Abstract

A new two-dimensional recursive filter for recovering degraded images is proposed that is based on particle-filter theory. The main contribution of this work lies in evolving a framework that has the potential to recover images suffering from a general class of degradations such as system nonlinearity and non-Gaussian observation noise. Samples of the prior probability distribution of the original image are obtained by propagating the samples through an appropriate state model. Given the measurement model and the degraded image, the weights of the samples are computed. The samples and their corresponding weights are used to calculate the conditional mean that yields an estimate of the original image. The proposed method is validated by demonstrating its effectiveness in recovering images degraded by film-grain noise. Synthetic as well as real examples are considered for this purpose. Performance is also compared with that of an existing scheme.

© 2005 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. M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14, 24–45 (1997).
    [CrossRef]
  3. M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. (Bellingham) 29, 393–414 (1990).
    [CrossRef]
  4. R. L. Lagendijk, J. Biemond, Iterative Identification and Restoration of Images (Kluwer Academic, Boston, Mass., 1991).
  5. J. W. Woods, C. H. Radewan, “Kalman filtering in two dimensions,” IEEE Trans. Inf. Theory 23, 473–482 (1977).
    [CrossRef]
  6. J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust., Speech, Signal Process. 29, 188–197 (1981).
    [CrossRef]
  7. A. J. Patti, M. K. Ozkan, A. M. Tekalp, M. I. Sezan, “New approaches for space-variant image restoration,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1993), pp. 261–264.
  8. D. Angwin, H. Kaufman, “Image restoration using a reduced order model Kalman filter,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1988), pp. 1000–1003.
  9. B. R. Hunt, “Bayesian methods in nonlinear digital image restoration,” IEEE Trans. Comput. 26, 219–229 (1977).
    [CrossRef]
  10. M. E. Zervakis, A. N. Venetsanopoulos, “Iterative least squares estimators in non-linear image restoration,” IEEE Trans. Signal Process. 40, 927–945 (1992).
    [CrossRef]
  11. M. K. Ng, “Nonlinear image restoration using FFT-based conjugate gradient methods,” in Proceedings of the International Conference on Image Processing (IEEE Press, Piscataway, N.J., 1995), pp. 41–44.
  12. A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall of India, New Delhi, 2000).
  13. N. J. Gordon, D. J. Salmond, A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F Image Vision Signal Process. 140, 107–113 (1993).
    [CrossRef]
  14. S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, “A tutorial on particle filters for on-line nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188 (2002).
    [CrossRef]
  15. D. L. Alspach, H. W. Sorenson, “Nonlinear Bayesian estimation using Gaussian sum approximation,” IEEE Trans. Autom. Control 17, 439–447 (1972).
    [CrossRef]
  16. M. Isard, A. Blake, “Visual tracking by stochastic propagation of conditional density,” in Proceedings of the Fourth European Conference on Computer Vision (Springer-Verlag, Berlin, 1996), pp. 343–356.
  17. A. Doucet, S. Godsill, C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
    [CrossRef]
  18. B. D. Anderson, J. B. Moore, Optimal Filtering (Prentice Hall, Englewood Cliffs, N.J., 1979).
  19. F. Naderi, A. Sawchuk, “Estimation of images degraded by film-grain noise,” Appl. Opt. 17, 1228–1237 (1978).
    [CrossRef] [PubMed]
  20. G. K. Froehlich, J. F. Walkup, T. F. Krille, “Estimation in signal-dependent film-grain noise,” Appl. Opt. 20, 3619–4626 (1981).
    [CrossRef] [PubMed]
  21. A. M. Tekalp, G. Pavlovic, “Restoration in the presence of multiplicative noise with application to scanned photographic images,” IEEE Trans. Signal Process. 39, 2132–2136 (1991).
    [CrossRef]
  22. D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1984).
  23. A. F. M. Smith, A. E. Gelfand, “Bayesian statistics without tears: a sampling-resampling perspective,” Am. Stat. 46, 84–88 (1992).
  24. P. Brodatz, Textures: A Photographic Album for Artists and Designers (Dover, New York, 1966).

2002

S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, “A tutorial on particle filters for on-line nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188 (2002).
[CrossRef]

2000

A. Doucet, S. Godsill, C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

1997

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14, 24–45 (1997).
[CrossRef]

1993

N. J. Gordon, D. J. Salmond, A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F Image Vision Signal Process. 140, 107–113 (1993).
[CrossRef]

1992

M. E. Zervakis, A. N. Venetsanopoulos, “Iterative least squares estimators in non-linear image restoration,” IEEE Trans. Signal Process. 40, 927–945 (1992).
[CrossRef]

A. F. M. Smith, A. E. Gelfand, “Bayesian statistics without tears: a sampling-resampling perspective,” Am. Stat. 46, 84–88 (1992).

1991

A. M. Tekalp, G. Pavlovic, “Restoration in the presence of multiplicative noise with application to scanned photographic images,” IEEE Trans. Signal Process. 39, 2132–2136 (1991).
[CrossRef]

1990

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. (Bellingham) 29, 393–414 (1990).
[CrossRef]

1981

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust., Speech, Signal Process. 29, 188–197 (1981).
[CrossRef]

G. K. Froehlich, J. F. Walkup, T. F. Krille, “Estimation in signal-dependent film-grain noise,” Appl. Opt. 20, 3619–4626 (1981).
[CrossRef] [PubMed]

1978

1977

B. R. Hunt, “Bayesian methods in nonlinear digital image restoration,” IEEE Trans. Comput. 26, 219–229 (1977).
[CrossRef]

J. W. Woods, C. H. Radewan, “Kalman filtering in two dimensions,” IEEE Trans. Inf. Theory 23, 473–482 (1977).
[CrossRef]

1972

D. L. Alspach, H. W. Sorenson, “Nonlinear Bayesian estimation using Gaussian sum approximation,” IEEE Trans. Autom. Control 17, 439–447 (1972).
[CrossRef]

Alspach, D. L.

D. L. Alspach, H. W. Sorenson, “Nonlinear Bayesian estimation using Gaussian sum approximation,” IEEE Trans. Autom. Control 17, 439–447 (1972).
[CrossRef]

Anderson, B. D.

B. D. Anderson, J. B. Moore, Optimal Filtering (Prentice Hall, Englewood Cliffs, N.J., 1979).

Andrews, H. C.

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

Andrieu, C.

A. Doucet, S. Godsill, C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

Angwin, D.

D. Angwin, H. Kaufman, “Image restoration using a reduced order model Kalman filter,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1988), pp. 1000–1003.

Arulampalam, S.

S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, “A tutorial on particle filters for on-line nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188 (2002).
[CrossRef]

Banham, M. R.

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14, 24–45 (1997).
[CrossRef]

Biemond, J.

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

Blake, A.

M. Isard, A. Blake, “Visual tracking by stochastic propagation of conditional density,” in Proceedings of the Fourth European Conference on Computer Vision (Springer-Verlag, Berlin, 1996), pp. 343–356.

Brodatz, P.

P. Brodatz, Textures: A Photographic Album for Artists and Designers (Dover, New York, 1966).

Clapp, T.

S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, “A tutorial on particle filters for on-line nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188 (2002).
[CrossRef]

Doucet, A.

A. Doucet, S. Godsill, C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

Dudgeon, D. E.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1984).

Froehlich, G. K.

Gelfand, A. E.

A. F. M. Smith, A. E. Gelfand, “Bayesian statistics without tears: a sampling-resampling perspective,” Am. Stat. 46, 84–88 (1992).

Godsill, S.

A. Doucet, S. Godsill, C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

Gordon, N.

S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, “A tutorial on particle filters for on-line nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188 (2002).
[CrossRef]

Gordon, N. J.

N. J. Gordon, D. J. Salmond, A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F Image Vision Signal Process. 140, 107–113 (1993).
[CrossRef]

Hunt, B. R.

B. R. Hunt, “Bayesian methods in nonlinear digital image restoration,” IEEE Trans. Comput. 26, 219–229 (1977).
[CrossRef]

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

Ingle, V. K.

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust., Speech, Signal Process. 29, 188–197 (1981).
[CrossRef]

Isard, M.

M. Isard, A. Blake, “Visual tracking by stochastic propagation of conditional density,” in Proceedings of the Fourth European Conference on Computer Vision (Springer-Verlag, Berlin, 1996), pp. 343–356.

Jain, A. K.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall of India, New Delhi, 2000).

Katsaggelos, A. K.

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14, 24–45 (1997).
[CrossRef]

Kaufman, H.

D. Angwin, H. Kaufman, “Image restoration using a reduced order model Kalman filter,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1988), pp. 1000–1003.

Krille, T. F.

Lagendijk, R. L.

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

Maskell, S.

S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, “A tutorial on particle filters for on-line nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188 (2002).
[CrossRef]

Mersereau, R. M.

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1984).

Moore, J. B.

B. D. Anderson, J. B. Moore, Optimal Filtering (Prentice Hall, Englewood Cliffs, N.J., 1979).

Naderi, F.

Ng, M. K.

M. K. Ng, “Nonlinear image restoration using FFT-based conjugate gradient methods,” in Proceedings of the International Conference on Image Processing (IEEE Press, Piscataway, N.J., 1995), pp. 41–44.

Ozkan, M. K.

A. J. Patti, M. K. Ozkan, A. M. Tekalp, M. I. Sezan, “New approaches for space-variant image restoration,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1993), pp. 261–264.

Patti, A. J.

A. J. Patti, M. K. Ozkan, A. M. Tekalp, M. I. Sezan, “New approaches for space-variant image restoration,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1993), pp. 261–264.

Pavlovic, G.

A. M. Tekalp, G. Pavlovic, “Restoration in the presence of multiplicative noise with application to scanned photographic images,” IEEE Trans. Signal Process. 39, 2132–2136 (1991).
[CrossRef]

Radewan, C. H.

J. W. Woods, C. H. Radewan, “Kalman filtering in two dimensions,” IEEE Trans. Inf. Theory 23, 473–482 (1977).
[CrossRef]

Salmond, D. J.

N. J. Gordon, D. J. Salmond, A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F Image Vision Signal Process. 140, 107–113 (1993).
[CrossRef]

Sawchuk, A.

Sezan, M. I.

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. (Bellingham) 29, 393–414 (1990).
[CrossRef]

A. J. Patti, M. K. Ozkan, A. M. Tekalp, M. I. Sezan, “New approaches for space-variant image restoration,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1993), pp. 261–264.

Smith, A. F. M.

N. J. Gordon, D. J. Salmond, A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F Image Vision Signal Process. 140, 107–113 (1993).
[CrossRef]

A. F. M. Smith, A. E. Gelfand, “Bayesian statistics without tears: a sampling-resampling perspective,” Am. Stat. 46, 84–88 (1992).

Sorenson, H. W.

D. L. Alspach, H. W. Sorenson, “Nonlinear Bayesian estimation using Gaussian sum approximation,” IEEE Trans. Autom. Control 17, 439–447 (1972).
[CrossRef]

Tekalp, A. M.

A. M. Tekalp, G. Pavlovic, “Restoration in the presence of multiplicative noise with application to scanned photographic images,” IEEE Trans. Signal Process. 39, 2132–2136 (1991).
[CrossRef]

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. (Bellingham) 29, 393–414 (1990).
[CrossRef]

A. J. Patti, M. K. Ozkan, A. M. Tekalp, M. I. Sezan, “New approaches for space-variant image restoration,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1993), pp. 261–264.

Venetsanopoulos, A. N.

M. E. Zervakis, A. N. Venetsanopoulos, “Iterative least squares estimators in non-linear image restoration,” IEEE Trans. Signal Process. 40, 927–945 (1992).
[CrossRef]

Walkup, J. F.

Woods, J. W.

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust., Speech, Signal Process. 29, 188–197 (1981).
[CrossRef]

J. W. Woods, C. H. Radewan, “Kalman filtering in two dimensions,” IEEE Trans. Inf. Theory 23, 473–482 (1977).
[CrossRef]

Zervakis, M. E.

M. E. Zervakis, A. N. Venetsanopoulos, “Iterative least squares estimators in non-linear image restoration,” IEEE Trans. Signal Process. 40, 927–945 (1992).
[CrossRef]

Am. Stat.

A. F. M. Smith, A. E. Gelfand, “Bayesian statistics without tears: a sampling-resampling perspective,” Am. Stat. 46, 84–88 (1992).

Appl. Opt.

IEE Proc. F Image Vision Signal Process.

N. J. Gordon, D. J. Salmond, A. F. M. Smith, “Novel approach to nonlinear/non-Gaussian Bayesian state estimation,” IEE Proc. F Image Vision Signal Process. 140, 107–113 (1993).
[CrossRef]

IEEE Signal Process. Mag.

M. R. Banham, A. K. Katsaggelos, “Digital image restoration,” IEEE Signal Process. Mag. 14, 24–45 (1997).
[CrossRef]

IEEE Trans. Acoust., Speech, Signal Process.

J. W. Woods, V. K. Ingle, “Kalman filtering in two dimensions: further results,” IEEE Trans. Acoust., Speech, Signal Process. 29, 188–197 (1981).
[CrossRef]

IEEE Trans. Autom. Control

D. L. Alspach, H. W. Sorenson, “Nonlinear Bayesian estimation using Gaussian sum approximation,” IEEE Trans. Autom. Control 17, 439–447 (1972).
[CrossRef]

IEEE Trans. Comput.

B. R. Hunt, “Bayesian methods in nonlinear digital image restoration,” IEEE Trans. Comput. 26, 219–229 (1977).
[CrossRef]

IEEE Trans. Inf. Theory

J. W. Woods, C. H. Radewan, “Kalman filtering in two dimensions,” IEEE Trans. Inf. Theory 23, 473–482 (1977).
[CrossRef]

IEEE Trans. Signal Process.

M. E. Zervakis, A. N. Venetsanopoulos, “Iterative least squares estimators in non-linear image restoration,” IEEE Trans. Signal Process. 40, 927–945 (1992).
[CrossRef]

S. Arulampalam, S. Maskell, N. Gordon, T. Clapp, “A tutorial on particle filters for on-line nonlinear/non-Gaussian Bayesian tracking,” IEEE Trans. Signal Process. 50, 174–188 (2002).
[CrossRef]

A. M. Tekalp, G. Pavlovic, “Restoration in the presence of multiplicative noise with application to scanned photographic images,” IEEE Trans. Signal Process. 39, 2132–2136 (1991).
[CrossRef]

Opt. Eng. (Bellingham)

M. I. Sezan, A. M. Tekalp, “Survey of recent developments in digital image restoration,” Opt. Eng. (Bellingham) 29, 393–414 (1990).
[CrossRef]

Stat. Comput.

A. Doucet, S. Godsill, C. Andrieu, “On sequential Monte Carlo sampling methods for Bayesian filtering,” Stat. Comput. 10, 197–208 (2000).
[CrossRef]

Other

B. D. Anderson, J. B. Moore, Optimal Filtering (Prentice Hall, Englewood Cliffs, N.J., 1979).

M. Isard, A. Blake, “Visual tracking by stochastic propagation of conditional density,” in Proceedings of the Fourth European Conference on Computer Vision (Springer-Verlag, Berlin, 1996), pp. 343–356.

A. J. Patti, M. K. Ozkan, A. M. Tekalp, M. I. Sezan, “New approaches for space-variant image restoration,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1993), pp. 261–264.

D. Angwin, H. Kaufman, “Image restoration using a reduced order model Kalman filter,” in Proceedings of the International Conference on Acoustics, Speech, and Signal Processing (IEEE Press, Piscataway, N.J., 1988), pp. 1000–1003.

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

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

M. K. Ng, “Nonlinear image restoration using FFT-based conjugate gradient methods,” in Proceedings of the International Conference on Image Processing (IEEE Press, Piscataway, N.J., 1995), pp. 41–44.

A. K. Jain, Fundamentals of Digital Image Processing (Prentice Hall of India, New Delhi, 2000).

D. E. Dudgeon, R. M. Mersereau, Multidimensional Digital Signal Processing (Prentice Hall, Englewood Cliffs, N.J., 1984).

P. Brodatz, Textures: A Photographic Album for Artists and Designers (Dover, New York, 1966).

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Figures (6)

Fig. 1
Fig. 1

(a) 2-D N × N field for recursive estimation. (b) State vector s ( m ,   n ) .

Fig. 2
Fig. 2

(a) Original “Bark” image. (b) Nonlinearly degraded image for α = 5 and SNR = 7   dB in the exposure domain. Image recovered with (c) traditional Wiener filter ( ISNR = 0.10   dB ) , (d) median filter of size 3 × 3   ( ISNR = 1.98   dB ) , (e) MLW filter ( ISNR = 2.35   dB ) , and (f) particle filter for N = 500   ( ISNR = 3.62   dB ) . Plot (g) shows the diversity of the particles at locations (181, 150) and (181, 151).

Fig. 3
Fig. 3

(a) Original “Pentagon” image. Degraded image in (b) the density domain and (c) the exposure domain ( σ v deg 2 = 0.088 ) . (d) Image recovered with the MLW filter ( ISNR = 2.82   dB , and σ v rec 2 = 0.046 ) . Image recovered with the particle filter with (e) N = 5   ( ISNR = 1.16   dB ) , and (f) N = 500 ( ISNR = 3.86   dB , and σ v rec 2 = 0.026 ) .

Fig. 4
Fig. 4

(a) Original “Peppers” image. Degraded image in (b) the density domain and (c) the exposure domain ( σ v deg 2 = 0.153 ) . (d) Image recovered with the MLW filter ( ISNR = 3.24   dB , and σ v rec 2 = 0.078 ) . Image recovered with the particle filter with (e) N = 5   ( ISNR = 2.06   dB ) , and (f) N = 500 ( ISNR = 4.65   dB , and σ v rec 2 = 0.047 ) .

Fig. 5
Fig. 5

(a) ISNR is plotted as a function of the number of samples of the particle filter for different values of SNR. (b) ISNR is plotted as a function of the input SNR. (c) Sensitivity of the particle filter output to uncertainty in α ( SNR = 13   dB ) . The true value of α is 5.

Fig. 6
Fig. 6

(a) Noisy “Boat” image corrupted by real film-grain noise ( σ v deg 2 = 0.073 ) . (b) Image recovered with the particle filter corresponding to N = 500   ( σ v rec 2 = 0.03 ) . (c) “Sky” image degraded by real film-grain noise ( σ v deg 2 = 0.758 ) . (d) Image recovered with the particle filter with 500 samples ( σ v rec 2 = 0.396 ) .

Equations (39)

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

s ( m ,   n ) = ( i , j ) Γ c i , j s ( m - i ,   n - j ) + 1 - ( i , j )   Γ c i , j s ¯ + w ( m ,   n ) ,
Γ = { ( m - i ,   n - j ) | ( - M i M ,   1 j M ) ( 0 ,   1 j M ) } ,
d ( m ,   n ) = g [ s ( m ,   n ) ] v ( m ,   n ) ,
d ( m ,   n ) = α   log 10 [ s ( m ,   n ) ] + β + v ( m ,   n ) ,
e ( m ,   n ) = v e ( m ,   n ) s ( m ,   n ) ,
s ( m ,   n ) = f m - 1 , n ( s ( m - 1 ,   n ) ,   w ( m - 1 ,   n ) ) ,
y ( m ,   n ) = h m , n ( s ( m ,   n ) ,   v ( m ,   n ) ) ,
p ( s ( m ,   n ) | χ ( m - 1 ,   n ) ) = p ( s ( m ,   n ) | s ( m - 1 ,   n ) ) .
p ( s ( m ,   n ) | Υ m - 1 , n )
= p ( s ( m ,   n ) | Υ m - 1 , n , s ( m - 1 ,   n ) ) × p ( s ( m - 1 ,   n ) | Υ m - 1 , n ) d s ( m - 1 ,   n ) .
p ( s ( m ,   n ) | Υ m - 1 , n ,   s ( m - 1 ,   n ) )
= p ( s ( m ,   n ) | s ( m - 1 ,   n ) ) ,
p ( s ( m ,   n ) | Υ m - 1 , n ) = p ( s ( m ,   n ) | s ( m - 1 ,   n ) ) × p ( s ( m - 1 ,   n ) | Υ m - 1 , n ) × d s ( m - 1 ,   n ) .
p ( s ( m ,   n ) | s ( m - 1 ,   n ) )
= p ( s ( m ,   n ) | s ( m - 1 ,   n ) ,   w ( m - 1 ,   n ) ) × p ( w ( m - 1 ,   n ) | s ( m - 1 ,   n ) ) d w ( m - 1 ,   n ) .
p ( w ( m - 1 ,   n ) | s ( m - 1 ,   n ) ) = p ( w ( m - 1 ,   n ) ) .
p ( s ( m ,   n ) | s ( m - 1 ,   n ) )
= p ( s ( m ,   n ) | s ( m - 1 ,   n ) ,   w ( m - 1 ,   n ) ) × p ( w ( m - 1 ,   n ) ) d w ( m - 1 ,   n ) ,
p ( s ( m ,   n ) | Υ m , n ) = p ( s ( m ,   n ) | Υ m - 1 , n ,   y ( m ,   n ) )
= p ( y ( m ,   n ) | s ( m ,   n ) ,   Υ m - 1 , n ) p ( s ( m ,   n ) ,   Υ m - 1 , n ) p ( y ( m ,   n ) ,   Υ m - 1 , n ) .
p ( y ( m ,   n ) | Υ m - 1 , n ,   s ( m ,   n ) ) = p ( y ( m ,   n ) | s ( m ,   n ) ) .
p ( s ( m ,   n ) | Υ m , n )
= p ( y ( m ,   n ) | s ( m ,   n ) ) p ( s ( m ,   n ) ,   Υ m - 1 , n ) p ( y ( m ,   n ) ,   Υ m - 1 , n ) = p ( y ( m ,   n ) | s ( m ,   n ) ) p ( s ( m ,   n ) | Υ m - 1 , n ) p ( y ( m ,   n ) | Υ m - 1 , n ) .
p ( y ( m ,   n ) | Υ m - 1 , n ) = p ( y ( m ,   n ) | Υ m - 1 , n ,   s ( m ,   n ) )
× p ( s ( m ,   n ) | Υ m - 1 , n ) d s ( m ,   n ) .
p ( s ( m ,   n ) | ( s ( m - 1 ,   n ) = s ( k ) ( m - 1 ,   n ) ) )
s pr ( k ) ( m ,   n ) = ( i , j ) Γ c i , j s ( k ) ( m - i ,   n - j )
+ 1 - ( i , j ) Γ c i , j s ¯ + w ( k ) ( m - 1 ,   n ) ,
p ( s ( m ,   n ) | Υ m , n )
= p ( y ( m ,   n ) | s ( m ,   n ) ) p ( s ( m ,   n ) | Υ m - 1 , n ) p ( y ( m ,   n ) | Υ m - 1 , n ) ,
Ω m , n ( k ) = p ( y ( m ,   n ) | s pr ( k ) ( m ,   n ) ) i = 1 N p ( y ( m ,   n ) | s pr ( i ) ( m ,   n ) ) , k = 1 , 2 , , N .
Ω m , n ( k ) = p v ( d ( m ,   n ) - α   log 10 [ s ( k ) ( m ,   n ) ] - β ) i = 1 N p v ( d ( m ,   n ) - α   log 10 [ s ( i ) ( m ,   n ) ] - β ) .
{ s ( j ) ( m ,   n ) = s pr ( k ) ( m ,   n ) } = Ω m , n ( k ) .
E [ f ( s ( m ,   n ) ) ] = k = 1 N Ω m , n ( k ) f ( s pr ( k ) ( m ,   n ) ) .
SNR = 10   log 10 variance of the degraded image variance of the observation noise dB .
ISNR = 10   log 10 i , j [ d ( i ,   j ) - s ( i ,   j ) ] 2 i , j [ r ( i ,   j ) - s ( i ,   j ) ] 2 dB ,
E [ ρ v e ( m ,   n ) ] = ρ   exp σ v 2 ln 2   10 2 α 2 ,
E [ ρ 2 v e 2 ( m ,   n ) ] = ρ   exp 2 σ v 2 ln 2   10 α 2 .
σ v 2 = α 2 ln 2   10 ln E [ ρ 2 v e 2 ( m ,   n ) ] { E [ ρ v e ( m ,   n ) ] } 2 .

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