Abstract

Iterative image deconvolution algorithms generally lack objective criteria for deciding when to terminate iterations, often relying on ad hoc metrics for determining optimal performance. A statistical-information-based analysis of the popular Richardson–Lucy iterative deblurring algorithm is presented after clarification of the detailed nature of noise amplification and resolution recovery as the algorithm iterates. Monitoring the information content of the reconstructed image furnishes an alternative criterion for assessing and stopping such an iterative algorithm. It is straightforward to implement prior knowledge and other conditioning tools in this statistical approach.

© 2002 Optical Society of America

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References

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  1. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).
  2. W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).
  3. R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer, Norwell, Mass., 1991).
  4. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).
  5. E. S. Meinel, “Origins of linear and nonlinear recursive restoration algorithms,” J. Opt. Soc. Am. A 3, 787–799 (1986).
    [CrossRef]
  6. A. S. Carasso, “Linear and nonlinear image deblurring: a documented study,” SIAM J. Numer. Anal. 36, 1659–1689 (1999).
    [CrossRef]
  7. B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
    [CrossRef]
  8. B. R. Hunt, “Superresolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
    [CrossRef]
  9. D. G. Sheppard, B. R. Hunt, M. W. Marcellin, “Iterative multiframe superresolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A 15, 978–992 (1998).
    [CrossRef]
  10. W. H. Richardson, “Bayesian-based iterative method of image restoration,” J. Opt. Soc. Am. 62, 55–59 (1972).
    [CrossRef]
  11. L. B. Lucy, “An iterative technique for the rectification of observed distributions,” Astron. J. 79, 745–754 (1974).
    [CrossRef]
  12. D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
    [CrossRef] [PubMed]
  13. D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
    [CrossRef]
  14. D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).
  15. H. J. Trussell, “Convergence criteria of iterative restoration methods,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 129–136 (1983).
    [CrossRef]
  16. J. Llacer, E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imaging 8, 186–193 (1989).
    [CrossRef]
  17. J. Llacer, “On the validity of hypothesis testing for feasibility of image reconstruction,” IEEE Trans. Med. Imaging 9, 226–230 (1990).
    [CrossRef]
  18. S. J. Reeves, “Generalized cross-validation as a stopping rule for the Richardson–Lucy algorithm,” Int. J. Imaging Syst. Technol. 6, 387–391 (1995).
    [CrossRef]
  19. P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. (London) A 247, 369–407 (1955).
    [CrossRef]
  20. C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. (New York) 57, 245–285 (1991).
  21. F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London Ser. A 354, 2193–2248 (1996).
    [CrossRef]
  22. S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
    [CrossRef]
  23. S. Prasad, “Information theoretic perspective on the formation, detection and processing of images from a seeing limited telescope,” in Proceedings of the AMOS Technical Conference (Maui Economic Development Board, Maui, HI, 1999), pp. 339–349.
  24. J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
    [CrossRef]
  25. C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
    [CrossRef]
  26. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
    [CrossRef] [PubMed]
  27. S. P. Luttrell, “The use of transinformation in the design of data sampling schemes for inverse problems,” Inverse Probl. 1, 199–218 (1985).
    [CrossRef]
  28. Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. R. Statist. Soc. B 55, 569–612 (1993).
  29. D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
    [CrossRef]
  30. See, e.g., T. J. Holmes, “Maximum likelihood image restoration adapted for incoherent optical imaging,” J. Opt. Soc. Am. A 5, 666–673 (1988).
    [CrossRef]

2000 (1)

S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
[CrossRef]

1999 (1)

A. S. Carasso, “Linear and nonlinear image deblurring: a documented study,” SIAM J. Numer. Anal. 36, 1659–1689 (1999).
[CrossRef]

1998 (2)

D. G. Sheppard, B. R. Hunt, M. W. Marcellin, “Iterative multiframe superresolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A 15, 978–992 (1998).
[CrossRef]

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

1996 (1)

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London Ser. A 354, 2193–2248 (1996).
[CrossRef]

1995 (2)

S. J. Reeves, “Generalized cross-validation as a stopping rule for the Richardson–Lucy algorithm,” Int. J. Imaging Syst. Technol. 6, 387–391 (1995).
[CrossRef]

B. R. Hunt, “Superresolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

1994 (1)

B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
[CrossRef]

1993 (1)

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. R. Statist. Soc. B 55, 569–612 (1993).

1992 (1)

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

1991 (1)

C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. (New York) 57, 245–285 (1991).

1990 (1)

J. Llacer, “On the validity of hypothesis testing for feasibility of image reconstruction,” IEEE Trans. Med. Imaging 9, 226–230 (1990).
[CrossRef]

1989 (1)

J. Llacer, E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imaging 8, 186–193 (1989).
[CrossRef]

1988 (1)

1987 (1)

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
[CrossRef] [PubMed]

1986 (1)

1985 (2)

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

S. P. Luttrell, “The use of transinformation in the design of data sampling schemes for inverse problems,” Inverse Probl. 1, 199–218 (1985).
[CrossRef]

1983 (1)

H. J. Trussell, “Convergence criteria of iterative restoration methods,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 129–136 (1983).
[CrossRef]

1974 (1)

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

1972 (2)

1955 (1)

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. (London) A 247, 369–407 (1955).
[CrossRef]

1948 (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Andrews, H. C.

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Biemond, J.

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer, Norwell, Mass., 1991).

Blahut, R. E.

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

Carasso, A. S.

A. S. Carasso, “Linear and nonlinear image deblurring: a documented study,” SIAM J. Numer. Anal. 36, 1659–1689 (1999).
[CrossRef]

Fales, C. L.

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London Ser. A 354, 2193–2248 (1996).
[CrossRef]

C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. (New York) 57, 245–285 (1991).

Fellgett, P. B.

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. (London) A 247, 369–407 (1955).
[CrossRef]

Frieden, B. R.

Holmes, T. J.

Huck, F. O.

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London Ser. A 354, 2193–2248 (1996).
[CrossRef]

C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. (New York) 57, 245–285 (1991).

Hunt, B. R.

D. G. Sheppard, B. R. Hunt, M. W. Marcellin, “Iterative multiframe superresolution algorithms for atmospheric-turbulence-degraded imagery,” J. Opt. Soc. Am. A 15, 978–992 (1998).
[CrossRef]

B. R. Hunt, “Superresolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
[CrossRef]

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

Lagendijk, R. L.

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer, Norwell, Mass., 1991).

Lee, D.

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. R. Statist. Soc. B 55, 569–612 (1993).

Linfoot, E. H.

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. (London) A 247, 369–407 (1955).
[CrossRef]

Llacer, J.

J. Llacer, “On the validity of hypothesis testing for feasibility of image reconstruction,” IEEE Trans. Med. Imaging 9, 226–230 (1990).
[CrossRef]

J. Llacer, E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imaging 8, 186–193 (1989).
[CrossRef]

Lucy, L. B.

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

Luttrell, S. P.

S. P. Luttrell, “The use of transinformation in the design of data sampling schemes for inverse problems,” Inverse Probl. 1, 199–218 (1985).
[CrossRef]

Marcellin, M. W.

Meinel, E. S.

Miller, M. I.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
[CrossRef] [PubMed]

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).

O’Sullivan, J. A.

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Politte, D. G.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
[CrossRef] [PubMed]

Prasad, S.

S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
[CrossRef]

S. Prasad, “Information theoretic perspective on the formation, detection and processing of images from a seeing limited telescope,” in Proceedings of the AMOS Technical Conference (Maui Economic Development Board, Maui, HI, 1999), pp. 339–349.

Pratt, W. K.

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

Rahman, Z.

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London Ser. A 354, 2193–2248 (1996).
[CrossRef]

Reeves, S. J.

S. J. Reeves, “Generalized cross-validation as a stopping rule for the Richardson–Lucy algorithm,” Int. J. Imaging Syst. Technol. 6, 387–391 (1995).
[CrossRef]

Richardson, W. H.

Roggemann, M. C.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

Schulz, T. J.

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Shannon, C.

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

Sheppard, D. G.

Snyder, D. L.

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
[CrossRef] [PubMed]

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).

Thomas, L. J.

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
[CrossRef] [PubMed]

Trussell, H. J.

H. J. Trussell, “Convergence criteria of iterative restoration methods,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 129–136 (1983).
[CrossRef]

Vardi, Y.

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. R. Statist. Soc. B 55, 569–612 (1993).

Veklerov, E.

J. Llacer, E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imaging 8, 186–193 (1989).
[CrossRef]

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

Astron. J. (1)

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

Bell Syst. Tech. J. (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423, 623–656 (1948).
[CrossRef]

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

H. J. Trussell, “Convergence criteria of iterative restoration methods,” IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 129–136 (1983).
[CrossRef]

IEEE Trans. Inf. Theory (1)

J. A. O’Sullivan, R. E. Blahut, D. L. Snyder, “Information-theoretic image formation,” IEEE Trans. Inf. Theory 44, 2094–2123 (1998).
[CrossRef]

IEEE Trans. Med. Imaging (3)

J. Llacer, E. Veklerov, “Feasible images and practical stopping rules for iterative algorithms in emission tomography,” IEEE Trans. Med. Imaging 8, 186–193 (1989).
[CrossRef]

J. Llacer, “On the validity of hypothesis testing for feasibility of image reconstruction,” IEEE Trans. Med. Imaging 9, 226–230 (1990).
[CrossRef]

D. L. Snyder, M. I. Miller, L. J. Thomas, D. G. Politte, “Noise and edge artifacts in maximum-likelihood reconstructions for emission tomography,” IEEE Trans. Med. Imaging 6, 228–238 (1987).
[CrossRef] [PubMed]

IEEE Trans. Nucl. Sci. (1)

D. L. Snyder, M. I. Miller, “The use of sieves to stabilize images produced with the EM algorithm for emission tomography,” IEEE Trans. Nucl. Sci. NS-32, 3864–3872 (1985).
[CrossRef]

IEEE Trans. Signal Process. (1)

D. L. Snyder, T. J. Schulz, J. A. O’Sullivan, “Deblurring subject to nonnegativity constraints,” IEEE Trans. Signal Process. 40, 1143–1150 (1992).
[CrossRef]

Inf. Sci. (New York) (1)

C. L. Fales, F. O. Huck, “An information theory of image gathering,” Inf. Sci. (New York) 57, 245–285 (1991).

Int. J. Imaging Syst. Technol. (2)

S. J. Reeves, “Generalized cross-validation as a stopping rule for the Richardson–Lucy algorithm,” Int. J. Imaging Syst. Technol. 6, 387–391 (1995).
[CrossRef]

B. R. Hunt, “Superresolution of images: algorithms, principles, performance,” Int. J. Imaging Syst. Technol. 6, 297–304 (1995).
[CrossRef]

Int. J. Mod. Phys. C (1)

B. R. Hunt, “Prospects for image restoration,” Int. J. Mod. Phys. C 5, 151–178 (1994).
[CrossRef]

Inverse Probl. (1)

S. P. Luttrell, “The use of transinformation in the design of data sampling schemes for inverse problems,” Inverse Probl. 1, 199–218 (1985).
[CrossRef]

J. Opt. Soc. Am. (2)

J. Opt. Soc. Am. A (3)

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

Y. Vardi, D. Lee, “From image deblurring to optimal investments: maximum likelihood solutions for positive linear inverse problems,” J. R. Statist. Soc. B 55, 569–612 (1993).

Opt. Commun. (1)

S. Prasad, “Information capacity of a seeing-limited imaging system,” Opt. Commun. 177, 119–134 (2000).
[CrossRef]

Philos. Trans. R. Soc. (London) A (1)

P. B. Fellgett, E. H. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. (London) A 247, 369–407 (1955).
[CrossRef]

Philos. Trans. R. Soc. London Ser. A (1)

F. O. Huck, C. L. Fales, Z. Rahman, “An information theory of visual communication,” Philos. Trans. R. Soc. London Ser. A 354, 2193–2248 (1996).
[CrossRef]

SIAM J. Numer. Anal. (1)

A. S. Carasso, “Linear and nonlinear image deblurring: a documented study,” SIAM J. Numer. Anal. 36, 1659–1689 (1999).
[CrossRef]

Other (6)

H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

W. K. Pratt, Digital Image Processing, 2nd ed. (Wiley, New York, 1991).

R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer, Norwell, Mass., 1991).

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC, Boca Raton, Fla., 1996).

D. L. Snyder, M. I. Miller, Random Point Processes in Time and Space, 2nd ed. (Springer-Verlag, New York, 1991).

S. Prasad, “Information theoretic perspective on the formation, detection and processing of images from a seeing limited telescope,” in Proceedings of the AMOS Technical Conference (Maui Economic Development Board, Maui, HI, 1999), pp. 339–349.

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Figures (8)

Fig. 1
Fig. 1

Gray-scale representation of the object ensemble. The ensemble consists of 10 objects, each comprised of 20 Gaussian stars of randomly chosen widths and locations distributed over a 64×64 pixelated array.

Fig. 2
Fig. 2

Blurred version of the object ensemble as would be obtained from a noise-free low-pass imaging system. The blurring psf is chosen to be the diffraction limited psf for a circular pupil, with a cutoff equal to 16 pixel units in the spatial-frequency plane.

Fig. 3
Fig. 3

Pixel-by-pixel distribution of information entropy in the amplitude spectrum of the object ensemble.

Fig. 4
Fig. 4

Pixel-by-pixel distribution of information entropy in the amplitude spectrum of the blurred object ensemble. Note the absence of information for spatial frequencies that exceed in magnitude the cutoff of 16 units.

Fig. 5
Fig. 5

Typical sequence of RL restorations at various iteration numbers. The picture shows the very first object in Fig. 1 being restored from image data that come from the corresponding blurred object in Fig. 2 corrupted by a single frame of additive noise with parameter W being 0.1.

Fig. 6
Fig. 6

Mutual information versus iteration number for the case of additive noise, in four contiguous spatial-frequency bands defined in the text. The noise level W is 0.1 (peak SNR of ∼128).

Fig. 7
Fig. 7

Same as Fig. 6 except for a lower noise level W=0.01 (peak SNR of ∼1280).

Fig. 8
Fig. 8

Same as Fig. 6 except for Poisson image data. The peak SNR in the blurred image data is ∼50.

Equations (24)

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

HO(a)=-ipO(ai)log2 pO(ai)=-log2 pO(a),
HO|I(a|b)=-log2 pO|I(a|b),
I(a; b)=HO(a)-HO|I(a|b).
i(x)=h(x, x)o(x)d2x.
ip=qhpqoq.
phpq=1.
op(n+1)=op(n)qhqpiqrhqror(n),
op(1)=1P2qiq,
iqrhqror(n)=uq(n)=1+wq(n)
op(n+1)-op(n)=op(n)qhqpwq(n).
Fμ=pfpexp(-i2πpμ/P),
Oμ(n+1)-Oμ(n)=1P2μOμ(n)Hμ-μ*Wμ-μ(n).
Oμ(n+1)=1P2μOμ-μ(n)Hμ*Uμ(n).
CO(n+1)(μ, μ)1P4ννHν*Hν×[CO(n)(μ-ν, μ-ν)Uν(n)Uν(n)*+CU(n)(ν, ν)Oμ-ν(n)Oμ-ν(n)*],
Cf(n)(μ, μ)=δfμ(n)δfμ(n)*
δ(fg)=δfg+δgf,
NO(n+1)(μ)1P4ν|Hν|2[NO(n)(μ-ν)|Uν(n)|2+NU(n)(ν)|Oμ-ν(n)|2],
uq(n)1+1P2|μ|>μ0Uμ(n)exp(i2πqμ/P).
Uμ(n)P2forμ=00for0<|μ|<μ0Uμ(n)for|μ|>μ0.
NO(n+1)(μ)-NO(n)(μ)1P4ν|Hν|2[NO(n)(μ-ν)|Uν(n)|2+NU(n)(ν)|Oμ-ν(n)|2].
WO(n+1)-WO(n)λnWO(n)+nSO(n),
λn1P4ν|Hν|2|Uν(n)|2,
n1P4ν|Hν|2NU(n)(ν)
SO(n)=μ|Oμ(n)|2.

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