Abstract

We present efficient algorithms for image restoration by means of Good’s roughness penalty. We assume Gaussian or Poisson statistics for the noise and derive an algorithm for each case. Performance is tested by simulated three-dimensional imaging with a fluorescence confocal laser scanning microscope. Results are compared with those for algorithms that use Gaussian or entropy penalty terms, which we derived previously [J. Opt. Soc. Am. A 14, 1696 (1997)]. The algorithms based on Good’s roughness yield superior results. An example is given of the restoration of an image of a biological specimen.

© 1998 Optical Society of America

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References

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  1. D. A. Agard, Y. Hiraoka, P. Shaw, J. W. Sedat, “Fluorescence microscopy in three dimensions,” in Methods in Cell Biology, D. L. Taylor, Y. Wang, eds. (Academic, San Diego, Calif., 1989), Vol. 3, pp. 353–377.
  2. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data via the EM algorithm,” J. R. Stat. Soc. B 39, 1–38 (1977).
  3. T. J. Holmes, “Maximum-likelihood image restoration adapted for noncoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]
  4. T. J. Holmes, Y.-H. Liu, “Acceleration of maximum-likelihood image restoration for fluorescence microscopy and other noncoherent imagery,” J. Opt. Soc. Am. A 8, 893–907 (1991).
    [CrossRef]
  5. W. A. Carrington, “Image restoration in 3D microscopy with limited data,” in Bioimaging and Two-Dimensional Spectroscopy, L. C. Smith, ed., Proc. SPIE1205, 72–83 (1990).
    [CrossRef]
  6. W. A. Carrington, R. M. Lynch, E. D. W. Moore, G. Isenberg, K. E. Fogarty, F. S. Fay, “Superresolution three-dimensional images of fluorescence in cells with minimal light exposure,” Science 268, 1483–1487 (1995).
    [CrossRef] [PubMed]
  7. R. L. Lagendijk, “Iterative Identification and Restoration of Images,” Ph.D. dissertation (Delft Technical University, Delft, The Netherlands, 1990).
  8. H. T. M. van der Voort, K. C. Strasters, “Restoration of confocal images for quantitative image analysis,” J. Microsc. 178, 165–181 (1995).
    [CrossRef]
  9. P. J. Verveer, T. M. Jovin, “Acceleration of the ICTM image restoration algorithm,” J. Microsc. 188, 191–195 (1997).
    [CrossRef]
  10. 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]
  11. 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]
  12. M. Bertero, P. Boccacci, “Regularization methods in image restoration: an application to HST images,” Int. J. Imaging Syst. Technol. 6, 376–386 (1995).
    [CrossRef]
  13. T. M. Jovin, D. J. Arndt-Jovin, “Luminescence digital imaging microscopy,” Annu. Rev. Biophys. Chem. 18, 271–308 (1989).
    [CrossRef]
  14. I. J. Good, “Nonparametric roughness penalties for probability densities,” Biometrika 58, 255–277 (1971).
    [CrossRef]
  15. D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
    [CrossRef]
  16. L. J. van Vliet, “Grey-scale measurements in multi-dimensional digitized images,” Ph.D. dissertation, (Delft Technical University, Delft, The Netherlands, 1993).
  17. P. J. Verveer, G. M. P. van Kempen, T. M. Jovin, “Super-resolution MAP algorithms applied to fluorescence imaging,” in Three-Dimensional Microscopy: Image Acquisition and Processing IV, C. J. Cogswell, J.-A. Conchello, T. Wilson, eds., Proc. SPIE2984, 125–135 (1997).
    [CrossRef]
  18. H. T. M. van der Voort, G. J. Brakenhoff, “3-D image formation in a high-aperture fluorescence confocal microscope: a numerical analysis,” J. Microsc. 158, 43–54 (1990).
    [CrossRef]
  19. W. H. Press, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, UK, 1992).

1997 (2)

1995 (3)

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

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

1993 (1)

1991 (1)

1990 (1)

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

1989 (1)

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

1988 (1)

1980 (1)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

1977 (1)

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

1971 (1)

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

Agard, D. A.

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

Arndt-Jovin, D. J.

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

Bertero, M.

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

Boccacci, P.

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

Brakenhoff, G. J.

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

Carrington, W. A.

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

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

Dempster, A. P.

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

Fay, F. S.

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

Fogarty, K. E.

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

Good, I. J.

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

Hildreth, E.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

Hiraoka, Y.

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

Holmes, T. J.

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

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

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

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

Lagendijk, R. L.

R. L. Lagendijk, “Iterative Identification and Restoration of Images,” Ph.D. dissertation (Delft Technical University, Delft, The Netherlands, 1990).

Laird, N. M.

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

Liu, Y.-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]

Marr, D.

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

Miller, M. I.

Moore, E. D. W.

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

Press, W. H.

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

Rubin, D. B.

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

Sedat, J. W.

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

Shaw, P.

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

Strasters, K. C.

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

Teukolsky, S. A.

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

van der Voort, H. T. M.

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

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

van Kempen, G. M. P.

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

van Vliet, L. J.

L. J. van Vliet, “Grey-scale measurements in multi-dimensional digitized images,” Ph.D. dissertation, (Delft Technical University, Delft, The Netherlands, 1993).

Verveer, P. J.

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

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

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

Vetterling, W. T.

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

Annu. Rev. Biophys. Chem. (1)

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

Biometrika (1)

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

Int. J. Imaging Syst. Technol. (1)

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

J. Microsc. (3)

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

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

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

J. Opt. Soc. Am. A (4)

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

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

Proc. R. Soc. London, Ser. B (1)

D. Marr, E. Hildreth, “Theory of edge detection,” Proc. R. Soc. London, Ser. B 207, 187–217 (1980).
[CrossRef]

Science (1)

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

Other (6)

R. L. Lagendijk, “Iterative Identification and Restoration of Images,” Ph.D. dissertation (Delft Technical University, Delft, The Netherlands, 1990).

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

L. J. van Vliet, “Grey-scale measurements in multi-dimensional digitized images,” Ph.D. dissertation, (Delft Technical University, Delft, The Netherlands, 1993).

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

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

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

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Figures (6)

Fig. 1
Fig. 1

Simulated confocal fluorescence imaging. (a) Object, (b) simulated fluorescence confocal image distorted with Gaussian noise (SNR=50), (c) simulated fluorescence confocal image distorted with Poisson noise (β=5); (d)–(i) results of the algorithms: (d) MAPGG, (e) MAPGE, (f) MAPGR, (g) MAPPG, (h) MAPPE, (i) MAPPR. All panels show a single section from the center of the 3D image.

Fig. 2
Fig. 2

MSE as a function of the number of iterations for the MAPGG, the MAPGE, and the MAPGR algorithms. A simulated confocal image was used with Gaussian noise with SNR=50. The regularization parameter was chosen to minimize the MSE after 500 iterations.

Fig. 3
Fig. 3

MSE as a function of the number of iterations for the MAPPG, the MAPPE, and the MAPPR algorithms. A simulated confocal image was used with Poisson noise with β=5. The regularization parameter was chosen to minimize the MSE after 250 iterations.

Fig. 4
Fig. 4

MSE as a function of the SNR for the MAPGG, the MAPGE, and the MAPGR algorithms. A simulated confocal image was used with Gaussian noise with varying SNR. The regularization parameters were chosen by renormalizing the value used in Fig. 2 with Eq. (12). The number of iterations was 500.

Fig. 5
Fig. 5

MSE as a function of β for the MAPPG, the MAPPE, and the MAPPR algorithms. A simulated confocal image was used with Poisson noise with varying β. The regularization parameters were chosen by renormalizing the value used in Fig. 3 with Eq. (14). The number of iterations was 250.

Fig. 6
Fig. 6

Image of nuclear envelopes of a Drosophila melanogaster embryo and restorations using the MAPPG, the MAPPE, and the MAPPR algorithms. (a) original image, (b) MAPPG, (c) MAPPE, (d) MAPPR. All panels show a single section from the middle of the 3D image.

Tables (1)

Tables Icon

Table 1 Average Execution Times per Iterationa

Equations (27)

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g=N(Hf+b),
p(f|g)=p(g|f)p(f)p(g).
Ψ=L(f, g)+γP(f).
LG=Hf-g2/σ2,
LP=βi=1M[Hf]i-gT ln(Hf+b),
P=d=1N-+ 1f(u)f(u)ud2dud.
P=4d=1N-+x(u)ud2dud.
P=4d=1N-+x(u)udd(x(ud))=x()udx()-x(-)udx(-)-4d=1N-+x(u)dx(u)ud=12f()ud-f(-)ud-4d=1N-+x(u) 2x(u)ud2dud=-4d=1N-+x(u) 2x(u)ud2dud.
P=-4d=1NxTLdx=-4xTLx,
2x(u)ud2x(u1, , ud+Sd, , uN)-2x(u)+x(u1, , ud-Sd, , uN)Sd2,
ΨG=Hx2-g2-4γGxTLx,
γGσ2.
ΨP=i=1M[Hx2]i-gT ln(Hx2+b)-4γPxTLx,
γP1/β.
SNR=maxi(Hf)i-mini(Hf)iσ,
xk+1=xk+αkdk,
dk=βkdk-1-Ψ(xk),
βk=Ψ(xk)2Ψ(xk-1)2.
ΨG=4XHT(Hx2-g)-8γLx,
ΨP=2XHT1-gHx2+b-8γLx,
p=(d2)THTHd2,
q=4(d2)THTHXd,
r=4dTXHTHXd+2(d2)THT(Hx2-g)-4γdTLd,
s=4dTXHT(Hx2-g)-8γdTLx,
t=(x2)THT(Hx2-2g)+gTg-4γxTLx.
dΨPdα=2(Xd+αd2)THT1-gH(x+αd)2+b-8γdTL(x+αd),
d2ΨPdα2=2(d2)THT1-gH(x+αd)2+b+4gTH(Xd+αd2)H(x+αd)2+b2-8γdTLd.

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