Abstract

Film-grain noise describes the intrinsic noise produced by a photographic emulsion during the process of image recording and reproduction. In this paper we consider the restoration of images degraded by film-grain noise. First a detailed model for the over-all photographic imaging system is presented. The model includes linear blurring effects and the signal-dependent effect of film-grain noise. The accuracy of this model is tested by simulating images according to it and comparing the results to images of similar targets that were actually recorded on film. The restoration of images degraded by film-grain noise is then considered in the context of estimation theory. A discrete Wiener filer is developed which explicitly allows for the signal dependence of the noise. The filter adaptively alters its characteristics based on the nonstationary first order statistics of an image and is shown to have advantages over the conventional Wiener filter. Experimental results for modeling and the adaptive estimation filter are presented.

© 1978 Optical Society of America

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References

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  1. F. Naderi, Ph.D. Dissertation, Department of Electrical Engineering, U. Southern California, Los Angeles (September1976);USC Image Processing Institute Report 690 (1976).
  2. W. Thomas, SPSE Handbook of Photographic Science and Engineering (Wiley, New York, 1973), p. 768.
  3. C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966), Chap. 7.
  4. P. G. Nutting, Philos. Mag. 26, 423 (1913).
  5. J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 41.
  6. G. C. Higgins, K. F. Stultz, J. Opt. Soc. Am. 49, 925 (1959).
    [CrossRef]
  7. E. A. Trabka, E. C. Doerner, J. Appl. Photogr. Eng. 2, 1 (1976).
    [CrossRef]
  8. T. S. Huang, “Some Notes on Film-Grain Noise,” Appendix 14, in Restoration of Atmospherically Degraded Images, NSF Summer Study Report, Woods Hole, Mass. (1966), pp. 105–109.
  9. D. H. Kelly, J. Opt. Soc. Am. 59, 269 (1960).
    [CrossRef]
  10. H. W. Lorber, IBM J. Res. Dev. 14, 515 (1970).
    [CrossRef]
  11. N. E. Nahi, Estimation Theory and Applications (Wiley, New York, 1969), Chap. 2.
  12. M. Naraghi, Ph.D. Dissertation, Department of Electrical Engineering, U. Southern California, Los Angeles (June1975);USC Image Processing Institute Report 580 (1975).
  13. B. R. Hunt, Proc. IEEE 63, 693 (1975).
    [CrossRef]
  14. J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).
    [CrossRef]
  15. K. Kondo, Y. Ichioka, T. Suzuki, Appl. Opt. 16, 2554 (1977).
    [CrossRef] [PubMed]
  16. B. R. Hunt, T. M. Cannon, IEEE Trans. Syst. Man Cybern. SMC-6, 876 (1976).
  17. W. K. Pratt, IEEE Trans. Comput. C-21, 636 (1972).
    [CrossRef]
  18. W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).
  19. H. C. Andrews, B. R. Hunt, Digital Image Restoration (Prentice-Hall, Englewood Cliffs, N.J., 1977).

1977 (1)

1976 (2)

B. R. Hunt, T. M. Cannon, IEEE Trans. Syst. Man Cybern. SMC-6, 876 (1976).

E. A. Trabka, E. C. Doerner, J. Appl. Photogr. Eng. 2, 1 (1976).
[CrossRef]

1975 (1)

B. R. Hunt, Proc. IEEE 63, 693 (1975).
[CrossRef]

1974 (1)

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).
[CrossRef]

1972 (1)

W. K. Pratt, IEEE Trans. Comput. C-21, 636 (1972).
[CrossRef]

1970 (1)

H. W. Lorber, IBM J. Res. Dev. 14, 515 (1970).
[CrossRef]

1960 (1)

D. H. Kelly, J. Opt. Soc. Am. 59, 269 (1960).
[CrossRef]

1959 (1)

1913 (1)

P. G. Nutting, Philos. Mag. 26, 423 (1913).

Andrews, H. C.

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

Cannon, T. M.

B. R. Hunt, T. M. Cannon, IEEE Trans. Syst. Man Cybern. SMC-6, 876 (1976).

Choens, R. C.

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).
[CrossRef]

Dainty, J. C.

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 41.

Doerner, E. C.

E. A. Trabka, E. C. Doerner, J. Appl. Photogr. Eng. 2, 1 (1976).
[CrossRef]

Higgins, G. C.

Huang, T. S.

T. S. Huang, “Some Notes on Film-Grain Noise,” Appendix 14, in Restoration of Atmospherically Degraded Images, NSF Summer Study Report, Woods Hole, Mass. (1966), pp. 105–109.

Hunt, B. R.

B. R. Hunt, T. M. Cannon, IEEE Trans. Syst. Man Cybern. SMC-6, 876 (1976).

B. R. Hunt, Proc. IEEE 63, 693 (1975).
[CrossRef]

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

Ichioka, Y.

James, T. H.

C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966), Chap. 7.

Kelly, D. H.

D. H. Kelly, J. Opt. Soc. Am. 59, 269 (1960).
[CrossRef]

Kondo, K.

Lorber, H. W.

H. W. Lorber, IBM J. Res. Dev. 14, 515 (1970).
[CrossRef]

Mees, C. E. K.

C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966), Chap. 7.

Naderi, F.

F. Naderi, Ph.D. Dissertation, Department of Electrical Engineering, U. Southern California, Los Angeles (September1976);USC Image Processing Institute Report 690 (1976).

Nahi, N. E.

N. E. Nahi, Estimation Theory and Applications (Wiley, New York, 1969), Chap. 2.

Naraghi, M.

M. Naraghi, Ph.D. Dissertation, Department of Electrical Engineering, U. Southern California, Los Angeles (June1975);USC Image Processing Institute Report 580 (1975).

Nutting, P. G.

P. G. Nutting, Philos. Mag. 26, 423 (1913).

Pratt, W. K.

W. K. Pratt, IEEE Trans. Comput. C-21, 636 (1972).
[CrossRef]

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).

Shaw, R.

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 41.

Stultz, K. F.

Suzuki, T.

Thomas, W.

W. Thomas, SPSE Handbook of Photographic Science and Engineering (Wiley, New York, 1973), p. 768.

Trabka, E. A.

E. A. Trabka, E. C. Doerner, J. Appl. Photogr. Eng. 2, 1 (1976).
[CrossRef]

Walkup, J. F.

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).
[CrossRef]

Appl. Opt. (1)

IBM J. Res. Dev. (1)

H. W. Lorber, IBM J. Res. Dev. 14, 515 (1970).
[CrossRef]

IEEE Trans. Comput. (1)

W. K. Pratt, IEEE Trans. Comput. C-21, 636 (1972).
[CrossRef]

IEEE Trans. Syst. Man Cybern. SMC-6 (1)

B. R. Hunt, T. M. Cannon, IEEE Trans. Syst. Man Cybern. SMC-6, 876 (1976).

J. Appl. Photogr. Eng. (1)

E. A. Trabka, E. C. Doerner, J. Appl. Photogr. Eng. 2, 1 (1976).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Eng. (1)

J. F. Walkup, R. C. Choens, Opt. Eng. 13, 258 (1974).
[CrossRef]

Philos. Mag. (1)

P. G. Nutting, Philos. Mag. 26, 423 (1913).

Proc. IEEE (1)

B. R. Hunt, Proc. IEEE 63, 693 (1975).
[CrossRef]

Other (9)

J. C. Dainty, R. Shaw, Image Science (Academic, New York, 1974), p. 41.

W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).

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

F. Naderi, Ph.D. Dissertation, Department of Electrical Engineering, U. Southern California, Los Angeles (September1976);USC Image Processing Institute Report 690 (1976).

W. Thomas, SPSE Handbook of Photographic Science and Engineering (Wiley, New York, 1973), p. 768.

C. E. K. Mees, T. H. James, The Theory of the Photographic Process (Macmillan, New York, 1966), Chap. 7.

T. S. Huang, “Some Notes on Film-Grain Noise,” Appendix 14, in Restoration of Atmospherically Degraded Images, NSF Summer Study Report, Woods Hole, Mass. (1966), pp. 105–109.

N. E. Nahi, Estimation Theory and Applications (Wiley, New York, 1969), Chap. 2.

M. Naraghi, Ph.D. Dissertation, Department of Electrical Engineering, U. Southern California, Los Angeles (June1975);USC Image Processing Institute Report 580 (1975).

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Figures (10)

Fig. 1
Fig. 1

An additive signal-dependent model for film-grain noise.

Fig. 2
Fig. 2

The target used for evaluation of the models.

Fig. 3
Fig. 3

Panatomic-X film digitized with a 2 × 2-μm2 aperture.

Fig. 4
Fig. 4

Image simulated according to the model of Fig. 1.

Fig. 5
Fig. 5

A more accurate model for the imaging system.

Fig. 6
Fig. 6

Image simulated according to the model of Fig. 5.

Fig. 7
Fig. 7

The discrete model for the imaging system.

Fig. 8
Fig. 8

Restoration of a 1-D signal: (a) ideal signal; (b) blurred signal; (c) nonlinear noise added; (d) observed signal; (e) estimated signal.

Fig. 9
Fig. 9

Two-dimensional restoration: (a) ideal image; (b) degraded image; (c) restored image.

Fig. 10
Fig. 10

Comparison with conventional Wiener filter: (a) ideal image; (b) degraded image; (c) image restored by conventional Wiener filter; (d) image restored by the nonstationary filter of Eq. (17).

Equations (22)

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D = g ( log 10 E ) ,
E = 10 g 1 ( D )
σ D = ( 0.43 a A ) 1 / 2 ( D ¯ ) P ,
σ D = k ( D ¯ ) 1 / 3 ,
D ¯ s = g ( log 10 E ) ,
D o = D ¯ s + n n N [ 0 , k 2 ( D ¯ s ) 2 / 3 ] ,
D o = D ¯ s + k ( D ¯ s ) 1 / 3 · n n N ( 0 , 1 ) ,
y = g ( s , n ) ,
s ̂ = E ( s | y ) = s P ( s | y ) d s ,
D o = D s + k D s P · n ,
y ¯ = g ( [ H ] · s ¯ ) + n ¯ ,
W ( ω 1 , ω 2 ) = Φ D s D s ( ω 1 , ω 2 ) Φ D s D s ( ω 1 , ω 2 ) + k 2 E [ D s ( ξ , η ) 2 P ] ,
E ( N ¯ ) = 0 ,
C NN = E ( N ¯ · N ¯ T ) = [ I ] ,
̂ = [ W ] D ¯ o .
[ W ] = [ C ID ¯ o ] [ C D ¯ o D ¯ o ] 1 ,
e = E ( ̂ ) T ( ̂ ) .
[ W ] = α [ C II ¯ ] [ H 1 ] T ( α 2 [ H 1 ] [ C II ¯ ] [ H 1 ] T + [ C N N ] ) 1 [ H 2 ] 1
[ C N N ] Δ E { [ F ] NN ¯ T [ F ] T }
[ C II ¯ ] = E [ II ¯ T ] m m T
Diagonal of [ C N N ] = α k 2 · E ( [ H 1 ] + B ¯ ) 2 P .
Diagonal of [ C N N ] = α k 2 [ H 1 ] m + α k 2 B ¯ .

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