Abstract

We address the problem of space-invariant image restoration when the blurring operator is not known exactly, a situation that arises regularly in practice. To account for this uncertainty, we model the point-spread function as the sum of a known deterministic component and an unknown random one. Such an approach has been studied before, but the problem of estimating the parameters of the restoration filter to our knowledge has not been addressed systematically. We propose an approach based on a Gaussian statistical assumption and derive an iterative, expectation–maximization algorithm that simultaneously restores the image and estimates the required filter parameters. We obtain two versions of the algorithm based on two different models for the statistics of the image. The computations are performed in the discrete Fourier transform domain; thus they are computationally efficient even for large images. We examine the convergence properties of the resulting estimators and evaluate their performance experimentally.

© 2000 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 4, 43–64 (1996).
    [CrossRef]
  2. D. Kundur, D. Hatzinakos, “Blind deconvolution revisited,” IEEE Signal Process. Mag. 6, 61–63 (1996).
    [CrossRef]
  3. D. Slepian, “Linear least-squares filtering of distorted images,” J. Opt. Soc. Am. 57, 918–922 (1967).
    [CrossRef]
  4. R. K. Ward, B. E. A. Saleh, “Restoration of images distorted by systems of random impulse response,” J. Opt. Soc. Am. A 3, 1254–1259 (1985).
    [CrossRef]
  5. R. K. Ward, B. E. A. Saleh, “Deblurring random blur,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-10, 1494–1498 (1987).
    [CrossRef]
  6. L. Guan, R. K. Ward, “Deblurring random time-varying blur,” J. Opt. Soc. Am. A 6, 1727–1737 (1989).
    [CrossRef] [PubMed]
  7. L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust., Speech, Signal Process. 10, 589–592 (1989).
    [CrossRef]
  8. P. L. Combettes, H. J. Trussell, “Methods for digital restoration of signals degraded by stochastic impulse response,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-3, 393–401 (1989).
    [CrossRef]
  9. V. Z. Mesarović, N. P. Galatsanos, A. K. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 8, 1096–1108 (1995).
    [CrossRef]
  10. V. Z. Mesarović, N. Galatsanos, M. N. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm for tomographic reconstruction,” in Conference Record of the IEEE Nuclear Science Symposium & Medical Imaging Conference (Piscataway, N.J., 1995), pp. 1257–1261.
  11. E. J. Hoffman, M. E. Phelps, Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart (Raven, New York, 1986).
  12. A. P. Dempster, N. M. Laird, D. B. Rubin, “Maximum likelihood from incomplete data,” J. R. Stat. Soc. B 39, 1–38 (1977).
  13. R. Molina, “On the hierarchical Bayesian approach to image restoration: applications to astronomical images,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 1122–1188 (1994).
    [CrossRef]
  14. R. Molina, A. K. Katsaggelos, J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 2, 231–245 (1999).
    [CrossRef]
  15. S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1993).
  16. V. Z. Mesarović, N. P. Galatsanos, M. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm,” presented at the 30th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., November 3–6, 1996.
  17. V. Z. Mesarović, “Image restoration under point spread function uncertainties,” Ph.D. dissertation (Illinois Institute of Technology, Chicago, Ill., 1997).
  18. H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).
  19. N. Galatsanos, V. Mesarovic, R. Molina, A. Katsaggelos, “Hyperparameter estimation using hyperpriors for hierarchical Bayesian image restoration from partially-known blurs,” in Bayesian Inference for Inverse Problems, A. M. Djafari, ed., Proc. SPIE3459, 337–348 (1998).
    [CrossRef]
  20. N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 3, 322–336 (1992).
    [CrossRef]
  21. A. K. Katsaggelos, K. T. Lay, “Maximum likelihood identification and restoration of images using the expectation–maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, Berlin, 1991), pp. 143–176.
  22. R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identi-fication and restoration of noisy blurred images using the expectation–maximization algorithm,” IEEE Trans. Acoust., Speech, Signal Process. 7, 1180–1191 (1990).
    [CrossRef]
  23. A. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 8, 1892–1898 (1991).
    [CrossRef]
  24. A. Hillery, “Parameter estimation for image restoration,” Ph.D. dissertation (University of Wisconsin–Madison, Madison, Wis., 1991).

1999

R. Molina, A. K. Katsaggelos, J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 2, 231–245 (1999).
[CrossRef]

1996

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 4, 43–64 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind deconvolution revisited,” IEEE Signal Process. Mag. 6, 61–63 (1996).
[CrossRef]

1995

V. Z. Mesarović, N. P. Galatsanos, A. K. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 8, 1096–1108 (1995).
[CrossRef]

1994

R. Molina, “On the hierarchical Bayesian approach to image restoration: applications to astronomical images,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 1122–1188 (1994).
[CrossRef]

1992

N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 3, 322–336 (1992).
[CrossRef]

1991

A. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 8, 1892–1898 (1991).
[CrossRef]

1990

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identi-fication and restoration of noisy blurred images using the expectation–maximization algorithm,” IEEE Trans. Acoust., Speech, Signal Process. 7, 1180–1191 (1990).
[CrossRef]

1989

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust., Speech, Signal Process. 10, 589–592 (1989).
[CrossRef]

P. L. Combettes, H. J. Trussell, “Methods for digital restoration of signals degraded by stochastic impulse response,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-3, 393–401 (1989).
[CrossRef]

L. Guan, R. K. Ward, “Deblurring random time-varying blur,” J. Opt. Soc. Am. A 6, 1727–1737 (1989).
[CrossRef] [PubMed]

1987

R. K. Ward, B. E. A. Saleh, “Deblurring random blur,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-10, 1494–1498 (1987).
[CrossRef]

1985

R. K. Ward, B. E. A. Saleh, “Restoration of images distorted by systems of random impulse response,” J. Opt. Soc. Am. A 3, 1254–1259 (1985).
[CrossRef]

1977

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

1967

Andrews, H.

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).

Biemond, J.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identi-fication and restoration of noisy blurred images using the expectation–maximization algorithm,” IEEE Trans. Acoust., Speech, Signal Process. 7, 1180–1191 (1990).
[CrossRef]

Boekee, D. E.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identi-fication and restoration of noisy blurred images using the expectation–maximization algorithm,” IEEE Trans. Acoust., Speech, Signal Process. 7, 1180–1191 (1990).
[CrossRef]

Chin, R. T.

A. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 8, 1892–1898 (1991).
[CrossRef]

Combettes, P. L.

P. L. Combettes, H. J. Trussell, “Methods for digital restoration of signals degraded by stochastic impulse response,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-3, 393–401 (1989).
[CrossRef]

Dempster, A. P.

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

Galatsanos, N.

V. Z. Mesarović, N. Galatsanos, M. N. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm for tomographic reconstruction,” in Conference Record of the IEEE Nuclear Science Symposium & Medical Imaging Conference (Piscataway, N.J., 1995), pp. 1257–1261.

N. Galatsanos, V. Mesarovic, R. Molina, A. Katsaggelos, “Hyperparameter estimation using hyperpriors for hierarchical Bayesian image restoration from partially-known blurs,” in Bayesian Inference for Inverse Problems, A. M. Djafari, ed., Proc. SPIE3459, 337–348 (1998).
[CrossRef]

Galatsanos, N. P.

V. Z. Mesarović, N. P. Galatsanos, A. K. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 8, 1096–1108 (1995).
[CrossRef]

N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 3, 322–336 (1992).
[CrossRef]

V. Z. Mesarović, N. P. Galatsanos, M. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm,” presented at the 30th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., November 3–6, 1996.

Guan, L.

L. Guan, R. K. Ward, “Deblurring random time-varying blur,” J. Opt. Soc. Am. A 6, 1727–1737 (1989).
[CrossRef] [PubMed]

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust., Speech, Signal Process. 10, 589–592 (1989).
[CrossRef]

Hatzinakos, D.

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 4, 43–64 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind deconvolution revisited,” IEEE Signal Process. Mag. 6, 61–63 (1996).
[CrossRef]

Hillery, A.

A. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 8, 1892–1898 (1991).
[CrossRef]

A. Hillery, “Parameter estimation for image restoration,” Ph.D. dissertation (University of Wisconsin–Madison, Madison, Wis., 1991).

Hoffman, E. J.

E. J. Hoffman, M. E. Phelps, Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart (Raven, New York, 1986).

Hunt, B.

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).

Katsaggelos, A.

N. Galatsanos, V. Mesarovic, R. Molina, A. Katsaggelos, “Hyperparameter estimation using hyperpriors for hierarchical Bayesian image restoration from partially-known blurs,” in Bayesian Inference for Inverse Problems, A. M. Djafari, ed., Proc. SPIE3459, 337–348 (1998).
[CrossRef]

Katsaggelos, A. K.

R. Molina, A. K. Katsaggelos, J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 2, 231–245 (1999).
[CrossRef]

V. Z. Mesarović, N. P. Galatsanos, A. K. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 8, 1096–1108 (1995).
[CrossRef]

N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 3, 322–336 (1992).
[CrossRef]

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood identification and restoration of images using the expectation–maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, Berlin, 1991), pp. 143–176.

Kay, S. M.

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1993).

Kundur, D.

D. Kundur, D. Hatzinakos, “Blind deconvolution revisited,” IEEE Signal Process. Mag. 6, 61–63 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 4, 43–64 (1996).
[CrossRef]

Lagendijk, R. L.

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identi-fication and restoration of noisy blurred images using the expectation–maximization algorithm,” IEEE Trans. Acoust., Speech, Signal Process. 7, 1180–1191 (1990).
[CrossRef]

Laird, N. M.

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

Lay, K. T.

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood identification and restoration of images using the expectation–maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, Berlin, 1991), pp. 143–176.

Mateos, J.

R. Molina, A. K. Katsaggelos, J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 2, 231–245 (1999).
[CrossRef]

Mesarovic, V.

N. Galatsanos, V. Mesarovic, R. Molina, A. Katsaggelos, “Hyperparameter estimation using hyperpriors for hierarchical Bayesian image restoration from partially-known blurs,” in Bayesian Inference for Inverse Problems, A. M. Djafari, ed., Proc. SPIE3459, 337–348 (1998).
[CrossRef]

Mesarovic, V. Z.

V. Z. Mesarović, N. P. Galatsanos, A. K. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 8, 1096–1108 (1995).
[CrossRef]

V. Z. Mesarović, N. P. Galatsanos, M. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm,” presented at the 30th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., November 3–6, 1996.

V. Z. Mesarović, “Image restoration under point spread function uncertainties,” Ph.D. dissertation (Illinois Institute of Technology, Chicago, Ill., 1997).

V. Z. Mesarović, N. Galatsanos, M. N. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm for tomographic reconstruction,” in Conference Record of the IEEE Nuclear Science Symposium & Medical Imaging Conference (Piscataway, N.J., 1995), pp. 1257–1261.

Molina, R.

R. Molina, A. K. Katsaggelos, J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 2, 231–245 (1999).
[CrossRef]

R. Molina, “On the hierarchical Bayesian approach to image restoration: applications to astronomical images,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 1122–1188 (1994).
[CrossRef]

N. Galatsanos, V. Mesarovic, R. Molina, A. Katsaggelos, “Hyperparameter estimation using hyperpriors for hierarchical Bayesian image restoration from partially-known blurs,” in Bayesian Inference for Inverse Problems, A. M. Djafari, ed., Proc. SPIE3459, 337–348 (1998).
[CrossRef]

Phelps, M. E.

E. J. Hoffman, M. E. Phelps, Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart (Raven, New York, 1986).

Rubin, D. B.

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

Saleh, B. E. A.

R. K. Ward, B. E. A. Saleh, “Deblurring random blur,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-10, 1494–1498 (1987).
[CrossRef]

R. K. Ward, B. E. A. Saleh, “Restoration of images distorted by systems of random impulse response,” J. Opt. Soc. Am. A 3, 1254–1259 (1985).
[CrossRef]

Slepian, D.

Trussell, H. J.

P. L. Combettes, H. J. Trussell, “Methods for digital restoration of signals degraded by stochastic impulse response,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-3, 393–401 (1989).
[CrossRef]

Ward, R. K.

L. Guan, R. K. Ward, “Deblurring random time-varying blur,” J. Opt. Soc. Am. A 6, 1727–1737 (1989).
[CrossRef] [PubMed]

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust., Speech, Signal Process. 10, 589–592 (1989).
[CrossRef]

R. K. Ward, B. E. A. Saleh, “Deblurring random blur,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-10, 1494–1498 (1987).
[CrossRef]

R. K. Ward, B. E. A. Saleh, “Restoration of images distorted by systems of random impulse response,” J. Opt. Soc. Am. A 3, 1254–1259 (1985).
[CrossRef]

Wernick, M.

V. Z. Mesarović, N. P. Galatsanos, M. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm,” presented at the 30th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., November 3–6, 1996.

Wernick, M. N.

V. Z. Mesarović, N. Galatsanos, M. N. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm for tomographic reconstruction,” in Conference Record of the IEEE Nuclear Science Symposium & Medical Imaging Conference (Piscataway, N.J., 1995), pp. 1257–1261.

IEEE Signal Process. Mag.

D. Kundur, D. Hatzinakos, “Blind image deconvolution,” IEEE Signal Process. Mag. 4, 43–64 (1996).
[CrossRef]

D. Kundur, D. Hatzinakos, “Blind deconvolution revisited,” IEEE Signal Process. Mag. 6, 61–63 (1996).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

L. Guan, R. K. Ward, “Restoration of randomly blurred images by the Wiener filter,” IEEE Trans. Acoust., Speech, Signal Process. 10, 589–592 (1989).
[CrossRef]

P. L. Combettes, H. J. Trussell, “Methods for digital restoration of signals degraded by stochastic impulse response,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-3, 393–401 (1989).
[CrossRef]

R. K. Ward, B. E. A. Saleh, “Deblurring random blur,” IEEE Trans. Acoust., Speech, Signal Process. ASSP-10, 1494–1498 (1987).
[CrossRef]

R. L. Lagendijk, J. Biemond, D. E. Boekee, “Identi-fication and restoration of noisy blurred images using the expectation–maximization algorithm,” IEEE Trans. Acoust., Speech, Signal Process. 7, 1180–1191 (1990).
[CrossRef]

IEEE Trans. Image Process.

N. P. Galatsanos, A. K. Katsaggelos, “Methods for choosing the regularization parameter and estimating the noise variance in image restoration and their relation,” IEEE Trans. Image Process. 3, 322–336 (1992).
[CrossRef]

R. Molina, A. K. Katsaggelos, J. Mateos, “Bayesian and regularization methods for hyperparameter estimation in image restoration,” IEEE Trans. Image Process. 2, 231–245 (1999).
[CrossRef]

V. Z. Mesarović, N. P. Galatsanos, A. K. Katsaggelos, “Regularized constrained total least squares image restoration,” IEEE Trans. Image Process. 8, 1096–1108 (1995).
[CrossRef]

IEEE Trans. Pattern. Anal. Mach. Intell.

R. Molina, “On the hierarchical Bayesian approach to image restoration: applications to astronomical images,” IEEE Trans. Pattern. Anal. Mach. Intell. 11, 1122–1188 (1994).
[CrossRef]

IEEE Trans. Signal Process.

A. Hillery, R. T. Chin, “Iterative Wiener filters for image restoration,” IEEE Trans. Signal Process. 8, 1892–1898 (1991).
[CrossRef]

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

R. K. Ward, B. E. A. Saleh, “Restoration of images distorted by systems of random impulse response,” J. Opt. Soc. Am. A 3, 1254–1259 (1985).
[CrossRef]

L. Guan, R. K. Ward, “Deblurring random time-varying blur,” J. Opt. Soc. Am. A 6, 1727–1737 (1989).
[CrossRef] [PubMed]

J. R. Stat. Soc. B

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

Other

S. M. Kay, Fundamentals of Statistical Signal Processing (Prentice-Hall, Englewood Cliffs, NJ, 1993).

V. Z. Mesarović, N. P. Galatsanos, M. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm,” presented at the 30th Asilomar Conference on Signals, Systems, and Computers, Pacific Grove, Calif., November 3–6, 1996.

V. Z. Mesarović, “Image restoration under point spread function uncertainties,” Ph.D. dissertation (Illinois Institute of Technology, Chicago, Ill., 1997).

H. Andrews, B. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, NJ, 1977).

N. Galatsanos, V. Mesarovic, R. Molina, A. Katsaggelos, “Hyperparameter estimation using hyperpriors for hierarchical Bayesian image restoration from partially-known blurs,” in Bayesian Inference for Inverse Problems, A. M. Djafari, ed., Proc. SPIE3459, 337–348 (1998).
[CrossRef]

V. Z. Mesarović, N. Galatsanos, M. N. Wernick, “Restoration from partially-known blur using an expectation–maximization algorithm for tomographic reconstruction,” in Conference Record of the IEEE Nuclear Science Symposium & Medical Imaging Conference (Piscataway, N.J., 1995), pp. 1257–1261.

E. J. Hoffman, M. E. Phelps, Positron Emission Tomography and Autoradiography: Principles and Applications for the Brain and Heart (Raven, New York, 1986).

A. Hillery, “Parameter estimation for image restoration,” Ph.D. dissertation (University of Wisconsin–Madison, Madison, Wis., 1991).

A. K. Katsaggelos, K. T. Lay, “Maximum likelihood identification and restoration of images using the expectation–maximization algorithm,” in Digital Image Restoration, A. K. Katsaggelos, ed. (Springer-Verlag, Berlin, 1991), pp. 143–176.

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (6)

Fig. 1
Fig. 1

Experiment 1: Restoration with all statistics assumed known. (a) Constant-γ MSE plot and (b) constant-β MSE plot. Pluses, EM for full model; squares, LMMSE in Ref. 4.

Fig. 2
Fig. 2

Experiment 1: Restoration and simultaneous estimation of the image statistics. (a) Constant-β MSE plot. (b) Constant-γ MSE plot. Crosses, LMMSE in Ref. 4; pluses and squares, EM for full model and SAR model, respectively; triangles, ideal LMMSE.

Fig. 3
Fig. 3

Experiment 1: Restoration and simultaneous estimation of the image statistics. (a) Degraded image by SNRh=20 dB, SNRg=11 dB; (b) LMMSE in Ref. 4; (c) EM with full Rf model; (d) EM with SAR model; (e) ideal LMMSE.

Fig. 4
Fig. 4

Experiment 2: Restoration and simultaneous estimation of the PSF noise and the image statistics. Constant-γ MSE plot. Pluses, EM for full model; squares, EM for SAR model; crosses, ideal LMMSE.

Fig. 5
Fig. 5

Experiment 2: Restoration and simultaneous estimation of the additive noise and the image statistics. (a) Constant-β MSE plot for partially known blur case. (b) Constant-β MSE plot for perfectly known blur case (β=0). Symbols same as for Fig. 4.

Fig. 6
Fig. 6

Experiment 2: Restoration and simultaneous estimation of the additive noise and the image statistics. (a) Degraded image by SNRh=20 dB, SNRg=11 dB, (b) EM with full Rf model, (c) EM with SAR model, (d) ideal LMMSE.

Equations (96)

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

h=h¯+Δh,
g=Hf+Δg,
H=H¯+ΔH
E{fft}=Rf
Rf=(αQtQ)-1,
0101-(4+ϵ)1010,
P(g|Ψ)={det(2π[H¯RfH¯t+RT(f)])}-1/2×exp{-12gt[H¯RfH¯t+RT(f)]-1g},
RT(f)=E{(ΔHf+Δg)(ΔHf+Δg)t}.
g=Tz.
P(z|Ψ)=[det(2πRz)]-1/2 exp(-12ztRz-1z),
log P(z|Ψ)-{log[det(Rz)]+ztRz-1z}.
mz|g=E{z|g}=RzgRg-1g
Rz|g=E{zzt|g}=Rz-RzgRg-1Rgz,
E{log P(z|Ψ)|g, Ψ(n)}-[log[det(Rz)]+tr(Rz-1{Rz|g(n)+mz|g(n)[mz|g(n)]t})],
F(Ψ; Ψ(n))=log[det(Rz)]+tr[Rz-1Rz|g(n)]+[mz|g(n)]tRz-1mz|g(n).
z=fg.
g=(0I)fg=Tz,
Rz=E{zzt}=RfRfH¯tH¯RfH¯RfH¯t+RT(f),
Rz-1=Rf-1+H¯tRT-1(f)H¯-H¯tRT-1(f)-RT-1(f)H¯RT-1(f).
F(Ψ; Ψ(n))=log[det(Rf)]+tr[Rf-1Rf|g(n)]+[mf|g(n)]tRf-1mf|g(n)+log{det[RT(f)]}+tr[H¯tRT-1(f)H¯Rf|g(n)]+[H¯mf|g(n)-g]tRT-1(f)[H¯mf|g(n)-g],
mf/g(n)=Rf(n)H¯t[H¯Rf(n)H¯t+RT(n)(f)]-1g,
Rf/g(n)=Rf(n)-Rf(n)H¯t[H¯Rf(n)H¯t+RT(n)(f)]-1H¯tRf(n).
F(Ψ; Ψ(n))=i=0N-1 log[Sf(i)]+1Sf(i)Sf|g(n)(i)+1N|Mf|g(n)(i)|2+log[NβSf(i)+γ]+1NβSf(i)+γ|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2,
Mf|g(n)(i)=H¯*(i)Sf(n)(i)|H¯(i)|2Sf(n)(i)+Nβ(n)Sf(n)(i)+γ(n)G(i),
Sf|g(n)(i)=Sf(n)(i)[Nβ(n)Sf(n)(i)+γ(n)]|H¯(i)|2Sf(n)(i)+Nβ(n)Sf(n)(i)+γ(n),
Sf(n+1)(i)=Sf|g(n)(i)+1N|Mf|g(n)(i)|2,
β(n+1)=β(n) v(β(n))u(β(n)),
γ(n+1)=γ(n) v(γ(n))u(γ(n)),
FLMMSE(n)(i)=H¯*(i)Sf(n)(i)|H¯(i)|2Sf(n)(i)+βSf(n)(i)+γG(i),
Sf(n+1)(i)=1N|FLMMSE(n)(i)|2.
F(Ψ;Ψ(n))=log[det(α-1I)]+tr[αQtQRf|g(n)]+[mf|g(n)]tαQtQmf|g(n)+log{det[RT(f)]}+tr[H¯tRT-1(f)H¯Rf|g(n)]
+[H¯mf|g(n)-g]tRT-1(f)[H¯mf|g(n)-g],
mf/g(n)=[α(n)QtQ]-1H¯t{H¯[α(n)QtQ]-1H¯t+RT(n)(f)}-1g,
Rf/g(n)=[α(n)QtQ]-1-[α(n)QtQ]-1H¯t{H¯[α(n)QtQ]-1H¯t+RT(n)(f)}-1H¯t[α(n)QtQ]-1.
F(Ψ; Ψ(n))=i=0N-1[-log(α)]+α|Q(i)|2[Sf|g(n)(i)+1N|Mf|g(n)(i)|2]+logNβ 1α|Q(i)|2+γ+1Nβ 1α|Q(i)|2+γ|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2,
Mf|g(n)(i)=H¯*(i) 1α(n)|H¯(i)|2 1α(n)+Nβ(n) 1α(n)+γ(n)|Q(i)|2G(i),
Sf|g(n)(i)=1α(n)|Q(i)|2Nβ(n)α(n)+γ(n)|Q(i)|2|H¯(i)|2α(n)+Nβ(n)α(n)+γ(n)|Q(i)|2,
1α(n+1)=1Ni=0N-1|Q(i)|2Sf|g(n)(i)+1N|Mf|g(n)(i)|2,
β(n+1)=β(n) v(β(n))u(β(n)),
γ(n+1)=γ(n) v(γ(n))u(γ(n)),
Sf(n+1)=Sf(n)[NβSf(n)+γ]|H¯|2Sf(n)+NβSf(n)+γ+[Sf(n)]2|H¯|2 1N|G|2[|H¯|2Sf(n)+NβSf(n)+γ]2.
1N|G|2=|H¯|2Sf+NβSf+γ,
S˜f=Sf(n+1)=Sf(n).
S˜f2(|H¯|2+Nβ)2+S˜f2(|H˜|2+Nβ)γ-|H¯|2 1N|G|2
+γ2=0,S˜f=0.
MSE=1Nf-fˆ22,
SNRh=h¯2Nβ,
SNRg=f2Nγ,
h(i, j)=c exp-i2+j22×32
fori, j=-15,-14, ,-1, 0, 1, , 14, 15,
RT(f)=EΔh{ΔHRfΔHt}+RΔg=(EΔh{ΔHtΔH})Rf+RΔg=EΔhi=0N-1(ΔH)i(ΔHt)iRf+RΔg,
(ΔH)i(n)=Δh(n-i)mod N.
(W(ΔH)i)(n)=(WΔh)(n)exp-j 2πNni,
W(ΔH)i=diag1, exp-j 2πNi,,exp-j 2πN(N-1)i(WΔh).
WEΔhi=0N-1(ΔH)i(ΔHH)iW-1=i=0N-1EΔhW(ΔH)i(ΔHH)iWH 1N
WEΔhi=0N-1(ΔH)i(ΔHH)iW-1
=i=0N-1EΔh(WΔh)(WΔh)H 1N
=i=0N-1WEΔh{ΔhΔhH}W-1=i=0N-1WRΔhW-1
=i=0N-1 diag(β, , β)=N diag(β ,, β),
WRT(f)W-1=diag(NβSf(0)+γ, NβSf(1)+γ, , NβSf(N-1)+γ),
NβSf(i)+γ.
RT(f)=βRf+γI.
βSf(i)+γ.
1Sf(n+1)(i)-1[Sf(n+1)(i)]2Sf|g(n)(i)+1N|Mf|g(n)(i)|2+Nβ(n)Nβ(n)Sf(n+1)(i)+γ(n)-Nβ(n)[Nβ(n)Sf(n+1)(i)+γ(n)]2|H¯(i)|2Sf|g(n)(i)+1N|H¯Mf|g(n)(i)-G(i)|2=0.
Sf(n+1)(i)=Sf|g(n)(i)+1N|Mf|g(n)(i)|2.
i=0N-1 NSf(n)(i)Nβ(n+1)Sf(n)(i)+γ(n)
=i=0N-1 NSf(n)(i)[Nβ(n+1)Sf(n)(i)+γ(n)]2×|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2.
u(β)=i=0N-1 NSf(n)(i)NβSf(n)(i)+γ(n),
v(β)=i=0N-1 NSf(n)(i)[NβSf(n)(i)+γ(n)]2|H¯(i)|2Sf|g(n)(i)+1N|H¯Mf|g(n)(i)-G(i)|2.
u(β)=v(β)
β(n+1)=β(n) v(β(n))u(β(n)),
i=0N-1 1Nβ(n)Sf(n)(i)+γ(n+1)
=i=0N-1 1[Nβ(n)Sf(n)(i)+γ(n+1)]2×|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2.
γ(n+1)=γ(n) v(γ(n))u(γ(n)),
u(γ(n))=i=0N-1 1Nβ(n)Sf(n)(i)+γ(n),
v(γ(n))=i=0N-1 1[Nβ(n)Sf(n)(i)+γ(n)]2|H¯(i)|2Sf|g(n)(i)+1N|H¯Mf|g(n)(i)-G(i)|2.
-Nα(n+1)+i=0N-1|Q(i)|2Sf|g(n)(i)+1N|Mf|g(n)(i)|2
-i=0N-1 Nβ(n)Nα(n+1)β(n)+(α2)(n+1)|Q(i)|2γ(n)
+i=0N-1 Nβ(n)|Q(i)|2(α2)(n+1)Nβ(n)α(n+1)+γ(n)|Q(i)|22
×|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2=0.
1α(n+1)=1Ni=0N-1|Q(i)|2Sf|g(n)(i)+1N|Mf|g(n)(i)|2.
i=0N-1 Nα(n)Nβ(n+1)α(n)+γ(n)|Q(i)|2=i=0N-1 N|Q(i)|2α(n)Nβ(n+1)α(n)+γ(n)|Q(i)|22|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2,
i=0N-1 |Q(i)|2Nβ(n)α(n)+γ(n+1)|Q(i)|2
=i=0N-1 |Q(i)|4Nβ(n)α(n)+γ(n+1)|Q(i)|22|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2.
β(n+1)=β(n) v(β(n))u(β(n)),
γ(n+1)=γ(n) v(γ(n))u(γ(n)),
u(β(n))=i=0N-1 Nα(n)Nβ(n)α(n)+γ(n)|Q(i)|2,
v(β(n))=i=0N-1 N|Q(i)|2α(n)Nβ(n)α(n)+γ(n)|Q(i)|22|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2,
u(γ(n))=i=0N-1 |Q(i)|2Nβ(n)α(n)+γ(n)|Q(i)|2,
v(γ(n))=i=0N-1 |Q(i)|4Nβ(n)α(n)+γ(n)|Q(i)|22|H¯(i)|2Sf|g(n)(i)+1N|H¯(i)Mf|g(n)(i)-G(i)|2.
Sf(n+1)Sf(n)=2|H¯|2 1N|G|2[Sf(n)]2[|H¯|2Sf(n)+NβSf(n)+γ]3γ+γ[2NβSf(n)+γ]+Nβ[Sf(n)]2[|H¯|2Sf(n)+NβSf(n)+γ]20
Sf(n+1)=Sf(n)(|H¯|2+Nβ)Sf(n)+γ×1/N|G|2(|H¯|2+Nβ)Sf(n)+γ|H¯|2Sf(n)+NβSf(n)+γ,
(|H¯|2+Nβ)Sf(n)+γ1N|G|2,
Sf(n+1)Sf(n).
(|H¯|2+Nβ)Sf(n)+γ1N|G|2,
Sf(n+1)Sf(n),

Metrics