Abstract

We derive an algorithm for maximum-likelihood image estimation on the basis of the expectation-maximization (EM) formalism by using a new approximate model for depth-varying image formation for optical sectioning microscopy. This new strata-based model incorporates spherical aberration that worsens as the microscope is focused deeper under the cover slip and is the result of the refractive-index mismatch between the immersion medium and the mounting medium of the specimen. Images of a specimen with known geometry and refractive index show that the model captures the main features of the image. We analyze the performance of the depth-variant EM algorithm with simulations, which show that the algorithm can compensate for image degradation changing with depth.

© 2004 Optical Society of America

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2003 (1)

2002 (3)

J. B. Pawley, “Limitations on optical sectioning in live-cell confocal microscopy,” Scanning 24, 241–246 (2002).
[CrossRef] [PubMed]

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

2001 (2)

J. Markham, J.-A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 18, 1062–1071 (2001).
[CrossRef]

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

1999 (2)

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]

J. Markham, J.-A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
[CrossRef]

1998 (2)

1997 (2)

1996 (1)

1995 (2)

M. Faisal, A. D. Lanterman, D. L. Snyder, R. L. White, “Implementation of a modified Richardson–Lucy method for image restoration on a massively parallel computer to compensate for space-variant point spread of a charge-coupled-device camera,” J. Opt. Soc. Am. A 12, 2593–2603 (1995).
[CrossRef]

W. A. Carrington, E. D. Lynch, 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 (3)

1993 (3)

1992 (2)

1991 (1)

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

1990 (1)

1989 (2)

1988 (1)

1985 (1)

1984 (1)

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

1978 (1)

H. J. Trussell, B. R. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 608–609 (1978).
[CrossRef]

1977 (1)

A. D. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–37 (1977).

Agard, D. A.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[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]

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

Albert, O.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

L. R. Sherman, O. Albert, C. F. Schmidt, G. V. Vdovin, G. A. Mourou, T. B. Norris, “Adaptive compensation of aberrations in ultrafast 3D microscopy using a deformable mirror,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE3919, 9–13 (2000).
[CrossRef]

Bille, J.

Boden, A. F.

Booth, M. J.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Burns, D.

Carrington, W. A.

W. A. Carrington, E. D. Lynch, 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, K. 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.

Conchello, J.-A.

J. Markham, J.-A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 18, 1062–1071 (2001).
[CrossRef]

J. Markham, J.-A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
[CrossRef]

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 three-dimensional reconstruction from confocal scanning microscope images. II. Depth discrimination versus 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. 1: Deterministic and maximum likelihood reconstructions,” Appl. Opt. 19, 3795–3804 (1990).
[CrossRef]

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

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, ed., Proc. SPIE2302, 369–378 (1994).
[CrossRef]

C. Preza, J.-A. Conchello, “Image estimation accounting for point-spread function depth variation in three-dimensional fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing X, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE4964, 135–142 (2003).
[CrossRef]

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]

J.-A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” in Proceedings of the IS&T/SPIE Symposium on Electronic Imaging, Science and Technology, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).

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]

Dempster, A. D.

A. D. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–37 (1977).

Dudgeon, D. E.

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

Erhardt, A.

Faisal, M.

Fay, F. S.

W. A. Carrington, E. D. Lynch, 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, K. 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, E. D. Lynch, 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, K. 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.

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, S. F.

S. F. 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 9, 154–66 (1992).
[CrossRef] [PubMed]

S. F. Gibson, F. Lanni, “Measured and analytical point spread functions of the optical microscope for use in 3-D optical serial sectioning microscopy,” in Optical Microscopy for Biology (Wiley-Liss, New York, 1990).

Girkin, J. M.

Gustafsson, M. G. L.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Hammoud, A. M.

Hanisch, R. J.

Hansen, E. W.

Hanser, B.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

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] [PubMed]

Holmes, T. J.

Hunt, B. R.

H. J. Trussell, B. R. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 608–609 (1978).
[CrossRef]

Isenberg, G.

W. A. Carrington, E. D. Lynch, 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]

Juskaitis, R.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Kam, Z.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Kim, J. J.

Komitowski, D.

Laird, N. M.

A. D. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–37 (1977).

Lanni, F.

S. F. 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 9, 154–66 (1992).
[CrossRef] [PubMed]

S. F. Gibson, F. Lanni, “Measured and analytical point spread functions of the optical microscope for use in 3-D optical serial sectioning microscopy,” in Optical Microscopy for Biology (Wiley-Liss, New York, 1990).

Lanterman, A. D.

Leung, H.

H. Leung, “The importance of mounting medium’s refractive index and cover glass thickness for reducing optical aberrations on confocal laser scanning microscope (CLSM),” Microsc. Soc. Can. Bull. 21, 19–24 (1993).

Lynch, E. D.

W. A. Carrington, E. D. Lynch, 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.

Marsh, P. N.

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 computational optical sectioning microscopy,” in Proceedings of the IS&T/SPIE Symposium on Electronic Imaging, Science and Technology, C. J. Cogswell, G. S. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).

Mersereau, R. M.

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

Miller, M. I.

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]

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]

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).

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]

Mo, J.

Moore, W.

W. A. Carrington, E. D. Lynch, 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]

Mourou, G. A.

L. R. Sherman, O. Albert, C. F. Schmidt, G. V. Vdovin, G. A. Mourou, T. B. Norris, “Adaptive compensation of aberrations in ultrafast 3D microscopy using a deformable mirror,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE3919, 9–13 (2000).
[CrossRef]

Nagy, J. G.

J. G. Nagy, D. P. O’Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063–1082 (1998).
[CrossRef]

J. G. Nagy, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, “Space-varying restoration of optical images,” J. Opt. Soc. Am. A 14, 3162–3174 (1997).
[CrossRef]

Neil, M. A. A.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Norris, T. B.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

L. R. Sherman, O. Albert, C. F. Schmidt, G. V. Vdovin, G. A. Mourou, T. B. Norris, “Adaptive compensation of aberrations in ultrafast 3D microscopy using a deformable mirror,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE3919, 9–13 (2000).
[CrossRef]

O’Leary, D. P.

J. G. Nagy, D. P. O’Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063–1082 (1998).
[CrossRef]

Pauca, V. P.

Pawley, J. B.

J. B. Pawley, “Limitations on optical sectioning in live-cell confocal microscopy,” Scanning 24, 241–246 (2002).
[CrossRef] [PubMed]

Plemmons, R. J.

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, J.-A. Conchello, “Image estimation accounting for point-spread function depth variation in three-dimensional fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing X, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE4964, 135–142 (2003).
[CrossRef]

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]

Redding, D. C.

Rubin, D. B.

A. D. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–37 (1977).

Schmidt, C. F.

L. R. Sherman, O. Albert, C. F. Schmidt, G. V. Vdovin, G. A. Mourou, T. B. Norris, “Adaptive compensation of aberrations in ultrafast 3D microscopy using a deformable mirror,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE3919, 9–13 (2000).
[CrossRef]

Sedat, J. W.

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[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]

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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] [PubMed]

Sherman, L.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Sherman, L. R.

L. R. Sherman, O. Albert, C. F. Schmidt, G. V. Vdovin, G. A. Mourou, T. B. Norris, “Adaptive compensation of aberrations in ultrafast 3D microscopy using a deformable mirror,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE3919, 9–13 (2000).
[CrossRef]

Snyder, D. L.

Stefanou, S. S.

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

Thomas, L. J.

Torgersen, T. C.

Trussell, H. J.

H. J. Trussell, B. R. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 608–609 (1978).
[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 (1994).
[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 (1994).
[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 (1994).
[CrossRef]

Vdovin, G. V.

L. R. Sherman, O. Albert, C. F. Schmidt, G. V. Vdovin, G. A. Mourou, T. B. Norris, “Adaptive compensation of aberrations in ultrafast 3D microscopy using a deformable mirror,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE3919, 9–13 (2000).
[CrossRef]

Verveer, P. 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]

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 (1994).
[CrossRef]

White, R. L.

Wilson, T.

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Ye, J. Y.

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

Zinser, G.

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Appl. Opt. (3)

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

H. J. Trussell, B. R. Hunt, “Image restoration of space-variant blurs by sectioned methods,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-26, 608–609 (1978).
[CrossRef]

J. Microsc. (3)

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 (1994).
[CrossRef]

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]

L. Sherman, J. Y. Ye, O. Albert, T. B. Norris, “Adaptive correction of depth-induced aberrations in multiphoton scanning microscopy using a deformable mirror,” J. Microsc. 206, 65–71 (2002).
[CrossRef] [PubMed]

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M. Faisal, A. D. Lanterman, D. L. Snyder, R. L. White, “Implementation of a modified Richardson–Lucy method for image restoration on a massively parallel computer to compensate for space-variant point spread of a charge-coupled-device camera,” J. Opt. Soc. Am. A 12, 2593–2603 (1995).
[CrossRef]

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. Markham, J.-A. Conchello, “Parametric blind deconvolution: a robust method for the simultaneous estimation of image and blur,” J. Opt. Soc. Am. A 16, 2377–2391 (1999).
[CrossRef]

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]

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]

J. G. Nagy, V. P. Pauca, R. J. Plemmons, T. C. Torgersen, “Space-varying restoration of optical images,” J. Opt. Soc. Am. A 14, 3162–3174 (1997).
[CrossRef]

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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).
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S. F. 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 9, 154–66 (1992).
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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]

D. L. Snyder, A. M. Hammoud, R. L. White, “Image recovery from data acquired with a charge-coupled-device camera,” J. Opt. Soc. Am. A 10, 1014–1023 (1993).
[CrossRef] [PubMed]

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]

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J. Markham, J.-A. Conchello, “Fast maximum-likelihood image-restoration algorithms for three-dimensional fluorescence microscopy,” J. Opt. Soc. Am. A 18, 1062–1071 (2001).
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J. R. Stat. Soc. Ser. B. Methodol. (1)

A. D. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. Ser. B. Methodol. 39, 1–37 (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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H. Leung, “The importance of mounting medium’s refractive index and cover glass thickness for reducing optical aberrations on confocal laser scanning microscope (CLSM),” Microsc. Soc. Can. Bull. 21, 19–24 (1993).

Opt. Express (1)

Proc. Natl. Acad. Sci. USA (2)

M. J. Booth, M. A. A. Neil, R. Juskaitis, T. Wilson, “Adaptive aberration correction in a confocal microscope,” Proc. Natl. Acad. Sci. USA 99, 5788–5792 (2002).
[CrossRef] [PubMed]

Z. Kam, B. Hanser, M. G. L. Gustafsson, D. A. Agard, J. W. Sedat, “Computational adaptive optics for live three-dimensional biological imaging,” Proc. Natl. Acad. Sci. USA 98, 3790–3795 (2001).
[CrossRef] [PubMed]

Scanning (1)

J. B. Pawley, “Limitations on optical sectioning in live-cell confocal microscopy,” Scanning 24, 241–246 (2002).
[CrossRef] [PubMed]

Science (1)

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SIAM J. Sci. Comput. (1)

J. G. Nagy, D. P. O’Leary, “Restoring images degraded by spatially variant blur,” SIAM J. Sci. Comput. 19, 1063–1082 (1998).
[CrossRef]

Other (12)

L. R. Sherman, O. Albert, C. F. Schmidt, G. V. Vdovin, G. A. Mourou, T. B. Norris, “Adaptive compensation of aberrations in ultrafast 3D microscopy using a deformable mirror,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing VII, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE3919, 9–13 (2000).
[CrossRef]

S. F. Gibson, F. Lanni, “Measured and analytical point spread functions of the optical microscope for use in 3-D optical serial sectioning microscopy,” in Optical Microscopy for Biology (Wiley-Liss, New York, 1990).

W. A. Carrington, K. 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.

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

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, ed., Proc. SPIE2302, 369–378 (1994).
[CrossRef]

J.-A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” in Proceedings of the IS&T/SPIE Symposium on Electronic Imaging, Science and Technology, 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. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 136–145 (1997).
[CrossRef]

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]

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).

The XCOSM deconvolution package is available at http://www.omrfcosm.omrf.org and http://www.essrl.wustl.edu/∼preza/xcosm .

C. Preza, J.-A. Conchello, “Image estimation accounting for point-spread function depth variation in three-dimensional fluorescence microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing X, J.-A. Conchello, C. J. Cogswell, T. Wilson, eds., Proc. SPIE4964, 135–142 (2003).
[CrossRef]

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

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Figures (10)

Fig. 1
Fig. 1

xz-section images through the center of (a) the three-sphere test object; (b) the image of the three-sphere object with a 60×/1.4 NA oil-immersion lens predicted by the depth-variant stratum-based model; and images of single-sphere test objects with the sphere centered at depths of (c) 0.225, (d) 1.125, and (e) 2.025 μm below the cover slip, obtained by the convolution of each test object with the appropriate PSF at each depth. The spheres have a diameter equal to 0.45 μm.

Fig. 2
Fig. 2

Comparison of xz-section images through the center of a measured image (left) and a simulated image (right) of the 4-μm bead phantom. Vertical profiles through the images are shown in Fig. 3.

Fig. 3
Fig. 3

Vertical profiles through the center of the images shown in Fig. 2. A plane indicates an optical section along the z axis.

Fig. 4
Fig. 4

xz slices of (a) the true annulus specimen, (b) its simulated image, (c) and (d) specimen estimated with the DV-EM algorithm after 100,000 and 20,000 iterations, respectively, (e) and (f) specimen estimate with the SI-EM algorithm and a PSF at zero depth after 100,000 and 20,000 iterations, respectively, (g) and (h) specimen estimate with the SI-EM algorithm and a PSF at 1.2-μm depth after 100,000 and 20,000 iterations, respectively. Images shown are 96 × 96 pixels.

Fig. 5
Fig. 5

Vertical profiles through the middle of xz-section images of the annulus test object (truth) and the estimated images computed with the DV-EM algorithm after a different number of iterations (20,000 and 100,000) shown in Figs. 4(a), 4(c), and 4(d).

Fig. 6
Fig. 6

IDIV discrepancy measure between the true and the estimated objects computed at each iteration of the DV-EM algorithm and the SI-EM algorithm for the annulus test object. For the SI-EM algorithm, results are shown obtained with a single LSF defined at a depth of 0 μm (0) and 1.2 μm (1.2). The y axis is normalized by the constant 107, and the x axis is displayed on a logarithmic scale.

Fig. 7
Fig. 7

xz images of the (a) true and (b) estimated two-disk objects computed from the synthetic image with the DV-EM (left) and the SI-EM algorithm (right) after (c) 50,000 DV-EM iterations, (d) 50,000 SI-EM iterations, (e) 10,000 DV-EM iterations, (f) 10,000 SI-EM iterations, (g) 2,000 DV-EM iterations, and (h) 2000 SI-EM iterations. The size of the images shown is 64×64 pixels.

Fig. 8
Fig. 8

Vertical profiles through the middle (x=1.74 μm) of the xz-section images of the two-disc object (truth) and the estimated images computed with the DV-EM algorithm after a different number of iterations (2000, 10,000, and 50,000) shown in Figs. 7(a), 7(c), 7(e), and 7(g).

Fig. 9
Fig. 9

IDIV discrepancy measure between the true and the estimated objects computed at each iteration of the DV-EM algorithm and the SI-EM algorithm for the two-disk test object. The y axis is normalized by the constant 107, and the x axis is displayed on a logarithmic scale.

Fig. 10
Fig. 10

Visibility measure computed at each iteration of the DV-EM algorithm and the SI-EM algorithm for the two-disk test object.

Equations (28)

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

g(xi)=Oh(ri-ro, zi, zo)s(xo)dxo,
Om={xo=(xo, yo, zo):Zmzo<Zm+1};
m=1, 2, 3,, M,
s(xo)=m=1Msm(xo),
sm(xo)=s(xo)for xoOm0otherwise.
hm(xo)=h(ro, Zm, zo),m=1, 2, 3,,M+1,
h˜m(xi, xo)=am(zo)hm(xi-xo)+[1-am(zo)]hm+1(xi-xo),
am(zo)=Zm+1-zoZm+1-ZmforzoOm0otherwise.
gm(xi)=Oh˜m(xi, xo)sm(xo)dxo,m=1, 2, 3,,M,
g(xi)m=1Mgm(xi),
μ(dxi, dxo)
=h˜m(xi-xo)sm(xo)dxidxoforxoOm0otherwise.
Lcd=-IOμ(dxi, dxo)+IOln[s(xo)]N(dxi, dxo),
Q[s|s^(k)]=-Omsm(xo)Ih˜m(xi-xo)dxidxo+Oln[s(xo)]ms^m(k)(xo)×Ih˜m(xi-xo) g(xi)g^(k)(xi)dxidxo,
s^(k+1)(xo)=argmax Q[s|s^(k)],s(xo)0,
s^m(k+1)(xo)=1HM(zo) s^m(k)(xo)Ih˜m(xi-xo) g(xi)g^(k)(xi)dxi1
forxoOm,
HM(zo)=m{am(zo)Hm+[1-am(zo)]Hm+1},
Hm=Ihm(xi-xo)dxi.
s^(k+1)(xo)=s^(k)(xo)HM(zo)am(zo)Ihm(xi-xo)r(k)(xi)dxi+[1-am(zo)]Ihm+1(xi-xo)×r(k)(xi)dxiforxoOm,
r(k)(xi)=g(xi)/g^(k)(xi)
dm(k)(xo)=Ihm(xi-xo)r(k)(xi)dxi.
s^(k+1)(xo)=s^(k)(xo)HM {am(zo)dm(k)(xo)+[1-am(zo)]dm+1(k)(xo)}=s^(k)(xo)HM d˜m(k)(xo)forxoOm,
d˜m(k)(xo)=am(zo)dm(k)(xo)+[1-am(zo)]dm+1(k)(xo)
forxoOm.
f(x, z)=-h(x, y, z)dy,
IDIV=i=1Jsilnsis^i+s^i-si,
V=s¯b-d¯s¯b+d¯,

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