Abstract

We consider the restoration of images degraded by a class of signal-uncorrelated noise, which is possibly signal-dependent. Some adaptive noise smoothing filters, which assume a nonstationary mean, nonstationary variance image model implicitly or explicitly, are reviewed, and their performances are compared by the mean-squares errors (MSES) and by the human subjective judgment. We also present a new noise smoothing technique which is called the noise updating repeated Wiener (NURW) filter. Explicit noise variance updating formulas are derived for the NURW filter. The performance is improved both in the MSE sense and in the vicinity of edges by subjective observation.

© 1986 Optical Society of America

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  1. A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vols. 1 and 2 (Academic, New York, 1982).
  2. D. S. Lebedev, L. I. Mirkin, “Digital Nonlinear Smoothing of Images,” Institute for Information Transmission Problems, Academy of Sciences, U.S.S.R. (1975), pp. 159–157.
  3. V. K. Ingle, J. W. Woods, “Multiple Model Recursive Estimation of Images,” in Proceedings IEEE, ICASSP 79 (Washington, DC, Apr.1979), pp. 642–645.
  4. B. R. Hunt, T. M. Cannon, “Nonstationary Assumptions for Gaussian Models of Images,” IEEE Trans. System, Man Cybernet. 6, 876 (1976).
  5. B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comp. C-26, 219 (1977).
  6. H. T. Trussell, B. R. Hunt, “Sectioned Methods for Image Restoration,” IEEE Trans. Acoust., Speech Signal Process ASSP-26, 157 (1978).
  7. R. Chellappa, R. L. Kashyap, “Digital Image Restoration Using Spatial Interaction Models,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 461 (1982).
  8. D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise,” IEEE Trans. Pattern Anal. Machine Intell PAMI-7, 165 (1985).
  9. D. T. Kuan, “Nonstationary 2-D Recursive Restoration of Images with Signal-Dependent Noise with Application to Speckle Reduction,” Ph.D. Thesis, U. Southern California, Los Angeles (1982).
  10. S-S. Jiang, “Image Restoration and Speckle Suppression Using the Noise Updating Repeated Wiener Filter,” Ph.D. Thesis, U. Southern California, Los Angeles (1986).
  11. J. S. Lee, “Digital Image Enhancement and Noise Filtering by Use of Local Statistics,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-2, 165 (1980).
  12. P. C. Chan, J. S. Lim, “One-Dimensional Processing for Adaptive Image Restoration,” in Proceedings IEEE, ICASSP (San Diego, 1983), pp. 37.3.1–37.3.4.
  13. W. K. Pratt, Digital Image Processing (Wiley-Interscience, New York, 1978).
  14. A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).
  15. J. S. Lee, “Refined Noise Filtering Using Local Statistics,” Comput. Graphics Image Process. 15, 380 (1981).
  16. G. S. Robinson, “Edge Detection by Compass Gradient Masks,” in Comput. Graphics Image Process. 6, 492 (1977).

1985 (1)

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise,” IEEE Trans. Pattern Anal. Machine Intell PAMI-7, 165 (1985).

1982 (1)

R. Chellappa, R. L. Kashyap, “Digital Image Restoration Using Spatial Interaction Models,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 461 (1982).

1981 (1)

J. S. Lee, “Refined Noise Filtering Using Local Statistics,” Comput. Graphics Image Process. 15, 380 (1981).

1980 (1)

J. S. Lee, “Digital Image Enhancement and Noise Filtering by Use of Local Statistics,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-2, 165 (1980).

1978 (1)

H. T. Trussell, B. R. Hunt, “Sectioned Methods for Image Restoration,” IEEE Trans. Acoust., Speech Signal Process ASSP-26, 157 (1978).

1977 (2)

B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comp. C-26, 219 (1977).

G. S. Robinson, “Edge Detection by Compass Gradient Masks,” in Comput. Graphics Image Process. 6, 492 (1977).

1976 (1)

B. R. Hunt, T. M. Cannon, “Nonstationary Assumptions for Gaussian Models of Images,” IEEE Trans. System, Man Cybernet. 6, 876 (1976).

Cannon, T. M.

B. R. Hunt, T. M. Cannon, “Nonstationary Assumptions for Gaussian Models of Images,” IEEE Trans. System, Man Cybernet. 6, 876 (1976).

Chan, P. C.

P. C. Chan, J. S. Lim, “One-Dimensional Processing for Adaptive Image Restoration,” in Proceedings IEEE, ICASSP (San Diego, 1983), pp. 37.3.1–37.3.4.

Chavel, P.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise,” IEEE Trans. Pattern Anal. Machine Intell PAMI-7, 165 (1985).

Chellappa, R.

R. Chellappa, R. L. Kashyap, “Digital Image Restoration Using Spatial Interaction Models,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 461 (1982).

Hunt, B. R.

H. T. Trussell, B. R. Hunt, “Sectioned Methods for Image Restoration,” IEEE Trans. Acoust., Speech Signal Process ASSP-26, 157 (1978).

B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comp. C-26, 219 (1977).

B. R. Hunt, T. M. Cannon, “Nonstationary Assumptions for Gaussian Models of Images,” IEEE Trans. System, Man Cybernet. 6, 876 (1976).

Ingle, V. K.

V. K. Ingle, J. W. Woods, “Multiple Model Recursive Estimation of Images,” in Proceedings IEEE, ICASSP 79 (Washington, DC, Apr.1979), pp. 642–645.

Jiang, S-S.

S-S. Jiang, “Image Restoration and Speckle Suppression Using the Noise Updating Repeated Wiener Filter,” Ph.D. Thesis, U. Southern California, Los Angeles (1986).

Kak, A. C.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vols. 1 and 2 (Academic, New York, 1982).

Kashyap, R. L.

R. Chellappa, R. L. Kashyap, “Digital Image Restoration Using Spatial Interaction Models,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 461 (1982).

Kuan, D. T.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise,” IEEE Trans. Pattern Anal. Machine Intell PAMI-7, 165 (1985).

D. T. Kuan, “Nonstationary 2-D Recursive Restoration of Images with Signal-Dependent Noise with Application to Speckle Reduction,” Ph.D. Thesis, U. Southern California, Los Angeles (1982).

Lebedev, D. S.

D. S. Lebedev, L. I. Mirkin, “Digital Nonlinear Smoothing of Images,” Institute for Information Transmission Problems, Academy of Sciences, U.S.S.R. (1975), pp. 159–157.

Lee, J. S.

J. S. Lee, “Refined Noise Filtering Using Local Statistics,” Comput. Graphics Image Process. 15, 380 (1981).

J. S. Lee, “Digital Image Enhancement and Noise Filtering by Use of Local Statistics,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-2, 165 (1980).

Lim, J. S.

P. C. Chan, J. S. Lim, “One-Dimensional Processing for Adaptive Image Restoration,” in Proceedings IEEE, ICASSP (San Diego, 1983), pp. 37.3.1–37.3.4.

Melsa, J. L.

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

Mirkin, L. I.

D. S. Lebedev, L. I. Mirkin, “Digital Nonlinear Smoothing of Images,” Institute for Information Transmission Problems, Academy of Sciences, U.S.S.R. (1975), pp. 159–157.

Pratt, W. K.

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

Robinson, G. S.

G. S. Robinson, “Edge Detection by Compass Gradient Masks,” in Comput. Graphics Image Process. 6, 492 (1977).

Rosenfeld, A.

A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vols. 1 and 2 (Academic, New York, 1982).

Sage, A. P.

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

Sawchuk, A. A.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise,” IEEE Trans. Pattern Anal. Machine Intell PAMI-7, 165 (1985).

Strand, T. C.

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise,” IEEE Trans. Pattern Anal. Machine Intell PAMI-7, 165 (1985).

Trussell, H. T.

H. T. Trussell, B. R. Hunt, “Sectioned Methods for Image Restoration,” IEEE Trans. Acoust., Speech Signal Process ASSP-26, 157 (1978).

Woods, J. W.

V. K. Ingle, J. W. Woods, “Multiple Model Recursive Estimation of Images,” in Proceedings IEEE, ICASSP 79 (Washington, DC, Apr.1979), pp. 642–645.

Comput. Graphics Image Process. (2)

J. S. Lee, “Refined Noise Filtering Using Local Statistics,” Comput. Graphics Image Process. 15, 380 (1981).

G. S. Robinson, “Edge Detection by Compass Gradient Masks,” in Comput. Graphics Image Process. 6, 492 (1977).

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

R. Chellappa, R. L. Kashyap, “Digital Image Restoration Using Spatial Interaction Models,” IEEE Trans. Acoust. Speech Signal Process. ASSP-30, 461 (1982).

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

H. T. Trussell, B. R. Hunt, “Sectioned Methods for Image Restoration,” IEEE Trans. Acoust., Speech Signal Process ASSP-26, 157 (1978).

IEEE Trans. Comp. (1)

B. R. Hunt, “Bayesian Methods in Nonlinear Digital Image Restoration,” IEEE Trans. Comp. C-26, 219 (1977).

IEEE Trans. Pattern Anal. Machine Intell (1)

D. T. Kuan, A. A. Sawchuk, T. C. Strand, P. Chavel, “Adaptive Noise Smoothing Filter for Images with Signal-Dependent Noise,” IEEE Trans. Pattern Anal. Machine Intell PAMI-7, 165 (1985).

IEEE Trans. Pattern Anal. Machine Intell. (1)

J. S. Lee, “Digital Image Enhancement and Noise Filtering by Use of Local Statistics,” IEEE Trans. Pattern Anal. Machine Intell. PAMI-2, 165 (1980).

IEEE Trans. System, Man Cybernet. (1)

B. R. Hunt, T. M. Cannon, “Nonstationary Assumptions for Gaussian Models of Images,” IEEE Trans. System, Man Cybernet. 6, 876 (1976).

Other (8)

A. Rosenfeld, A. C. Kak, Digital Picture Processing, Vols. 1 and 2 (Academic, New York, 1982).

D. S. Lebedev, L. I. Mirkin, “Digital Nonlinear Smoothing of Images,” Institute for Information Transmission Problems, Academy of Sciences, U.S.S.R. (1975), pp. 159–157.

V. K. Ingle, J. W. Woods, “Multiple Model Recursive Estimation of Images,” in Proceedings IEEE, ICASSP 79 (Washington, DC, Apr.1979), pp. 642–645.

D. T. Kuan, “Nonstationary 2-D Recursive Restoration of Images with Signal-Dependent Noise with Application to Speckle Reduction,” Ph.D. Thesis, U. Southern California, Los Angeles (1982).

S-S. Jiang, “Image Restoration and Speckle Suppression Using the Noise Updating Repeated Wiener Filter,” Ph.D. Thesis, U. Southern California, Los Angeles (1986).

P. C. Chan, J. S. Lim, “One-Dimensional Processing for Adaptive Image Restoration,” in Proceedings IEEE, ICASSP (San Diego, 1983), pp. 37.3.1–37.3.4.

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

A. P. Sage, J. L. Melsa, Estimation Theory with Applications to Communications and Control (McGraw-Hill, New York, 1971).

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Figures (10)

Fig. 1
Fig. 1

NMNV image model justification.

Fig. 2
Fig. 2

Locations of the four quadrant variances.

Fig. 3
Fig. 3

Comparison between an averaging filter and a median filter.

Fig. 4
Fig. 4

LLMMSE filters for additive noise with different local variance estimations (I).

Fig. 5
Fig. 5

LLMMSE filters for additive noise with different local variance estimations (II).

Fig. 6
Fig. 6

Restoration of additive noise with σ n 2 = 750 .

Fig. 7
Fig. 7

Restoration of Poisson noise with λ = 0.1.

Fig. 8
Fig. 8

Comparison between Kuan’s and Lee’s LLMMSE filters for multiplicative noise with σ n = 0.28.

Fig. 9
Fig. 9

Restoration of multiplicative noise with σ n = 0.14.

Fig. 10
Fig. 10

Restoration of multiplicative noise with σ n = 0.28.

Tables (1)

Tables Icon

Table I Comparison of MSE Improvements

Equations (72)

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f ( i , j ) = E [ f ( i , j ) ] + [ σ f 2 ( i , j ) ] 1 / 2 n ( i , j ) ,
w ( i , j ) = A + B { f ( i , j ) - m f ( i , j ) [ v f ( i , j ) ] 1 / 2 } ,
s ( i , j ) = m f ( i , j ) + v f ( i , j ) n ( i , j ) ,
g ( i , j ) = α f ( i , j ) + u ( i , j ) ,
g = α f + u .
f ^ LMMSE = E [ f ] + C f g C g - 1 ( g - E [ g ] ) ,
f ^ ( i , j ) = E [ f ( i , j ) ] + α σ f 2 ( i , j ) σ g 2 ( i , j ) { g ( i , j ) - E [ g ( i , j ) ] } ,
f ^ LLMMSE ( i , j ) = m g ( i , j ) α + v g ( i , j ) α v g ( i , j ) [ g ( i , j ) - m g ( i , j ) ]
m g ( i , j ) = 1 ( 2 m + 1 ) ( 2 n + 1 ) k = i - m i + m l = j - n j + n g ( k , l ) .
v g ( i , j ) = 1 ( 2 m + 1 ) ( 2 n + 1 ) k = i - m i + m l = j - n j + n [ g ( k , l ) - m g ( i , j ) ] 2 .
v g ( i , j ) = max [ v u ( i , j ) , v s ( i , j ) ] ,
g ( i , j ) = f ( i , j ) + n ( i , j ) ,
f ^ ( i , j ) = m g ( i , j ) + v g ( i , j ) - σ n 2 v g ( i , j ) [ g ( i , j ) - m g ( i , j ) ] .
g ( i , j ) = f ( i , j ) v ( i , j ) ,
g ( i , j ) v ¯ f ( i , j ) + { f ¯ ( i , j ) [ v ( i , j ) - v ¯ ] } ,
f ^ ( i , j ) = m g ( i , j ) v ¯ + v ¯ v f ( i , j ) m g 2 ( i , j ) v ¯ 2 σ v 2 + v ¯ 2 v f ( i , j ) [ g ( i , j ) - m g ( i , j ) ] ,
v f ( i , j ) v g ( i , j ) + m g 2 ( i , j ) σ v 2 + v ¯ 2 - m g 2 ( i , j ) v ¯ 2 .
f ^ ( i , j ) = m g ( i , j ) + v g ( i , j ) - m g 2 ( i , j ) σ v 2 m g 2 ( i , j ) σ v 4 + v g ( i , j ) [ g ( i , j ) - m g ( i , j ) ] .
m g ( i , j ) = 1 ( 2 m + 1 ) ( 2 n + 1 ) k = i - m i + m l = j - n j + n g ( k , l ) .
v g ( i , j ) = 1 ( 2 m + 1 ) ( 2 n + 1 ) k = i - m i + m l = j - n j + n c ( i - k , j - l ) × [ g ( k , l ) - m g ( k , l ) ] 2 .
g ( i , j ) = f ( i , j ) v ( i , j ) / v ¯ ,
g ( i , j ) = f ( i , j ) + u ( i , j ) ,
u ( i , j ) f ( i , j ) [ v ( i , j ) v ¯ - 1 ] .
σ u 2 ( i , j ) = [ σ v 2 v ¯ 2 g ¯ ( i , j ) + σ g 2 ( i , j ) 1 + σ v 2 v ¯ 2 ] .
f ^ ( i , j ) = m g ( i , j ) + v f ( i , j ) v f ( i , j ) + σ v 2 v ¯ 2 [ m g 2 ( i , j ) + v f ( i , j ) ] × [ g ( i , j ) - m g ( i , j ) ] ,
v f ( i , j ) v g ( i , j ) - σ v 2 v ¯ 2 m g 2 ( i , j ) 1 + σ v 2 v ¯ 2 .
f ^ ( i , j ) = m g ( i , j ) + v g ( i , j ) - σ v 2 v ¯ 2 m g 2 ( i , j ) ( 1 + σ v 2 v ¯ 2 ) v g ( i , j ) [ g ( i , j ) - m g ( i , j ) ] .
g ( i , j ) = Poisson [ λ , f ( i , j ) ] ,
P r [ g ( i , j ) f ( i , j ) , λ ] = [ λ f ( i , j ) ] g ( i , j ) exp [ - λ f ( i , j ) ] g ( i , j ) ! .
g ( i , j ) = g ( i , j ) / λ .
g ( i , j ) = f ( i , j ) + u ( i , j ) ,
u ( i , j ) g ( i , j ) - f ( i , j )
E [ g ( i , j ) ] = E [ f ( i , j ) ] ,
E [ u ( i , j ) ] = 0 ,
var [ u ( i , j ) ] = E [ f ( i , j ) ] / λ ,
var [ g ( i , j ) ] = E [ f ( i , j ) ] / λ + var [ f ( i , j ) ] ,
E [ u ( i , j ) f ( i , j ) ] = E [ u ( i , j ) ] E [ f ( i , j ) ] = 0.
f ^ ( i , j ) = m g ( i , j ) + v g ( i , j ) - 1 λ m g ( i , j ) v g ( i , j ) [ g ( i , j ) - m g ( i , j ) ] .
m g ( i , j ) = 1 ( 2 m + 1 ) 2 k = - m m l = - m m g ( i + k , j + l ) .
v g ( i , j ) = min [ v g 1 ( i , j ) , v g 2 ( i , j ) , v g 3 ( i , j ) , v g 4 ( i , j ) ] ,
v g 1 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i + k , j - l ) - m g ( i , j ) ] 2 ,
v g 2 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i - k , j - l ) - m g ( i , j ) ] 2 ,
v g 3 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i - k , j + l ) - m g ( i , j ) ] 2 ,
v g 4 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i + k , j + l ) - m g ( i , j ) ] 2 .
v g 1 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i + k , j - l ) - m g ( i + k , j - l ) ] 2 ,
v g 2 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i - k , j - l ) - m g ( i - k , j - l ) ] 2 ,
v g 3 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i - k , j + l ) - m g ( i - k , j + l ) ] 2 ,
v g 4 ( i , j ) = 1 w 2 k = 0 w - 1 l = 0 w - 1 [ g ( i + k , j + l ) - m g ( i + k , j + l ) ] 2 .
m g ( i , j ) = 1 2 m + 1 k = - m m g ( i + k , j ) .
v g ( i , j ) = 1 2 m + 1 k = - m m [ g ( i + k , j ) - m g ( i , j ) ] 2 .
v u 1 ( i , j ) = k = i - m i + m h 1 2 ( k ; i , j ) v u ( k , j ) .
m g ( i , j ) = 1 ( 2 m + 1 ) 2 k = - m m l = - m m g ( i + k , j + l ) .
v g ( i , j ) = min [ v g 1 ( i , j ) , v g 2 ( i , j ) , v g 3 ( i , j ) , v g 4 ( i , j ) ] ,
v g 1 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i + k , j ) - m g ( i , j ) ] 2 ,
v g 2 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i + k , j - k ) - m g ( i , j ) ] 2 ,
v g 3 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i , j + k ) - m g ( i , j ) ] 2 ,
v g 4 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i + k , j + k ) - m g ( i , j ) ] 2 .
v g 1 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i + k , j ) - m g ( i + k , j ) ] 2 ,
v g 2 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i + k , j - k ) - m g ( i + k , j - k ) ] 2 ,
v g 3 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i , j + k ) - m g ( i , j + k ) ] 2 ,
v g 4 ( i , j ) = 1 2 w + 1 k = - w w [ g ( i + k , j + k ) - m g ( i + k , j + k ) ] 2 .
g ( i , j ) = f ( i , j ) + u ( i , j ) ,
m g ( i , j ) = 1 ( 2 m + 1 ) 2 k = - m m l = - m m g ( i + k , j + l ) ,
m g = L g ,
g 1 = m f + C f g C g - 1 ( g - m g ) = L g + ( C g - C u ) C g - 1 ( g - L g ) = ( I - C u C g - 1 - C u C g - 1 L ) g f 1 + u 1 ,
f 1 ( I - C u C g - 1 + C u C g - 1 L ) f ,
u 1 ( I - C u C g - 1 + C u C g - 1 L ) u .
u 1 ( i , j ) = [ 1 - v u ( i , j ) v g ( i , j ) ] u ( i , j ) + v u ( i , j ) v g ( i , j ) 1 ( 2 m + 1 ) 2 k = - m m l = - m m u ( i + k , j + l ) = [ 1 - v u ( i , j ) v g ( i , j ) + v u ( i , j ) v g ( i , j ) 1 ( 2 m + 1 ) 2 ] u ( i , j ) + [ v u ( i , j ) v g ( i , j ) 1 ( 2 m + 1 ) 2 ] k = - m m l = - m m ( k , l ) ( 0 , 0 ) u ( i + k , j + l ) .
v u 1 ( i , j ) = [ 1 - v u ( i , j ) v g ( i , j ) + v u ( i , j ) v g ( i , j ) 1 ( 2 m + 1 ) 2 ] 2 v u ( i , j ) + [ v u ( i , j ) v g ( i , j ) 1 ( 2 m + 1 ) 2 ] 2 k = - m m l = - m m ( k , l ) ( 0 , 0 ) v u ( i + k , j + l ) ,
var ( a X + b Y ) = a 2 var ( X ) + b 2 var ( Y ) ,
m g ( i , j ) = 1 2 m + 1 k = - m m g ( i + k , j ) .
v u 1 = [ 1 - v u ( i , j ) v g ( i , j ) + v u ( i , j ) v g ( i , j ) 1 2 m + 1 ] 2 v u ( i , j ) + [ v u ( i , j ) v g ( i , j ) 1 2 m + 1 ] 2 k = - m k 0 m v u ( i + k , j ) .

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