Abstract

Blind-deconvolution microscopy, the simultaneous estimation of the specimen function and the point-spread function (PSF) of the microscope, is an underdetermined problem with nonunique solutions that are usually avoided by enforcing constraints on the specimen function and the PSF. We derived a maximum-likelihood-based method for blind deconvolution in which we assume a mathematical model for the PSF that depends on a small number of parameters (e.g., less than 20). The algorithm then estimates the unknown parameters together with the specimen function. The mathematical model ensures that all the constraints of the PSF are satisfied, and the maximum-likelihood approach ensures that the specimen is nonnegative. The method successfully estimates the PSF and removes out-of-focus blur. The PSF estimation is robust to aberrations in the PSF and to noise in the image.

© 1999 Optical Society of America

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  1. S. C. Gens, J. G. McNally, B. G. Pickard, “Resolution of binding sites for antibodies to integrin, vitronection and fibronectin on onion epidermis protoplasts and depectinated walls,” ASGSB Bull. 7, 42 (1993).
  2. J. S. Gens, K. W. Doolittle, J. G. McNally, B. G. Pickard, “Binding sites for antibodies to animal integrin, vitronectin and fibronectin in a plant model for mechanosensing,” Biophys. J. 66, A169 (1994).
  3. Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).
  4. J. G. McNally, “Computational optical-sectioning microscopy for 3D quantization of cell motion: results and challenges,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 342–351 (1994).
    [CrossRef]
  5. B. G. Pickard, “Contemplating the plasmalemmal control center model,” Protoplasma 182, 1–9 (1994).
    [CrossRef] [PubMed]
  6. B. G. Pickard, C. Reuzeau, K. W. Doolittle, J. G. McNally, “High resolution visualization in onion of distribution patterns of spectrin, talin and vinculin antigenicities,” ASGSB Bull. 8, 54 (1994).
  7. J. A. Conchello, “Fluorescence photobleaching correction for expectation maximization algorithm,” in Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson, C. J. Cogswell, eds., Proc. SPIE2412, 138–146 (1995).
    [CrossRef]
  8. J.-A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope im-ages. 2: depth discrimination vs. signal-to-noise ratio in partially confocal images,” Appl. Opt. 33, 3740–3750 (1994).
    [CrossRef] [PubMed]
  9. J. A. Conchello, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope images. 1: Deterministic and maximum likelihood reconstructions,” Appl. Opt. 29, 3795–3804 (1990).
    [CrossRef] [PubMed]
  10. J. A. Conchello, J. G. McNally, “Fast regularization technique for expectation maximization algorithm for computational optical sectioning microscopy,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).
    [CrossRef]
  11. C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
    [CrossRef] [PubMed]
  12. 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]
  13. D. A. Agard, “Optical sectioning microscopy,” Annu. Rev. Biophys. Bioeng. 13, 191–219 (1984).
    [CrossRef]
  14. 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).
  15. A. Erhardt, G. Zinser, D. Komitowski, J. Bille, “Reconstructing 3-D light-microscopic images by digital image processing,” Appl. Opt. 24, 194–200 (1985).
    [CrossRef] [PubMed]
  16. 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]
  17. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  18. 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]
  19. F. S. Gibson, F. Lanni, “Experimental test of an analytical model of aberration in an oil-immersion objective lens used in three-dimensional light microscopy,” J. Opt. Soc. Am. A 8, 1601–1613 (1991).
    [CrossRef]
  20. J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
    [CrossRef]
  21. J. Markham, J. A. Conchello, “Parametric blind deconvolution of microscopic images: further results,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, chairs/eds., Proc. SPIE3261, 38–49 (1998).
    [CrossRef]
  22. T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
    [CrossRef]
  23. V. Krishnamurthi, Y.-H. Liu, S. Bhattacharyya, J. N. Turner, T. J. Holmes, “Blind deconvolution of fluorescence micrographs by maximum-likelihood estimation,” Appl. Opt. 34, 6633–6647 (1995).
    [CrossRef] [PubMed]
  24. T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
    [CrossRef] [PubMed]
  25. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
    [CrossRef]
  26. Y. Yang, N. P. Galatsanos, H. Stark, “Projection-based blind deconvolution,” J. Opt. Soc. Am. A 11, 2401–2409 (1994).
    [CrossRef]
  27. R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
    [CrossRef]
  28. E. Thiébaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
    [CrossRef]
  29. G. B. Avinash, “Simultaneous blur and image restoration in 3D optical microscopy,” Zoological Studies 34 Suppl. I, 184–185 (1995).
  30. B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
    [CrossRef]
  31. A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
    [CrossRef]
  32. D. Snyder, M. I. Miller, Random Point Processes in Time and Space (Springer-Verlag, New York, 1991).
  33. M. H. DeGroot, Probability and Statistics, 2nd ed. (Addison Wesley, Reading, Mass., 1984), pp. 348–349.
  34. 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).
  35. R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987), p. 22.
  36. See Subsection 4.1 of Ref. 35.
  37. G. A. Seber, C. J. Wild, Nonlinear Regression (Wiley, New York, 1989), pp. 605–609.
  38. C. G. Broyden, “Quasi-Newton methods and their application to function maximization,” Math. Comput. 21, 368–381 (1967).
    [CrossRef]
  39. C. G. Broyden, “The convergence of a class of double-rank minimization algorithms. Part I,” J. Inst. Math. Appl. 6, 76–90 (1970);J. Inst. Math. Appl. “ Part II,” 222–231 (1970).
    [CrossRef]
  40. R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13, 317–322 (1970).
    [CrossRef]
  41. D. Goldfarb, “A family of variable metric methods derived by variational means,” Math. Comput. 24, 23–26 (1970).
    [CrossRef]
  42. D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 647–657 (1970).
    [CrossRef]
  43. See pp. 21–23 of Ref. 35.
  44. The netlib routines are available from the University of Tennessee, Knoxville, URL http://netlib2.cs.utk.edu .
  45. 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]
  46. The XCOSM deconvolution package is available from URL http://www.ibc.wustl.edu/bcl/xcosm/xcosm.html .
  47. 40×/1.0 NA is notation used widely to give the magnification and the numerical aperture of an objective. In this case the objective magnifies 40 times and has a numerical aperture of 1.0.
  48. M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 436.
  49. S. T. Buckland, K. P. Burnham, N. H. Augustin, “Model selection: an integral part of inference,” Biometrics 53, 603–618 (1997).
    [CrossRef]
  50. A. C. Atkinson, “A note on the generalized information criterion for choice of a model,” Biometrika 67, 413–418 (1980).
    [CrossRef]
  51. C. R. Rao, Y. Wu, “A strongly consistent procedure for model selection in a regression problem,” Biometrika 76, 369–374 (1989).
    [CrossRef]
  52. H. Akaike, “Fitting autoregressive models for prediction,” Ann. Inst. Statist. Math. 21, 243–247 (1969).
    [CrossRef]
  53. H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control. 19, 716–723 (1974).
    [CrossRef]
  54. G. Schwarz, “Estimating the dimension of a model,” Ann. Statist. 6, 461–464 (1978).
    [CrossRef]
  55. J. Rissanen, “Modeling by shortest data description,” Automatica 12, 94–104 (1991).
  56. S. Konishi, G. Kitagawa, “Generalised information criteria in model selection,” Biometrika 83, 875–890 (1996).
    [CrossRef]
  57. Z. Liang, J. R. MacFall, D. P. Harrington, “Parameter estimation and tissue segmentation from multispectral MR images,” IEEE Trans. Med. Imaging 13, 441–449 (1994).
    [CrossRef]
  58. N. Merhav, “The estimation of model order in exponen-tial families,” IEEE Trans. Inf. Theory 35, 1109–1114 (1989).
    [CrossRef]
  59. D. Hirshberg, N. Merhav, “Robust methods for model order estimation,” IEEE Trans. Signal Process. 44, 620–628 (1996).
    [CrossRef]
  60. G. Qian, H. R. Künsch, “Some notes on Rissanen’s stochastic complexity,” IEEE Trans. Inf. Theory 44, 782–786 (1998).
    [CrossRef]
  61. J. J. Rissanen, “Fisher information and stochastic complexity,” IEEE Trans. Inf. Theory 42, 40–47 (1996).
    [CrossRef]
  62. H. L. Van, Trees Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), p. 80.
  63. See Example 3.3.5, p. 141 of Ref. 32.
  64. A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.
  65. L. Kaufman, “Implementing and accelerating the EM algorithm for positron emission tomography,” IEEE Trans. Med. Imaging 6, 37–51 (1987).
    [CrossRef] [PubMed]

1998 (1)

G. Qian, H. R. Künsch, “Some notes on Rissanen’s stochastic complexity,” IEEE Trans. Inf. Theory 44, 782–786 (1998).
[CrossRef]

1997 (1)

S. T. Buckland, K. P. Burnham, N. H. Augustin, “Model selection: an integral part of inference,” Biometrics 53, 603–618 (1997).
[CrossRef]

1996 (3)

J. J. Rissanen, “Fisher information and stochastic complexity,” IEEE Trans. Inf. Theory 42, 40–47 (1996).
[CrossRef]

S. Konishi, G. Kitagawa, “Generalised information criteria in model selection,” Biometrika 83, 875–890 (1996).
[CrossRef]

D. Hirshberg, N. Merhav, “Robust methods for model order estimation,” IEEE Trans. Signal Process. 44, 620–628 (1996).
[CrossRef]

1995 (3)

1994 (7)

Y. Yang, N. P. Galatsanos, H. Stark, “Projection-based blind deconvolution,” J. Opt. Soc. Am. A 11, 2401–2409 (1994).
[CrossRef]

B. G. Pickard, “Contemplating the plasmalemmal control center model,” Protoplasma 182, 1–9 (1994).
[CrossRef] [PubMed]

B. G. Pickard, C. Reuzeau, K. W. Doolittle, J. G. McNally, “High resolution visualization in onion of distribution patterns of spectrin, talin and vinculin antigenicities,” ASGSB Bull. 8, 54 (1994).

J.-A. Conchello, J. J. Kim, E. W. Hansen, “Enhanced 3-D reconstruction from confocal scanning microscope im-ages. 2: depth discrimination vs. signal-to-noise ratio in partially confocal images,” Appl. Opt. 33, 3740–3750 (1994).
[CrossRef] [PubMed]

J. S. Gens, K. W. Doolittle, J. G. McNally, B. G. Pickard, “Binding sites for antibodies to animal integrin, vitronectin and fibronectin in a plant model for mechanosensing,” Biophys. J. 66, A169 (1994).

Z. Liang, J. R. MacFall, D. P. Harrington, “Parameter estimation and tissue segmentation from multispectral MR images,” IEEE Trans. Med. Imaging 13, 441–449 (1994).
[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]

1993 (4)

1992 (4)

1991 (4)

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).

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

J. Rissanen, “Modeling by shortest data description,” Automatica 12, 94–104 (1991).

1990 (2)

1989 (3)

C. R. Rao, Y. Wu, “A strongly consistent procedure for model selection in a regression problem,” Biometrika 76, 369–374 (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]

N. Merhav, “The estimation of model order in exponen-tial families,” IEEE Trans. Inf. Theory 35, 1109–1114 (1989).
[CrossRef]

1988 (1)

1987 (1)

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

1985 (1)

1984 (1)

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

1980 (1)

A. C. Atkinson, “A note on the generalized information criterion for choice of a model,” Biometrika 67, 413–418 (1980).
[CrossRef]

1978 (1)

G. Schwarz, “Estimating the dimension of a model,” Ann. Statist. 6, 461–464 (1978).
[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).

1974 (1)

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control. 19, 716–723 (1974).
[CrossRef]

1970 (4)

C. G. Broyden, “The convergence of a class of double-rank minimization algorithms. Part I,” J. Inst. Math. Appl. 6, 76–90 (1970);J. Inst. Math. Appl. “ Part II,” 222–231 (1970).
[CrossRef]

R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13, 317–322 (1970).
[CrossRef]

D. Goldfarb, “A family of variable metric methods derived by variational means,” Math. Comput. 24, 23–26 (1970).
[CrossRef]

D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 647–657 (1970).
[CrossRef]

1969 (1)

H. Akaike, “Fitting autoregressive models for prediction,” Ann. Inst. Statist. Math. 21, 243–247 (1969).
[CrossRef]

1967 (1)

C. G. Broyden, “Quasi-Newton methods and their application to function maximization,” Math. Comput. 21, 368–381 (1967).
[CrossRef]

Agard, D. A.

Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).

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

Akaike, H.

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control. 19, 716–723 (1974).
[CrossRef]

H. Akaike, “Fitting autoregressive models for prediction,” Ann. Inst. Statist. Math. 21, 243–247 (1969).
[CrossRef]

Atkinson, A. C.

A. C. Atkinson, “A note on the generalized information criterion for choice of a model,” Biometrika 67, 413–418 (1980).
[CrossRef]

Augustin, N. H.

S. T. Buckland, K. P. Burnham, N. H. Augustin, “Model selection: an integral part of inference,” Biometrics 53, 603–618 (1997).
[CrossRef]

Avinash, G. B.

G. B. Avinash, “Simultaneous blur and image restoration in 3D optical microscopy,” Zoological Studies 34 Suppl. I, 184–185 (1995).

Berry, D. S.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Bhattacharyya, S.

Bille, J.

Born, M.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 436.

Broyden, C. G.

C. G. Broyden, “The convergence of a class of double-rank minimization algorithms. Part I,” J. Inst. Math. Appl. 6, 76–90 (1970);J. Inst. Math. Appl. “ Part II,” 222–231 (1970).
[CrossRef]

C. G. Broyden, “Quasi-Newton methods and their application to function maximization,” Math. Comput. 21, 368–381 (1967).
[CrossRef]

Buckland, S. T.

S. T. Buckland, K. P. Burnham, N. H. Augustin, “Model selection: an integral part of inference,” Biometrics 53, 603–618 (1997).
[CrossRef]

Burnham, K. P.

S. T. Buckland, K. P. Burnham, N. H. Augustin, “Model selection: an integral part of inference,” Biometrics 53, 603–618 (1997).
[CrossRef]

Carrington, W. A.

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).

Conan, J.-M.

Conchello, J. A.

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. 29, 3795–3804 (1990).
[CrossRef] [PubMed]

J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
[CrossRef]

J. Markham, J. A. Conchello, “Parametric blind deconvolution of microscopic images: further results,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, chairs/eds., Proc. SPIE3261, 38–49 (1998).
[CrossRef]

J. A. Conchello, “Fluorescence photobleaching correction for expectation maximization algorithm,” in Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson, C. J. Cogswell, eds., Proc. SPIE2412, 138–146 (1995).
[CrossRef]

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

Conchello, J.-A.

DeGroot, M. H.

M. H. DeGroot, Probability and Statistics, 2nd ed. (Addison Wesley, Reading, Mass., 1984), pp. 348–349.

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).

Doolittle, K. W.

B. G. Pickard, C. Reuzeau, K. W. Doolittle, J. G. McNally, “High resolution visualization in onion of distribution patterns of spectrin, talin and vinculin antigenicities,” ASGSB Bull. 8, 54 (1994).

J. S. Gens, K. W. Doolittle, J. G. McNally, B. G. Pickard, “Binding sites for antibodies to animal integrin, vitronectin and fibronectin in a plant model for mechanosensing,” Biophys. J. 66, A169 (1994).

England, A. H.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Erhardt, A.

Faulkner, T. R.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Fay, F. S.

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).

Fienup, J. R.

Fletcher, R.

R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13, 317–322 (1970).
[CrossRef]

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987), p. 22.

Fogarty, K. E.

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).

Galatsanos, N. P.

Gens, J. S.

J. S. Gens, K. W. Doolittle, J. G. McNally, B. G. Pickard, “Binding sites for antibodies to animal integrin, vitronectin and fibronectin in a plant model for mechanosensing,” Biophys. J. 66, A169 (1994).

Gens, S. C.

S. C. Gens, J. G. McNally, B. G. Pickard, “Resolution of binding sites for antibodies to integrin, vitronection and fibronectin on onion epidermis protoplasts and depectinated walls,” ASGSB Bull. 7, 42 (1993).

Gibson, F. S.

Goldfarb, D.

D. Goldfarb, “A family of variable metric methods derived by variational means,” Math. Comput. 24, 23–26 (1970).
[CrossRef]

Green, W. A.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Hammoud, A. M.

Hansen, E. W.

Harrington, D. P.

Z. Liang, J. R. MacFall, D. P. Harrington, “Parameter estimation and tissue segmentation from multispectral MR images,” IEEE Trans. Med. Imaging 13, 441–449 (1994).
[CrossRef]

Hirshberg, D.

D. Hirshberg, N. Merhav, “Robust methods for model order estimation,” IEEE Trans. Signal Process. 44, 620–628 (1996).
[CrossRef]

Holden, J. T.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Holmes, T. J.

Joshi, S.

Kam, Z.

Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).

Katsaggelos, A. K.

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

Kaufman, L.

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

Kim, J. J.

Kitagawa, G.

S. Konishi, G. Kitagawa, “Generalised information criteria in model selection,” Biometrika 83, 875–890 (1996).
[CrossRef]

Komitowski, D.

Konishi, S.

S. Konishi, G. Kitagawa, “Generalised information criteria in model selection,” Biometrika 83, 875–890 (1996).
[CrossRef]

Krishnamurthi, V.

Künsch, H. R.

G. Qian, H. R. Künsch, “Some notes on Rissanen’s stochastic complexity,” IEEE Trans. Inf. Theory 44, 782–786 (1998).
[CrossRef]

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).

Lane, R. G.

Lanni, F.

Lay, K. T.

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

Leptin, M.

Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).

Liang, Z.

Z. Liang, J. R. MacFall, D. P. Harrington, “Parameter estimation and tissue segmentation from multispectral MR images,” IEEE Trans. Med. Imaging 13, 441–449 (1994).
[CrossRef]

Liu, Y.-H.

MacFall, J. R.

Z. Liang, J. R. MacFall, D. P. Harrington, “Parameter estimation and tissue segmentation from multispectral MR images,” IEEE Trans. Med. Imaging 13, 441–449 (1994).
[CrossRef]

Markham, J.

J. Markham, J. A. Conchello, “Parametric blind deconvolution of microscopic images: further results,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, chairs/eds., Proc. SPIE3261, 38–49 (1998).
[CrossRef]

McCallum, B. C.

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

McNally, J. G.

B. G. Pickard, C. Reuzeau, K. W. Doolittle, J. G. McNally, “High resolution visualization in onion of distribution patterns of spectrin, talin and vinculin antigenicities,” ASGSB Bull. 8, 54 (1994).

J. S. Gens, K. W. Doolittle, J. G. McNally, B. G. Pickard, “Binding sites for antibodies to animal integrin, vitronectin and fibronectin in a plant model for mechanosensing,” Biophys. J. 66, A169 (1994).

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]

S. C. Gens, J. G. McNally, B. G. Pickard, “Resolution of binding sites for antibodies to integrin, vitronection and fibronectin on onion epidermis protoplasts and depectinated walls,” ASGSB Bull. 7, 42 (1993).

C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized 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 Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 199–208 (1996).
[CrossRef]

J. G. McNally, “Computational optical-sectioning microscopy for 3D quantization of cell motion: results and challenges,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 342–351 (1994).
[CrossRef]

Merhav, N.

D. Hirshberg, N. Merhav, “Robust methods for model order estimation,” IEEE Trans. Signal Process. 44, 620–628 (1996).
[CrossRef]

N. Merhav, “The estimation of model order in exponen-tial families,” IEEE Trans. Inf. Theory 35, 1109–1114 (1989).
[CrossRef]

Middleton, D.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Miller, M. I.

Minden, J. S.

Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).

Parker, D. F.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Paxman, R. G.

Pickard, B. G.

J. S. Gens, K. W. Doolittle, J. G. McNally, B. G. Pickard, “Binding sites for antibodies to animal integrin, vitronectin and fibronectin in a plant model for mechanosensing,” Biophys. J. 66, A169 (1994).

B. G. Pickard, “Contemplating the plasmalemmal control center model,” Protoplasma 182, 1–9 (1994).
[CrossRef] [PubMed]

B. G. Pickard, C. Reuzeau, K. W. Doolittle, J. G. McNally, “High resolution visualization in onion of distribution patterns of spectrin, talin and vinculin antigenicities,” ASGSB Bull. 8, 54 (1994).

S. C. Gens, J. G. McNally, B. G. Pickard, “Resolution of binding sites for antibodies to integrin, vitronection and fibronectin on onion epidermis protoplasts and depectinated walls,” ASGSB Bull. 7, 42 (1993).

Preza, C.

Qian, G.

G. Qian, H. R. Künsch, “Some notes on Rissanen’s stochastic complexity,” IEEE Trans. Inf. Theory 44, 782–786 (1998).
[CrossRef]

Rao, C. R.

C. R. Rao, Y. Wu, “A strongly consistent procedure for model selection in a regression problem,” Biometrika 76, 369–374 (1989).
[CrossRef]

Reuzeau, C.

B. G. Pickard, C. Reuzeau, K. W. Doolittle, J. G. McNally, “High resolution visualization in onion of distribution patterns of spectrin, talin and vinculin antigenicities,” ASGSB Bull. 8, 54 (1994).

Rissanen, J.

J. Rissanen, “Modeling by shortest data description,” Automatica 12, 94–104 (1991).

Rissanen, J. J.

J. J. Rissanen, “Fisher information and stochastic complexity,” IEEE Trans. Inf. Theory 42, 40–47 (1996).
[CrossRef]

Rogers, T. G.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Rubin, D. B.

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).

Schulz, T. J.

Schwarz, G.

G. Schwarz, “Estimating the dimension of a model,” Ann. Statist. 6, 461–464 (1978).
[CrossRef]

Seber, G. A.

G. A. Seber, C. J. Wild, Nonlinear Regression (Wiley, New York, 1989), pp. 605–609.

Sedat, J. W.

Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).

Shanno, D. F.

D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 647–657 (1970).
[CrossRef]

Snyder, D.

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

Snyder, D. L.

Spencer, A. J. M.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

Stark, H.

Thiébaut, E.

Thomas, L. J.

Turner, J. N.

Van, H. L.

H. L. Van, Trees Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), p. 80.

White, R. L.

Wild, C. J.

G. A. Seber, C. J. Wild, Nonlinear Regression (Wiley, New York, 1989), pp. 605–609.

Wolf, E.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 436.

Wu, Y.

C. R. Rao, Y. Wu, “A strongly consistent procedure for model selection in a regression problem,” Biometrika 76, 369–374 (1989).
[CrossRef]

Yang, Y.

Yu, Q.

J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
[CrossRef]

Zinser, G.

Ann. Inst. Statist. Math. (1)

H. Akaike, “Fitting autoregressive models for prediction,” Ann. Inst. Statist. Math. 21, 243–247 (1969).
[CrossRef]

Ann. Statist. (1)

G. Schwarz, “Estimating the dimension of a model,” Ann. Statist. 6, 461–464 (1978).
[CrossRef]

Annu. Rev. Biophys. Bioeng. (1)

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

Appl. Opt. (4)

ASGSB Bull. (2)

B. G. Pickard, C. Reuzeau, K. W. Doolittle, J. G. McNally, “High resolution visualization in onion of distribution patterns of spectrin, talin and vinculin antigenicities,” ASGSB Bull. 8, 54 (1994).

S. C. Gens, J. G. McNally, B. G. Pickard, “Resolution of binding sites for antibodies to integrin, vitronection and fibronectin on onion epidermis protoplasts and depectinated walls,” ASGSB Bull. 7, 42 (1993).

Automatica (1)

J. Rissanen, “Modeling by shortest data description,” Automatica 12, 94–104 (1991).

Biometrics (1)

S. T. Buckland, K. P. Burnham, N. H. Augustin, “Model selection: an integral part of inference,” Biometrics 53, 603–618 (1997).
[CrossRef]

Biometrika (3)

A. C. Atkinson, “A note on the generalized information criterion for choice of a model,” Biometrika 67, 413–418 (1980).
[CrossRef]

C. R. Rao, Y. Wu, “A strongly consistent procedure for model selection in a regression problem,” Biometrika 76, 369–374 (1989).
[CrossRef]

S. Konishi, G. Kitagawa, “Generalised information criteria in model selection,” Biometrika 83, 875–890 (1996).
[CrossRef]

Biophys. J. (1)

J. S. Gens, K. W. Doolittle, J. G. McNally, B. G. Pickard, “Binding sites for antibodies to animal integrin, vitronectin and fibronectin in a plant model for mechanosensing,” Biophys. J. 66, A169 (1994).

Comput. J. (1)

R. Fletcher, “A new approach to variable metric algorithms,” Comput. J. 13, 317–322 (1970).
[CrossRef]

Development (Cambridge, UK) (1)

Z. Kam, J. S. Minden, D. A. Agard, J. W. Sedat, M. Leptin, “Drosophila gastrulation: analysis of cell shape changes in living embryos by three-dimensional fluorescence microscopy,” Development (Cambridge, UK) 112, 365–370 (1991).

IEEE Trans. Autom. Control. (1)

H. Akaike, “A new look at the statistical model identification,” IEEE Trans. Autom. Control. 19, 716–723 (1974).
[CrossRef]

IEEE Trans. Inf. Theory (3)

N. Merhav, “The estimation of model order in exponen-tial families,” IEEE Trans. Inf. Theory 35, 1109–1114 (1989).
[CrossRef]

G. Qian, H. R. Künsch, “Some notes on Rissanen’s stochastic complexity,” IEEE Trans. Inf. Theory 44, 782–786 (1998).
[CrossRef]

J. J. Rissanen, “Fisher information and stochastic complexity,” IEEE Trans. Inf. Theory 42, 40–47 (1996).
[CrossRef]

IEEE Trans. Med. Imaging (2)

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

Z. Liang, J. R. MacFall, D. P. Harrington, “Parameter estimation and tissue segmentation from multispectral MR images,” IEEE Trans. Med. Imaging 13, 441–449 (1994).
[CrossRef]

IEEE Trans. Signal Process. (2)

D. Hirshberg, N. Merhav, “Robust methods for model order estimation,” IEEE Trans. Signal Process. 44, 620–628 (1996).
[CrossRef]

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood blur identification and image restoration using the EM algorithm,” IEEE Trans. Signal Process. 39, 729–733 (1991).
[CrossRef]

J. Inst. Math. Appl. (1)

C. G. Broyden, “The convergence of a class of double-rank minimization algorithms. Part I,” J. Inst. Math. Appl. 6, 76–90 (1970);J. Inst. Math. Appl. “ Part II,” 222–231 (1970).
[CrossRef]

J. Opt. Soc. Am. A (13)

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]

T. J. Schulz, “Multiframe blind deconvolution of astronomical images,” J. Opt. Soc. Am. A 10, 1064–1073 (1993).
[CrossRef]

T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
[CrossRef] [PubMed]

R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
[CrossRef]

Y. Yang, N. P. Galatsanos, H. Stark, “Projection-based blind deconvolution,” J. Opt. Soc. Am. A 11, 2401–2409 (1994).
[CrossRef]

R. G. Paxman, T. J. Schulz, J. R. Fienup, “Joint estimation of object and aberrations by using phase diversity,” J. Opt. Soc. Am. A 9, 1072–1085 (1992).
[CrossRef]

E. Thiébaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
[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]

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

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]

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

C. Preza, M. I. Miller, L. J. Thomas, J. G. McNally, “Regularized method for reconstruction of three-dimensional microscopic objects from optical sections,” J. Opt. Soc. Am. A 9, 219–228 (1992).
[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]

J. R. Statist. Soc. B (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).

Math. Comput. (3)

D. Goldfarb, “A family of variable metric methods derived by variational means,” Math. Comput. 24, 23–26 (1970).
[CrossRef]

D. F. Shanno, “Conditioning of quasi-Newton methods for function minimization,” Math. Comput. 24, 647–657 (1970).
[CrossRef]

C. G. Broyden, “Quasi-Newton methods and their application to function maximization,” Math. Comput. 21, 368–381 (1967).
[CrossRef]

Opt. Commun. (1)

B. C. McCallum, “Blind deconvolution by simulated annealing,” Opt. Commun. 75, 101–105 (1990).
[CrossRef]

Protoplasma (1)

B. G. Pickard, “Contemplating the plasmalemmal control center model,” Protoplasma 182, 1–9 (1994).
[CrossRef] [PubMed]

Zoological Studies (1)

G. B. Avinash, “Simultaneous blur and image restoration in 3D optical microscopy,” Zoological Studies 34 Suppl. I, 184–185 (1995).

Other (19)

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

M. H. DeGroot, Probability and Statistics, 2nd ed. (Addison Wesley, Reading, Mass., 1984), pp. 348–349.

See pp. 21–23 of Ref. 35.

The netlib routines are available from the University of Tennessee, Knoxville, URL http://netlib2.cs.utk.edu .

R. Fletcher, Practical Methods of Optimization, 2nd ed. (Wiley, New York, 1987), p. 22.

See Subsection 4.1 of Ref. 35.

G. A. Seber, C. J. Wild, Nonlinear Regression (Wiley, New York, 1989), pp. 605–609.

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).

J. A. Conchello, Q. Yu, “Parametric blind deconvolution of fluorescence microscopy images: preliminary results,” in Three-Dimensional Microscopy: Image Acquisition and Processing III, C. J. Cogswell, G. Kino, T. Wilson, eds., Proc. SPIE2655, 164–174 (1996).
[CrossRef]

J. Markham, J. A. Conchello, “Parametric blind deconvolution of microscopic images: further results,” in Three-Dimensional and Multidimensional Microscopy: Image Acquisition and Processing V, C. J. Cogswell, J. A. Conchello, T. Wilson, chairs/eds., Proc. SPIE3261, 38–49 (1998).
[CrossRef]

J. G. McNally, “Computational optical-sectioning microscopy for 3D quantization of cell motion: results and challenges,” in Image Reconstruction and Restoration, T. J. Schulz, D. L. Snyder, eds., Proc. SPIE2302, 342–351 (1994).
[CrossRef]

J. A. Conchello, “Fluorescence photobleaching correction for expectation maximization algorithm,” in Three-Dimensional Microscopy: Image Acquisition and Processing II, T. Wilson, C. J. Cogswell, eds., Proc. SPIE2412, 138–146 (1995).
[CrossRef]

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

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

40×/1.0 NA is notation used widely to give the magnification and the numerical aperture of an objective. In this case the objective magnifies 40 times and has a numerical aperture of 1.0.

M. Born, E. Wolf, Principles of Optics, 6th ed. (Pergamon, New York, 1980), p. 436.

H. L. Van, Trees Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), p. 80.

See Example 3.3.5, p. 141 of Ref. 32.

A. J. M. Spencer, D. F. Parker, D. S. Berry, A. H. England, T. R. Faulkner, W. A. Green, J. T. Holden, D. Middleton, T. G. Rogers, Engineering Mathematics (Van Nostrand Reinhold, New York, 1977), Vol. 2, Chap. 8.

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Figures (13)

Fig. 1
Fig. 1

Schematic of the cross section of the second simulated test specimen. The intensity of the planes (lines in the figure) is twice the intensity of the elliptical shell. The vertical axes are 4.8 times the horizontal axes. The horizontal spokes are three times thicker than the vertical and diagonal ones. The figure on the left shows the dimensions of the test specimen; the figure on the right shows the appearance of the test specimen after the subsampling of the image in the z axis as described above.

Fig. 2
Fig. 2

Results from PBD of a simulated spherical shell of uniform fluorescence. Lateral (left) and axial (right) medial sections through the image (a) and estimated specimen function (b)–(d): (b) 200 PBD iterations, (c) 3000 PBD iterations, (d) 3000 iterations of the nonblind (NB) EM algorithm with the correct PSF. For display purposes the vertical cuts were scaled in z to obtain a 1:1 aspect ratio.

Fig. 3
Fig. 3

Lateral (top) and axial (bottom) intensity traces through the actual PSF (solid curves) and the PSF estimated by 3000 iterations of the PBD algorithm (dotted curves) with a spherical shell as the test object. For the plots, the PSF’s were normalized to h(0, 0)=1.

Fig. 4
Fig. 4

Image and restorations with a spoked-wheel test specimen. Only xz or vertical sections are shown. (a) Simulated image, (b) estimated specimen function after 3000 iterations of the nonblind EM algorithm with the correct PSF, (c) 500 PBD iterations, (d) 3000 PBD iterations.

Fig. 5
Fig. 5

Lateral (top) and axial (bottom) intensity traces across the actual PSF (solid curves) and the estimated PSF after 3000 iterations of the PBD algorithm with the spoked-wheel test specimen (dotted curves). For the plots, the PSF’s were normalized to h(0, 0)=1.0.

Fig. 6
Fig. 6

Lateral (top) and axial (bottom) intensity traces across an aberrated PSF for the actual PSF (solid curves) and the estimated PSF after 200 iterations of the PBD algorithm (dotted curves). For the plots, the PSF’s were normalized to have a maximum value of 1.0.

Fig. 7
Fig. 7

Lateral (left) and axial (right) medial sections through the image (top) and the estimated specimen function obtained with 200 iterations of the PBD algorithm (middle) and with 200 iterations of the nonblind EM algorithm with the correct PSF (bottom). The PSF suffered from spherical aberration that results from assuming that the spherical-shell test object was 20 μm under the coverslip in a watery medium. For display purposes the xz sections were scaled in z to obtain a 1:1 aspect ratio.

Fig. 8
Fig. 8

Image and estimated specimen function for a noisy process of image formation with three different SNR’s. Left, high-SNR=31.6 (maximum photon rate=1000 photons/pixel); center, intermediate SNR=10 (maximum photon rate=100 photons/pixel); right, High noise, low SNR=6.32 (maximum photon rate=40 photons/pixel). For each noise level, the left-hand columns show lateral (or xy) medial sections and the right-hand columns show axial (or xz) medial sections. Each column is divided in two half-images. The half-image on the left-hand side is the specimen function estimate obtained with the nonblind (NB) EM algorithm with the correct PSF. The half-image on the right-hand side shows the specimen function estimate obtained with the PBD algorithm. The top row shows the simulated image; the other three rows are the estimated specimen function after 200, 500, and 3,000 iterations (top to bottom). For display purposes, the xz sections were scaled to a 1:1 aspect ratio.

Fig. 9
Fig. 9

Lateral (top) and axial (bottom) intensity traces through the actual PSF (solid curves) and the PSF estimated from an image with high SNR with 200 iterations of the PBD algorithm (dashed curves line). The maximum photon rate in the image was 1000 photons/pixel (SNR31.6). For the plots, the PSF’s were normalized to h(0, 0)=1.

Fig. 10
Fig. 10

Lateral (top) and axial (bottom) intensity traces through the actual PSF (solid curves) and the PSF estimated from an image with high SNR with 500 iterations of the PBD algorithm (dashed curves). The maximum photon rate in the image was 1000 photons/pixel (SNR31.6). For the plots, the PSF’s were normalized to h(0, 0)=1.

Fig. 11
Fig. 11

Lateral (top) and axial (bottom) intensity traces through the actual PSF (solid curves) and the PSF estimated from an image with low SNR with 200 iterations of the PBD algorithm (dotted curves). The maximum photon rate in the image was 40 photons/pixel (SNR6.32). For the plots, the PSF’s were normalized to h(0, 0)=1.

Fig. 12
Fig. 12

Lateral (top) and axial (bottom) intensity traces through the actual PSF (solid curves) and the PSF estimated from an image with low SNR with 500 iterations of the PBD algorithm (dotted curves). The maximum photon rate in the image was 40 photons/pixel (SNR6.32). For the plots, the PSF’s were normalized to h(0, 0)=1.

Fig. 13
Fig. 13

Lateral (top) and axial (bottom) intensity traces through the actual PSF (solid curves) and the PSF estimated from an image with low SNR with 200 iterations of the PBD algorithm (dashed curves). The maximum photon rate in the image was 40 photons/pixel (SNR6.32). For these results the phase aberration B(ρ) is assumed to have two terms, and the misfocus-induced aberration C(ρ) is assumed to have six terms. For the plots, the PSF’s were normalized to h(0, 0)=1.

Tables (1)

Tables Icon

Table 1 Schedule for PF Updates and Tolerance with Iteration

Equations (53)

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

g(xi)=Oh(xi-xo)s(xo)dxo=hs,
Ho=Oh(xo)dxo,So=Os(xo)dxo
L[s(), Θ]=-IOh(xi-xo; Θ)s(xo)dxodxi+Ig(xi)logOh(xi-xo; Θ)×s(xo)dxodxi.
hnc(xo; Θ)=01J0[(2π/λ)aρro/zd]P(ρ, zo, Θ)ρdρ2,
P(ρ, zo, Θ)=A(ρ)exp[jW(ρ, zo)]
W(ρ, zo)=(2π/λ)OPD(ρ, zo)=zoC(ρ)+B(ρ),
A(ρ)n=0Na-1anχn(ρ),B(ρ)n=0Nb-1bnϕn(ρ),
C(ρ)n=0Nc-1cnψn(ρ),
a={a0, a1, a2,, aNa-1},
b={b0, b1, b2,, bNb-1},
c={c0, c1, c2,, cNc-1}.
χn(ρ)=ρn,ϕn(ρ)=ρ2n,ψn(ρ)=ρ2n,
forn=0, 1, 2, .
s^(k+1)(x)=s^(k)(x)H0×h^(k)(-x)g(x)h^(k)(x)s^(k)(x).
Θ^(n+1)=Θ^(n)+α(n)σ(n),
α(n)=argmaxα>0{L[Θ^(n)+ασ(n)]};
|(L(k)-L(k-1))/L(k)|<
σ(k)=ΘL(k).
σ(k)=-C(k)ΘL(k),
C(k+1)=C(k)+E(k),
E(k)=-C(k)γ(k)[γ(k)]TC(k)[γ(k)]TC(k)γ(k)+δ(k)[δ(k)]T[δ(k)]Tγ(k)+r(k)[r(k)]T,
r(k)={[γ(k)]TC(k)γ(k)}1/2δ(k)[δ(k)]Tγ(k)-C(k)γ(k)[γ(k)]TC(k)γ(k),
γ(k)=ΘL(k+1)-ΘL(k),
δ(k)=Θ^(k+1)-Θ^(k).
maxi{|[θi(k+1)-θi(k)]/θi(k+1)|}<
Λ(α)=L(Θ^(k)+ασ(k))-L(Θ^(k)).
p(xo; Θ)=Fνro{P(ν, zo; Θ)},
h(xo;Θ)=|p(xo, Θ)|2.
Lθj=-I[h^θjsˆ-g×(h^θjsˆ)/(hˆsˆ)]dxi,
h^θj=hˆ/θj;j=1, 2,, NΘ.
Lbi=-2νBϕi(ν)ImzoOP(ν, zo; Θ)R(ν, zo; Θ),
Lci=-2νBψi(ν)ImzoOzoP(ν, zo; Θ)R(ν, zo; Θ),
Lai=2νBχi(ν)RezoOexp[jW(ν, zo; Θ)]R(ν, zo; Θ),
R(ν, zo, Θ)=Froν{p*(xo; Θ)[d(xo)s(-xo)]};
d(x)=[g(x)/gˆ(x)]-1,
gˆ(x)=sˆ(x)hˆ(x)
B(ρ)=b4ρ4+b6ρ6+,
C(ρ)=c2ρ2+c4ρ4+,
v=(2πNA/λ)r=(2πNA/λ)(x2+y2)1/2,
u=z(2πNA2)/(λnoil)
Lcd(s, Θ)=-OIh(xi-xo; Θ)s(xo)dxidxo+OI log[h(xi-xo; Θ)s(xo)]×N(dxi, dxo).
Lcd(s, Θ)=-Odxos(xo)Ih(xi-xo)dxi+Odxo log[s(xo)]IN(dxi; dxo)+Odxo log[h(xi-xo)]IN(dxi; dxo).
Q(s, Θ|s^(k), Θ^(k))=-H0Odxos(xo)+Odxo log[s(xo)]×IE{N(xi; xo)|g, s^(k), Θ^(k)}dxi+OI log[h(xi-xo)]×E{N(xi; xo)|g, s^(k), Θ^(k)}dxidxo,
H0=Ih(xi-xo)dxi
Odxo log[h(xo)]IE{N(xi; xi-xo)|g, s^(k), Θ^(k)}dxi.
E{N(xi; xo)|g, s^(k), Θ^(k)}
=h^(k)(xi-xo)s^(k)(xo)g(xi)/g^(k)(xi),
g^(k)(xi)=h^(k)s^(k)
Q(s, Θ|s^(k), Θ^(k))=-H(0)Odxos(xo)+Odxo log[s(xo)]s^(k)(xo)×Ih^(k)(xi-xo) g(xi)g^(k)(xi) dxi+Odxo log[h(xo)]h^(k)(-xo)×Is^(k)(xi-xo) g(xi)g^(k)(xi) dxi.
Q=Qs+Qh,
Qs=-H(0)Odxos(xo)+Odxo log[s(xo)]s^(k)(xo)×Ih^(k)(xi-xo) g(xi)g^(k)(xi) dxi,
Qh=Odxo log[h(xo)]h^(k)(-xo)×Is^(k)(xi-xo) g(xi)g^(k)(xi) dxi.
{s^(k+1), Θ^(k+1)}=argmax{Q(s, Θ|g, s^(k), Θ^(k))},

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