a regularized linear method incorporating a smoothness prior,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 129–139 (1993).
[CrossRef]

Csiszár, I.

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

Dawson, S. V.

J. P. Butler, J. A. Reeds, S. V. Dawson, “Estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing,” SIAM J. Numer. Anal. 18, 381–397 (1981).
[CrossRef]

Dempster, A. P.

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

Erhardt, A.

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]

W. A. Carrington, F. E. Fogarty, F. S. Fay, “3D fluorescence imaging of single cells using image restoration,” in Noninvasive Techniques in Cell Biology, J. K. Fosket, S. Grinstein, eds. (Wiley–Liss, New York, 1990), pp. 53–72.

Fletcher, R.

M. Al-Baali, R. Fletcher, “On the order of convergence of preconditioned nonlinear conjugate gradient methods,” SIAM J. Sci. Comput. 17, 658–665 (1996).
[CrossRef]

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Fogarty, F. E.

W. A. Carrington, F. E. Fogarty, F. S. Fay, “3D fluorescence imaging of single cells using image restoration,” in Noninvasive Techniques in Cell Biology, J. K. Fosket, S. Grinstein, eds. (Wiley–Liss, New York, 1990), pp. 53–72.

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]

Frieden, B. R.

Gaskins, R. A.

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

Gemkow, M. J.

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. 193, 50–61 (1999).
[CrossRef]

Gibson, F. S.

Good, I. J.

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

Hansen, E. W.

Hecht, E.

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), p. 354.

Hell, S. W.

H. Kano, H. T. M. van der Voort, M. Schrader, G. M. P. van Kempen, S. W. Hell, “Avalanche photodiode detection with object scanning and image restoration provides 2–4 fold resolution increase in two-photon fluorescence microscopy,” Bioimaging 4, 187–197 (1996).
[CrossRef]

Hiraoka, Y.

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

Holmes, T. J.

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]

Joshi, S.

Jovin, T. M.

P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. 193, 50–61 (1999).
[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]

Kano, H.

H. Kano, H. T. M. van der Voort, M. Schrader, G. M. P. van Kempen, S. W. Hell, “Avalanche photodiode detection with object scanning and image restoration provides 2–4 fold resolution increase in two-photon fluorescence microscopy,” Bioimaging 4, 187–197 (1996).
[CrossRef]

Kaufman, L.

L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imaging MI-6, 37–51 (1987).
[CrossRef]

Kim, J. J.

Komitowski, D.

Laird, N. M.

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

Lanni, F.

Liu, Y. H.

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]

Markham, J.

J. Markham, J. A. Conchello, “Tradeoffs in regularized maximum-likelihood image restoration,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. Cogswell, J. A. Conchello, T. Wilson, eds., Proc. SPIE2984, 136–145 (1997).

McNally, J. G.

J. G. McNally, C. Preza, J. A. Conchello, L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11, 1056–1067 (1994).
[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]

J. A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for optical sectioning microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).

Miller, M. I.

S. Joshi, M. I. Miller, “Maximum a posteriori estimation with Good’s roughness for optical sectioning microscopy,” J. Opt. Soc. Am. A 10, 1078–1085 (1993).
[CrossRef] [PubMed]

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, M. I. Miller, J.-A. Conchello, “Image reconstruction for 3-D light microscopy with a regularized linear method incorporating a smoothness prior,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 129–139 (1993).
[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, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Ortega, J. M.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970), p. 259.

Pan, T.-S.

N. H. Clinthorne, T.-S. Pan, P.-C. Chiao, W. L. Rogers, “Preconditioning methods for improved convergence rates in iterative reconstructions,” IEEE Trans. Med. Imaging 12, 78–83 (1993).
[CrossRef] [PubMed]

Preza, C.

J. G. McNally, C. Preza, J. A. Conchello, L. J. Thomas, “Artifacts in computational optical-sectioning microscopy,” J. Opt. Soc. Am. A 11, 1056–1067 (1994).
[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]

C. Preza, M. I. Miller, J.-A. Conchello, “Image reconstruction for 3-D light microscopy with a regularized linear method incorporating a smoothness prior,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 129–139 (1993).
[CrossRef]

Reeds, J. A.

J. P. Butler, J. A. Reeds, S. V. Dawson, “Estimating solutions of first kind integral equations with nonnegative constraints and optimal smoothing,” SIAM J. Numer. Anal. 18, 381–397 (1981).
[CrossRef]

Reeves, C. M.

R. Fletcher, C. M. Reeves, “Function minimization by conjugate gradients,” Comput. J. 7, 149–154 (1964).
[CrossRef]

Rheinboldt, W. C.

J. M. Ortega, W. C. Rheinboldt, Iterative Solution of Nonlinear Equations in Several Variables (Academic, New York, 1970), p. 259.

Rogers, W. L.

N. H. Clinthorne, T.-S. Pan, P.-C. Chiao, W. L. Rogers, “Preconditioning methods for improved convergence rates in iterative reconstructions,” IEEE Trans. Med. Imaging 12, 78–83 (1993).
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H. Kano, H. T. M. van der Voort, M. Schrader, G. M. P. van Kempen, S. W. Hell, “Avalanche photodiode detection with object scanning and image restoration provides 2–4 fold resolution increase in two-photon fluorescence microscopy,” Bioimaging 4, 187–197 (1996).
[CrossRef]

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[CrossRef]

Seber, G. A. F.

G. A. F. Seber, C. J. Wild, Nonlinear Regression (Wiley, New York, 1989), p. 610.

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[CrossRef]

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

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P. J. Shaw, “Comparison of wide-field/deconvolution and confocal microscopy for 3D imaging,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 373–387.

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]

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[CrossRef]

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A. N. Tikhonov, V. Y. Arsenin, Solutions to Ill-Posed Problems (Wiley, New York, 1977), p. 70.

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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]

H. Kano, H. T. M. van der Voort, M. Schrader, G. M. P. van Kempen, S. W. Hell, “Avalanche photodiode detection with object scanning and image restoration provides 2–4 fold resolution increase in two-photon fluorescence microscopy,” Bioimaging 4, 187–197 (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]

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G. M. P. van Kempen, L. J. van Vliet, “Background estimation in nonlinear image restoration,” J. Opt. Soc. Am. A 17, 425–433 (2000).
[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]

H. Kano, H. T. M. van der Voort, M. Schrader, G. M. P. van Kempen, S. W. Hell, “Avalanche photodiode detection with object scanning and image restoration provides 2–4 fold resolution increase in two-photon fluorescence microscopy,” Bioimaging 4, 187–197 (1996).
[CrossRef]

van Vliet, L. J.

G. M. P. van Kempen, L. J. van Vliet, “Background estimation in nonlinear image restoration,” J. Opt. Soc. Am. A 17, 425–433 (2000).
[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]

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P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. 193, 50–61 (1999).
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[CrossRef]

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G. A. F. Seber, C. J. Wild, Nonlinear Regression (Wiley, New York, 1989), p. 610.

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E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), p. 354.

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[CrossRef] [PubMed]

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[CrossRef]

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H. T. M. van der Voort, K. C. Strasters, “Restoration of confocal images for quantitative image analysis,” J. Microsc. 178, 165–181 (1995).
[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).
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P. J. Verveer, M. J. Gemkow, T. M. Jovin, “A comparison of image restoration approaches applied to three-dimensional confocal and wide-field fluorescence microscopy,” J. Microsc. 193, 50–61 (1999).
[CrossRef]

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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, “Background estimation in nonlinear image restoration,” J. Opt. Soc. Am. A 17, 425–433 (2000).
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A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Statist. Soc. B 39, 1–38 (1977).

Methods Cell Biol. (1)

D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three-dimensions,” Methods Cell Biol. 30, 353–377 (1989).
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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).
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G. A. F. Seber, C. J. Wild, Nonlinear Regression (Wiley, New York, 1989), p. 610.

The XCOSM deconvolution package is available from URL http://www.ibc.wustl.edu/bcl/xcosm/xcosm.html .

A. N. Tikhonov, V. Y. Arsenin, Solutions to Ill-Posed Problems (Wiley, New York, 1977), p. 70.

J. A. Conchello, “Super-resolution and point spread function sensitivity analysis of the expectation-maximization algorithm for computational optical sectioning microscopy,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 369–378 (1994).
[CrossRef]

S. S. Stefanou, E. W. Hansen, “Restoration of edges under Poisson noise using convex constraints with application to 3D optical microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 232–242 (1997).

C. Preza, M. I. Miller, J.-A. Conchello, “Image reconstruction for 3-D light microscopy with a regularized linear method incorporating a smoothness prior,” in Biomedical Image Processing and Biomedical Visualization, R. S. Acharya, D. B. Goldgof, eds., Proc. SPIE1905, 129–139 (1993).
[CrossRef]

P. J. Shaw, “Comparison of wide-field/deconvolution and confocal microscopy for 3D imaging,” in Handbook of Biological Confocal Microscopy, 2nd ed., J. B. Pawley, ed. (Plenum, New York, 1995), pp. 373–387.

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J. Markham, J. A. Conchello, “Tradeoffs in regularized maximum-likelihood image restoration,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. Cogswell, J. A. Conchello, T. Wilson, eds., Proc. SPIE2984, 136–145 (1997).

E. Hecht, A. Zajac, Optics (Addison-Wesley, Reading, Mass., 1974), p. 354.

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Figures (6)

Fig. 1
Fig. 1

Restored images of a spherical shell test specimen from three algorithms without regularization. Lateral (left) and axial (right) medial sections (a) through the image, and estimated specimen functions after (b) 20,000 EM-ML iterations, (c) 300 EM-ML iterations, (d) 300 IDIV iterations, and (e) 300 WLS iterations. In the true object, the intensity of the cross bars is the same as that of the shell.

Fig. 2
Fig. 2

Number of iterations required for two test methods, IDIV and WLS, to reach a given log-likelihood value (as computed by the EM-ML algorithm) in relation to the number required by the EM-ML algorithm to reach the same log-likelihood value for the spherical shell test specimen. The curves can be fitted by the functions IML=1.3(IID)1.9 and IML=2.0(IWLS)1.7.

Fig. 3
Fig. 3

Median sections from the restored shell images. Only xz or vertical sections are shown for (left) 30 iterations of IDIV and (right) 1000 iterations of EM-ML. Elongation for the two images appears to be approximately the same, and the log-likelihood values for the two images are quite similar.

Fig. 4
Fig. 4

Value of the I-divergence discrepancy measure computed from the restored and true shell specimens for the EM-ML algorithm, the IDIV method, and the WLS method as a function of the number of iterations. The x axis is displayed on a logarithmic scale.

Fig. 5
Fig. 5

Slices from images of cells of the yeast Histoplasma capsulatum. Lateral sections of (a) raw images and restored images from (b) 40 iterations of IDIV, (c) 40 iterations of WCAR with α=810-7, (d) 100 iterations of EM-ML, and (e) 1000 iterations of EM-ML. No penalty was used for any method except WCAR. The image restored by WLS is not shown, as it was not visibly different from the IDIV image.

Fig. 6
Fig. 6

Slices from images of tobacco cells. (a) xy sections of the raw image, and (b)–(e) images restored by penalized methods. Images are restored by (b) 20 iterations of IDIV with α=5×10-6, (c) 20 iterations of WLS with α=8×10-6, (d) 20 iterations of WCAR with α=1.2×10-5, and (e) 400 iterations of EM-ML with α=2×10-5.

Tables (1)

Tables Icon

Table 1 Average Values at Rayleigh Resolution (V=0.15) for Figures of Merit Computed from Restored Images of 20 Noisy Realizations of the Two-Line Resolution Object with SNR0=6.3a

Equations (18)

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

g(xi)=Oh(xi-xo)s(xo)dxo=hs,
g=Hs,
L(s|g)=-i=1M[gˆi-gi ln(gˆi)],
I(s)=i=1Mgi ln gigˆi+gˆi-gi+αsi2,
I(c)=2CHT1-ggˆ+2αCc,
Φ(s)=12[(g-Hs)TW(g-Hs)+αsTs],
Φ(c)=12[(g-HCc)TW(g-HCc)+αcTC2c],
Φ(c)=2C[-HTW(g-Hs)+αCc].
s=max(0, HTc),
Ψ(c)=12cT(HHT+αW-1)c-cTg,
Ψ(c)=(HHT+αW-1)c-g.
MSE=i=1Mwi(si-sˆi)2,
IDIV(s)=i=1Msi ln sisˆi+sˆi-si.
V=s¯l-s¯ms¯l+s¯m,
s¯l=12Nxx=1Nx[sˆ(x, 0, zu)+sˆ(x, 0, zl)]
s¯m=1Nx x=1Nx sˆ(x, 0, zm)
FWHM¯=12Nx x=1Nx[|yh1(x, zu)-yh2(x, zu)|+|yh1(x, zl)-yh2(x, zl)|],
IML=A(IID)p.

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