Abstract

A blind deconvolution algorithm based on the Richardson–Lucy deconvolution algorithm is presented. Its performance in the presence of noise is found to be superior to that of other blind deconvolution algorithms. Results are presented and compared with results obtained from implementation of a Weiner filter blind deconvolution algorithm. The algorithm is developed further to incorporate functional forms of the point-spread function with unknown parameters. In the presence of noise the point-spread function can be evaluated with 1.0% error, and the object can be reconstructed with a quality near that of the deconvolution process with a known point-spread function.

© 1995 Optical Society of America

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References

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  1. G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [Crossref] [PubMed]
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    [Crossref]
  3. B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
    [Crossref]
  4. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [Crossref]
  5. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [Crossref]
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    [Crossref]
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    [Crossref] [PubMed]
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1992 (1)

1989 (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[Crossref]

1988 (1)

1982 (2)

J. R. Feinup, “Phase retrieval algorithms: a comparison,” Appl. Opt. 21, 2758–2769 (1982).
[Crossref]

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstructions for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[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. 39, 1–38 (1977).

1974 (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[Crossref]

1972 (1)

Ayers, G. R.

Bates, R. H. T.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[Crossref]

Dainty, J. C.

Davey, B. L. K.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[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. 39, 1–38 (1977).

Feinup, J. R.

Flannery, B. P.

W. H. Press, B. P. Flannery, J. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Holmes, T. J.

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. 39, 1–38 (1977).

Lane, R. G.

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[Crossref]

Lucy, L. B.

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[Crossref]

Press, W. H.

W. H. Press, B. P. Flannery, J. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Richardson, W. H.

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. 39, 1–38 (1977).

Shepp, L. A.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstructions for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[Crossref]

Teukolsky, J. A.

W. H. Press, B. P. Flannery, J. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Vardi, Y.

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstructions for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[Crossref]

Vetterling, W. T.

W. H. Press, B. P. Flannery, J. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

Appl. Opt. (1)

Astron. J. (1)

L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
[Crossref]

IEEE Trans. Med. Imaging (1)

L. A. Shepp, Y. Vardi, “Maximum likelihood reconstructions for emission tomography,”IEEE Trans. Med. Imaging MI-1, 113–122 (1982).
[Crossref]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (1)

J. R. Stat. Soc. (1)

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

Opt. Commun. (1)

B. L. K. Davey, R. G. Lane, R. H. T. Bates, “Blind deconvolution of noisy complex-valued image,” Opt. Commun. 69, 353–356 (1989).
[Crossref]

Opt. Lett. (1)

Other (1)

W. H. Press, B. P. Flannery, J. A. Teukolsky, W. T. Vetterling, Numerical Recipes: the Art of Scientific Computing (Cambridge U. Press, Cambridge, 1988).

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Figures (13)

Fig. 1
Fig. 1

Blind deconvolution based on the Richardson–Lucy algorithm.

Fig. 2
Fig. 2

(a) Simulated object, (b) Gaussian PSF, and (c) their convolution with 1.5% Poissonian noise.

Fig. 3
Fig. 3

Blind deconvolution by the Weiner filter algorithm. Reconstructions of the object (left) and the PSF (right) with (a) zero noise and (b) 1.5% noise.

Fig. 4
Fig. 4

Blind deconvolution by the Richardson–Lucy algorithm. (a) Convolutions with 1.5% (left) and 10.0% (right) noise. (b) Reconstructions of the object (left) and the PSF (right) at the 1.5% noise level. (c) Reconstructions of the object (left) and the PSF (right) at the 10.0% noise level.

Fig. 5
Fig. 5

Semiblind deconvolution by a Weiner filter-based algorithm. (a) True object (left), random starting guess of the object (center), and noiseless convolution (right). (b) Object (left) and PSF (right) from the first iteration. (c) Object (left) and PSF (right) from the second iteration. (d) Object (left) and PSF (right) from the third iteration.

Fig. 6
Fig. 6

Semiblind deconvolution by the Richardson–Lucy-based algorithm. (a) Object (left) and convolution (right) with 20.0% noise. (b) Reconstruction of the object (left) and the fitted Gaussian PSF (right).

Fig. 7
Fig. 7

Comparison of Richardson–Lucy semiblind deconvolution with standard deconvolution algorithms. (a) Reconstruction by semiblind deconvolution with a 0.1-pixel step width. (b) Reconstruction by Fourier regularization. (c) Reconstruction by the Richardson–Lucy algorithm.

Fig. 8
Fig. 8

Many-variable semiblind deconvolution. (a) Object (left) and PSF (right). (b) Convolution with 1.0% noise.

Fig. 9
Fig. 9

Reconstructions of the image shown in Fig. 8. (a) Richardson–Lucy deconvolution after 1000 iterations. (b) Semiblind deconvolution after 15 iterations: object (left) and PSF (right).

Fig. 10
Fig. 10

Error graphs for the 1.0% noise image shown in Fig. 8. (a) Fitting parameters A2, C1, C2 with iteration number. (b) Percentage error in the PSF with iteration number.

Fig. 11
Fig. 11

Reconstructions of an image with 4.0% noise. (a) 4.0% noise image. (b) Richardson–Lucy deconvolution after 1000 iterations. (c) Semiblind deconvolution after 15 iterations: object (left) and PSF (right).

Fig. 12
Fig. 12

Error graphs for the 4.0% noise image. (a) Fitting parameters A2, C1, C2 with iteration number. (b) Percentage error in the PSF with iteration number.

Fig. 13
Fig. 13

Error graphs for a 6.0% noise image. (a) Fitting parameters A2, C1, C2 with iteration number. (b) Percentage error in the PSF with iteration number.

Equations (7)

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P ( x y ) = P ( y x ) P ( x ) P ( y x ) P ( x ) d x ,
f i + 1 ( x ) = g ( y , x ) c ( y ) d y g ( y , z ) f i ( z ) d z f i ( x ) ,
f i + 1 ( x ) = { [ c ( x ) f i ( x ) g ( x ) ] g ( - x ) } f i ( x ) ,
g i + 1 k ( x ) = { [ c ( x ) g i k ( x ) f k - 1 ( x ) ] f k - 1 ( - x ) } g i k ( x ) ,
f i + 1 k ( x ) = { [ c ( x ) f i k ( x ) g k ( x ) ] g k ( - x ) } f i k ( x ) .
y ( r ) = k [ A k r 2 exp ( 1.0 ) C k 2 + B k ] exp ( - r 2 C k 2 ) ,
A 1 = 0.0 , A 2 = 0.1 , B 1 = 1.0 , B 2 = 0.0 , C 1 = 1.0 , C 2 = 5.0.

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