Abstract

A transformation to convert signal-dependent noise corrupting an image to additive Gaussian signal-independent noise is derived in this paper. Wiener filtering techniques using a Markovian covariance model for the image signal are applied to the transformed data followed by an inverse transformation to restore the degraded image. An ad hoc technique using contrast manipulation to adaptively convert signal-dependent noise to signal-independent noise is also described. The results of the computer simulations designed to evaluate the performance of these techniques are also presented.

© 1983 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 (Wiley, New York, 1978).
  3. T. S. Huang, Ed., Picture Processing and Digital Filtering (Springer-Verlag, New York, 1975).
  4. G. K. Froehlich, J. F. Walkup, T. F. Krile, Appl. Opt. 20, 3619 (1981).
    [CrossRef] [PubMed]
  5. C. M. Lo, Estimation of Image Signals with Poisson Noise, Image Processing Institute, University of Southern California, Report No. 890 (1979).
  6. B. R. Hunt, H. J. Trussell, Proc. Soc. Photo-Opt. Instrum. Eng. 119, 269 (1977).
  7. H. H. Arsenault, C. Gendron, M. Denis, J. Opt. Soc. Am. 71, 91 (1981).
    [CrossRef]
  8. H. H. Arsenault, M. Denis, Opt. Lett. 6, 210 (1981).
    [CrossRef] [PubMed]
  9. P. R. Prucnal, B. E. A. Saleh, Opt. Lett. 6, 316 (1981).
    [CrossRef] [PubMed]
  10. G. K. Froehlich, J. F. Walkup, R. B. Asher, J. Opt. Soc. Am. 68, 1665 (1978).
    [CrossRef]
  11. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 126, 151, 212.
  12. H. H. Arsenault, M. Denis, Can. J. Phys. 61, 309 (1983).
    [CrossRef]
  13. R. Kasturi, “Adaptive Image Restoration in Signal-Dependent Noise, Ph.D. Dissertation (Texas Tech University, 1982).
  14. A. K. Jain, “Some New Techniques in Image Processing,” in Image Science Mathematics (Western Periodicals, North Hollywood, Calif., 1977), pp. 201–223.
  15. B. R. Hunt, T. M. Cannon, IEEE Trans. Syst. Man Cybern. SMC-6, 876 (1976).
  16. A. Arcese, P. H. Mengert, E. W. Trombini, IEEE Trans. Inf. Theory, IT-16, 534 (1970).
    [CrossRef]
  17. W. H. Beyer, Ed., CRC Standard Mathematical Tables (CRC Press, West Palm Beach, Fla., 1978), pp. 32, 33.
  18. C. F. Hall, “Digital Color Image Compression in a Perceptual Space,” Image Processing Institute, University of Southern California, Report No. 790 (1978), Chap. 8.

1983 (1)

H. H. Arsenault, M. Denis, Can. J. Phys. 61, 309 (1983).
[CrossRef]

1981 (4)

1978 (1)

1977 (1)

B. R. Hunt, H. J. Trussell, Proc. Soc. Photo-Opt. Instrum. Eng. 119, 269 (1977).

1976 (1)

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

1970 (1)

A. Arcese, P. H. Mengert, E. W. Trombini, IEEE Trans. Inf. Theory, IT-16, 534 (1970).
[CrossRef]

Andrews, H. C.

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

Arcese, A.

A. Arcese, P. H. Mengert, E. W. Trombini, IEEE Trans. Inf. Theory, IT-16, 534 (1970).
[CrossRef]

Arsenault, H. H.

Asher, R. B.

Cannon, T. M.

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

Denis, M.

Froehlich, G. K.

Gendron, C.

Hall, C. F.

C. F. Hall, “Digital Color Image Compression in a Perceptual Space,” Image Processing Institute, University of Southern California, Report No. 790 (1978), Chap. 8.

Hunt, B. R.

B. R. Hunt, H. J. Trussell, Proc. Soc. Photo-Opt. Instrum. Eng. 119, 269 (1977).

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

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

Jain, A. K.

A. K. Jain, “Some New Techniques in Image Processing,” in Image Science Mathematics (Western Periodicals, North Hollywood, Calif., 1977), pp. 201–223.

Kasturi, R.

R. Kasturi, “Adaptive Image Restoration in Signal-Dependent Noise, Ph.D. Dissertation (Texas Tech University, 1982).

Krile, T. F.

Lo, C. M.

C. M. Lo, Estimation of Image Signals with Poisson Noise, Image Processing Institute, University of Southern California, Report No. 890 (1979).

Mengert, P. H.

A. Arcese, P. H. Mengert, E. W. Trombini, IEEE Trans. Inf. Theory, IT-16, 534 (1970).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 126, 151, 212.

Pratt, W. K.

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

Prucnal, P. R.

Saleh, B. E. A.

Trombini, E. W.

A. Arcese, P. H. Mengert, E. W. Trombini, IEEE Trans. Inf. Theory, IT-16, 534 (1970).
[CrossRef]

Trussell, H. J.

B. R. Hunt, H. J. Trussell, Proc. Soc. Photo-Opt. Instrum. Eng. 119, 269 (1977).

Walkup, J. F.

Appl. Opt. (1)

Can. J. Phys. (1)

H. H. Arsenault, M. Denis, Can. J. Phys. 61, 309 (1983).
[CrossRef]

IEEE Trans. Inf. Theory (1)

A. Arcese, P. H. Mengert, E. W. Trombini, IEEE Trans. Inf. Theory, IT-16, 534 (1970).
[CrossRef]

IEEE Trans. Syst. Man Cybern. (1)

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

J. Opt. Soc. Am. (2)

Opt. Lett. (2)

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

B. R. Hunt, H. J. Trussell, Proc. Soc. Photo-Opt. Instrum. Eng. 119, 269 (1977).

Other (9)

R. Kasturi, “Adaptive Image Restoration in Signal-Dependent Noise, Ph.D. Dissertation (Texas Tech University, 1982).

A. K. Jain, “Some New Techniques in Image Processing,” in Image Science Mathematics (Western Periodicals, North Hollywood, Calif., 1977), pp. 201–223.

W. H. Beyer, Ed., CRC Standard Mathematical Tables (CRC Press, West Palm Beach, Fla., 1978), pp. 32, 33.

C. F. Hall, “Digital Color Image Compression in a Perceptual Space,” Image Processing Institute, University of Southern California, Report No. 790 (1978), Chap. 8.

A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965), pp. 126, 151, 212.

C. M. Lo, Estimation of Image Signals with Poisson Noise, Image Processing Institute, University of Southern California, Report No. 890 (1979).

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

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

T. S. Huang, Ed., Picture Processing and Digital Filtering (Springer-Verlag, New York, 1975).

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Figures (6)

Fig. 1
Fig. 1

Original image.

Fig. 2
Fig. 2

Image corrupted by signal-dependent noise with σ1 = 4.0 and σ2 = 20.0.

Fig. 3
Fig. 3

Image restored using normalizing transformation–Wiener filter estimator (correlation along vertical direction only).

Fig. 4
Fig. 4

Image restored using normalizing transformation–Wiener filter estimator (correlation along horizontal direction only).

Fig. 5
Fig. 5

Composite image obtained by combining Figs. 3 and 4.

Fig. 6
Fig. 6

Image restored using the ad hoc contrast manipulation technique and Wiener filter.

Tables (1)

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Table I Image Quality Comparisons

Equations (52)

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R = S + f ( S ) N 1 + N 2 ,
N 1 N ( 0 , σ 1 2 ) ,
N 2 N ( 0 , σ 2 2 ) ,
X R | S = s
X N { s , [ f ( s ) ] 2 σ 1 2 + σ 2 2 } .
y = g ( x ) = dx σ X
Y N ( μ Y , 1 ) ,
μ Y g ( μ X ) .
Y = V + Z ,
Z N ( 0 , 1 ) ,
V = μ Y g ( S ) .
S = g 1 ( V ) .
f ( s ) = s ,
y = g ( x ) = 2 σ 1 2 x σ 1 2 + σ 2 2 ,
υ 2 σ 1 2 s σ 1 2 + σ 2 2 .
s = ( υ σ 1 2 ) 2 ( σ 2 σ 1 ) 2 .
p V ( υ ) = p S ( s ) | g ( s ) | | s
g ( s ) = g ( s ) s = 1 s σ 1 2 + σ 2 2 .
p V ( υ ) = υ σ 1 2 2 2 π σ S exp { 0.5 [ ( υ σ 1 2 ) 2 ( σ 2 σ 1 ) 2 μ S σ S ] 2 } .
μ V g ( μ S ) + g ( μ S ) σ S 2 2 = μ S σ 1 2 + σ 2 2 [ 2 σ 1 2 σ 1 2 σ S 2 4 ( μ S σ 1 2 + σ 2 2 ) 2 ] ,
σ V 2 = [ g ( μ S ) ] 2 σ S 2 + { [ g ( μ S ) ] 2 2 + g ( μ S ) g ( μ S ) } σ S 4 = σ S 2 ( μ S σ 1 2 + σ 2 2 ) 3 [ ( μ S σ 1 2 + σ 2 2 ) 2 + 7 8 σ 1 4 σ S 2 ] .
S ] N ( μ s ] , [ K S ] ) ,
[ K S ] = σ S 2 [ C ] ,
X ] N ( s ] , [ K X ] ) ,
Y ] N ( V ] , [ I ] ) ,
[ K V ] = σ V 2 [ C ] ,
Y ] = V ] + Z ] ,
Z ] N ( 0 ] , [ I ] )
V ̂ ] = [ W ] Y ] + B ] ,
[ W ] = [ I + K V 1 ] 1 ,
B ] = [ I W ] μ V ] .
a 1 = 1 + 1 ( 1 ρ 2 ) σ V 2 ,
a 2 = 1 + ( 1 + ρ 2 ) ( 1 ρ 2 ) σ V 2 ,
a 3 = ρ ( 1 ρ 2 ) σ V 2 .
W M , j = U M , j , j = 1 , M , W M k , j = U M k , j T M k W M k + 1 , j , [ k = 1 , M 1 j = 1 , M k ] , W j , i = W i , j , [ i = 1 , M j = 1 , M ] ,
U 1 , 1 = 1 a 1 , U i , i = 1 a 2 a 3 2 U i 1 , i 1 , i = 2 , M 1 , U M , M = 1 a 1 a 3 2 U M 1 , M 1 , U i , i k = a 3 U i , i U i 1 , i k , [ k = 1 , M 1 i = k + 1 , M , ]
T i = a 3 U i , i i = 1 , M 1 .
μ ̂ S i , j = 1 N 2 k 1 r ( i + k ) , ( j + 1 ) ,
y i , j = 2 σ 1 2 r i , j σ 1 2 + σ 2 2 ;
S ̂ i , j = ( V ̂ i , j σ 1 2 ) 2 ( σ 2 σ 1 ) 2 ;
σ R 2 = σ S 2 + σ 2 2 + σ 1 2 E { [ f ( S ) ] 2 } .
Y = μ S + ( R μ S ) σ S 2 + σ 2 2 σ R ,
μ Y = μ S ,
σ Y 2 = σ S 2 + σ 2 2 .
Y = S + N 2 .
σ ̂ R i , j 2 = σ S 2 + σ 2 2 + σ 1 2 μ ̂ S i , j .
S ̂ i , j = σ S 2 Y i , j + σ 2 2 μ ̂ S i , j σ S 2 + σ 2 2 .
NMSE = i = 1 M j = 1 M ( S i , j S ̂ i , j ) 2 i = 1 M j = 1 M ( S i , j ) 2 ,
LMSE = i = 2 M 1 j = 2 M 1 ( G i , j G ̂ i , j ) 2 i = 2 M 1 j = 2 M 1 ( G i , j ) 2 ,
G i , j = S i + 1 , j + S i 1 , j + S i , j + 1 + S i , j 1 4 S i , j
PMSE = i = 1 M j = 1 M ( z i , j z ̂ i , j ) 2 i = 1 M j = 1 M ( z i , j ) 2 ,
z = ln ( S ) * h ,

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