Abstract

Many proposed image watermarking techniques are sensitive to geometric distortions, such as rotation, scaling and translation. Geometric distortions, even by slight amount, can make watermark decoder disable. In this paper, a geometric invariant blind image watermarking is designed by utilizing the invariant Tchebichef moments. The detailed construction of invariant Tchebichef moments is described. Watermark is generated independent to the original image and inserted into the perceptually significant invariant Tchebichef moments of original image. The watermark decoder extracts watermark blindly utilizing Independent Component Analysis (ICA). The computational aspects of the proposed watermarking are also discussed in detail. Experimental results have demonstrated that the proposed watermarking technique is robust against geometric distortions and other attacks performed by popular watermark benchmark-Stirmark, such as filtering, image compression, additive noise, random geometric distortions.

© 2007 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. S. Voloshynovskiy, S. Pereira, and T. Pun, University of Geneva. J.J. Eggers and J.K. Su, “Attacks on digital watermarks: classification, estimation-based attacks and benchmarks,” IEEE Communications Magazine 8, 2∼10 (2001)
  2. Choong-Hoon Lee and Heung-Kyu Lee, “Geometric attack resistant watermarking in wavelet transform domain,” OPTICS EXPRESS 13, 1307–1321 (2005)
    [Crossref] [PubMed]
  3. M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Transactions on Image Processing 13, 145–153 (2004)
    [Crossref] [PubMed]
  4. Y. Xin, S. Liao, and M. Pawlak, “Geometrically robust image watermarking via pseudo-Zernike moments,” IEEE Canadian Conference on Electrical and Computer Engineering 2, 939–942 (2004)
  5. Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition IV, 861–864 (2004)
  6. S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transations on Image Processing 9, 1123–1129 (2000)
    [Crossref]
  7. M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lecture Notes in Computer Science 1728, 238–250 (1999)
  8. P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” IEEE Int. Conf. Image Pro.480–492 (2002)
  9. J. O’Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66, 303–317 (1998)
    [Crossref]
  10. H. S. Kim and H. K, Lee. “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits and Systems for Vid. 13, 766–775 (2003)
    [Crossref]
  11. M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Amer. 70, 920–930 (1980)
    [Crossref]
  12. Chao Kan and Mandyam D. Srinath, “Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments,” Pattern Recognition 35, 143–154 (2002)
    [Crossref]
  13. R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans, Image Processing 10, 1357–1364 (2001)
    [Crossref]
  14. R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Transactions on Image Processing 13, 1055–1059 (2004)
    [Crossref] [PubMed]
  15. M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187 (1962)
  16. Hyvarinen A, and Oja E., “Independent component analysis: A tutorial. Notes for International Joint Conference on Neural Networks (IJCNN’99),” Washington D. C. , http://www.cis.hut.fi/projects/ica/ (1999)
  17. S. Pereira and T Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transactions on Image Processing 9, 1123∼1129 (2000)
    [Crossref]
  18. D. Yu, F. Sattar, and K.K. Ma, “Watermark detection and extraction using independent component analysis method,” EURASIP Journal on Applied Signal Processing 1, 92–104 (2002)

2005 (1)

Choong-Hoon Lee and Heung-Kyu Lee, “Geometric attack resistant watermarking in wavelet transform domain,” OPTICS EXPRESS 13, 1307–1321 (2005)
[Crossref] [PubMed]

2004 (4)

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Transactions on Image Processing 13, 145–153 (2004)
[Crossref] [PubMed]

Y. Xin, S. Liao, and M. Pawlak, “Geometrically robust image watermarking via pseudo-Zernike moments,” IEEE Canadian Conference on Electrical and Computer Engineering 2, 939–942 (2004)

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition IV, 861–864 (2004)

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Transactions on Image Processing 13, 1055–1059 (2004)
[Crossref] [PubMed]

2003 (1)

H. S. Kim and H. K, Lee. “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits and Systems for Vid. 13, 766–775 (2003)
[Crossref]

2002 (3)

P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” IEEE Int. Conf. Image Pro.480–492 (2002)

Chao Kan and Mandyam D. Srinath, “Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments,” Pattern Recognition 35, 143–154 (2002)
[Crossref]

D. Yu, F. Sattar, and K.K. Ma, “Watermark detection and extraction using independent component analysis method,” EURASIP Journal on Applied Signal Processing 1, 92–104 (2002)

2001 (2)

S. Voloshynovskiy, S. Pereira, and T. Pun, University of Geneva. J.J. Eggers and J.K. Su, “Attacks on digital watermarks: classification, estimation-based attacks and benchmarks,” IEEE Communications Magazine 8, 2∼10 (2001)

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans, Image Processing 10, 1357–1364 (2001)
[Crossref]

2000 (2)

S. Pereira and T Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transactions on Image Processing 9, 1123∼1129 (2000)
[Crossref]

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transations on Image Processing 9, 1123–1129 (2000)
[Crossref]

1999 (1)

M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lecture Notes in Computer Science 1728, 238–250 (1999)

1998 (1)

J. O’Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66, 303–317 (1998)
[Crossref]

1980 (1)

M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Amer. 70, 920–930 (1980)
[Crossref]

1962 (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187 (1962)

A,, Hyvarinen

Hyvarinen A, and Oja E., “Independent component analysis: A tutorial. Notes for International Joint Conference on Neural Networks (IJCNN’99),” Washington D. C. , http://www.cis.hut.fi/projects/ica/ (1999)

Alghoniemy, M.

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Transactions on Image Processing 13, 145–153 (2004)
[Crossref] [PubMed]

D. Srinath, Mandyam

Chao Kan and Mandyam D. Srinath, “Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments,” Pattern Recognition 35, 143–154 (2002)
[Crossref]

Dong, P.

P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” IEEE Int. Conf. Image Pro.480–492 (2002)

E., Oja

Hyvarinen A, and Oja E., “Independent component analysis: A tutorial. Notes for International Joint Conference on Neural Networks (IJCNN’99),” Washington D. C. , http://www.cis.hut.fi/projects/ica/ (1999)

Galasanos, N. P.

P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” IEEE Int. Conf. Image Pro.480–492 (2002)

Hu, M. K.

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187 (1962)

Kan, Chao

Chao Kan and Mandyam D. Srinath, “Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments,” Pattern Recognition 35, 143–154 (2002)
[Crossref]

Kim, H. S.

H. S. Kim and H. K, Lee. “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits and Systems for Vid. 13, 766–775 (2003)
[Crossref]

Kutter, M.

M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lecture Notes in Computer Science 1728, 238–250 (1999)

Lee, Choong-Hoon

Choong-Hoon Lee and Heung-Kyu Lee, “Geometric attack resistant watermarking in wavelet transform domain,” OPTICS EXPRESS 13, 1307–1321 (2005)
[Crossref] [PubMed]

Lee, H. K,

H. S. Kim and H. K, Lee. “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits and Systems for Vid. 13, 766–775 (2003)
[Crossref]

Lee, Heung-Kyu

Choong-Hoon Lee and Heung-Kyu Lee, “Geometric attack resistant watermarking in wavelet transform domain,” OPTICS EXPRESS 13, 1307–1321 (2005)
[Crossref] [PubMed]

Lee, P. A.

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans, Image Processing 10, 1357–1364 (2001)
[Crossref]

Liao, S.

Y. Xin, S. Liao, and M. Pawlak, “Geometrically robust image watermarking via pseudo-Zernike moments,” IEEE Canadian Conference on Electrical and Computer Engineering 2, 939–942 (2004)

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition IV, 861–864 (2004)

Ma, K.K.

D. Yu, F. Sattar, and K.K. Ma, “Watermark detection and extraction using independent component analysis method,” EURASIP Journal on Applied Signal Processing 1, 92–104 (2002)

Mukundan, R.

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Transactions on Image Processing 13, 1055–1059 (2004)
[Crossref] [PubMed]

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans, Image Processing 10, 1357–1364 (2001)
[Crossref]

O’Ruanaidh, J.

J. O’Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66, 303–317 (1998)
[Crossref]

Ong, S. H.

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans, Image Processing 10, 1357–1364 (2001)
[Crossref]

Pawlak, M.

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition IV, 861–864 (2004)

Y. Xin, S. Liao, and M. Pawlak, “Geometrically robust image watermarking via pseudo-Zernike moments,” IEEE Canadian Conference on Electrical and Computer Engineering 2, 939–942 (2004)

Pereira, S.

S. Voloshynovskiy, S. Pereira, and T. Pun, University of Geneva. J.J. Eggers and J.K. Su, “Attacks on digital watermarks: classification, estimation-based attacks and benchmarks,” IEEE Communications Magazine 8, 2∼10 (2001)

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transations on Image Processing 9, 1123–1129 (2000)
[Crossref]

S. Pereira and T Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transactions on Image Processing 9, 1123∼1129 (2000)
[Crossref]

Pun, T

S. Pereira and T Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transactions on Image Processing 9, 1123∼1129 (2000)
[Crossref]

Pun, T.

S. Voloshynovskiy, S. Pereira, and T. Pun, University of Geneva. J.J. Eggers and J.K. Su, “Attacks on digital watermarks: classification, estimation-based attacks and benchmarks,” IEEE Communications Magazine 8, 2∼10 (2001)

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transations on Image Processing 9, 1123–1129 (2000)
[Crossref]

J. O’Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66, 303–317 (1998)
[Crossref]

Sattar, F.

D. Yu, F. Sattar, and K.K. Ma, “Watermark detection and extraction using independent component analysis method,” EURASIP Journal on Applied Signal Processing 1, 92–104 (2002)

Teague, M. R.

M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Amer. 70, 920–930 (1980)
[Crossref]

Tewfik, A. H.

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Transactions on Image Processing 13, 145–153 (2004)
[Crossref] [PubMed]

Voloshynovskiy, S.

S. Voloshynovskiy, S. Pereira, and T. Pun, University of Geneva. J.J. Eggers and J.K. Su, “Attacks on digital watermarks: classification, estimation-based attacks and benchmarks,” IEEE Communications Magazine 8, 2∼10 (2001)

Xin, Y.

Y. Xin, S. Liao, and M. Pawlak, “Geometrically robust image watermarking via pseudo-Zernike moments,” IEEE Canadian Conference on Electrical and Computer Engineering 2, 939–942 (2004)

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition IV, 861–864 (2004)

Yu, D.

D. Yu, F. Sattar, and K.K. Ma, “Watermark detection and extraction using independent component analysis method,” EURASIP Journal on Applied Signal Processing 1, 92–104 (2002)

EURASIP Journal on Applied Signal Processing (1)

D. Yu, F. Sattar, and K.K. Ma, “Watermark detection and extraction using independent component analysis method,” EURASIP Journal on Applied Signal Processing 1, 92–104 (2002)

IEEE Canadian Conference on Electrical and Computer Engineering (1)

Y. Xin, S. Liao, and M. Pawlak, “Geometrically robust image watermarking via pseudo-Zernike moments,” IEEE Canadian Conference on Electrical and Computer Engineering 2, 939–942 (2004)

IEEE Communications Magazine (1)

S. Voloshynovskiy, S. Pereira, and T. Pun, University of Geneva. J.J. Eggers and J.K. Su, “Attacks on digital watermarks: classification, estimation-based attacks and benchmarks,” IEEE Communications Magazine 8, 2∼10 (2001)

IEEE Int. Conf. Image Pro. (1)

P. Dong and N. P. Galasanos, “Affine transform resistant watermarking based on image normalization,” IEEE Int. Conf. Image Pro.480–492 (2002)

IEEE Trans, Image Processing (1)

R. Mukundan, S. H. Ong, and P. A. Lee, “Image analysis by Tchebichef moments,” IEEE Trans, Image Processing 10, 1357–1364 (2001)
[Crossref]

IEEE Trans. Circuits and Systems for Vid. (1)

H. S. Kim and H. K, Lee. “Invariant image watermark using Zernike moments,” IEEE Trans. Circuits and Systems for Vid. 13, 766–775 (2003)
[Crossref]

IEEE Transactions on Image Processing (3)

M. Alghoniemy and A. H. Tewfik, “Geometric invariants in image watermarking,” IEEE Transactions on Image Processing 13, 145–153 (2004)
[Crossref] [PubMed]

S. Pereira and T Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transactions on Image Processing 9, 1123∼1129 (2000)
[Crossref]

R. Mukundan, “Some computational aspects of discrete orthonormal moments,” IEEE Transactions on Image Processing 13, 1055–1059 (2004)
[Crossref] [PubMed]

IEEE Transations on Image Processing (1)

S. Pereira and T. Pun, “Robust template matching for affine resistant image watermarks,” IEEE Transations on Image Processing 9, 1123–1129 (2000)
[Crossref]

International Conference on Pattern Recognition (1)

Y. Xin, S. Liao, and M. Pawlak, “A multibit geometrically robust image watermark based on Zernike moments,” International Conference on Pattern Recognition IV, 861–864 (2004)

IRE Trans. Information Theory (1)

M. K. Hu, “Visual pattern recognition by moment invariants,” IRE Trans. Information Theory IT-8, 179–187 (1962)

J. Opt. Soc. Amer. (1)

M. R. Teague, “Image analysis via the general theory of moments,” J. Opt. Soc. Amer. 70, 920–930 (1980)
[Crossref]

Lecture Notes in Computer Science (1)

M. Kutter, “Performance improvement of spread spectrum based image watermarking schemes through M-ary modulation,” Lecture Notes in Computer Science 1728, 238–250 (1999)

OPTICS EXPRESS (1)

Choong-Hoon Lee and Heung-Kyu Lee, “Geometric attack resistant watermarking in wavelet transform domain,” OPTICS EXPRESS 13, 1307–1321 (2005)
[Crossref] [PubMed]

Pattern Recognition (1)

Chao Kan and Mandyam D. Srinath, “Invariant character recognition with Zernike and orthogonal Fourier-Mellin moments,” Pattern Recognition 35, 143–154 (2002)
[Crossref]

Signal Processing (1)

J. O’Ruanaidh and T. Pun, “Rotation, scale and translation invariant spread spectrum digital image watermarking,” Signal Processing 66, 303–317 (1998)
[Crossref]

Other (1)

Hyvarinen A, and Oja E., “Independent component analysis: A tutorial. Notes for International Joint Conference on Neural Networks (IJCNN’99),” Washington D. C. , http://www.cis.hut.fi/projects/ica/ (1999)

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 (4)

Fig.  1.
Fig. 1.

Original image, watermark and watermarked image. (a)original image (b)watermark (c)permuted watermark (d)watermarked image

Fig. 2.
Fig. 2.

Experimental results of robust against geometric distortions. (a)NC due to scaling attack (b) NC due to rotation attacks

Fig. 3.
Fig. 3.

Experimental results of robustness against filtering. (a)NC due to low pass filtering (b)NC due to median filtering

Fig. 4.
Fig. 4.

NC comparison (scaling and rotation).

Tables (4)

Tables Icon

Table 1. Robust to additive noise

Tables Icon

Table 2. Robust to JPEG compression produced by Stirmark

Tables Icon

Table 3. Robustness against attacks produced by Stirmark

Tables Icon

Table 4. Experimental results compared to [17] and two commercial watermarking

Equations (35)

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

T pq = 1 ρ p N ρ q N x = 0 N 1 y = 0 N 1 t p ( x ) t p ( x ) f x y , p , q = 0,1,2 , , N 1
t n ( x ) = n ! k = 0 n ( 1 ) n k ( N 1 k n k ) ( n + k n ) ( x k )
f x y = p = 0 N 1 q = 0 N 1 T pq t p ( x ) t q ( y )
t ˜ n ( x ) = t n ( x ) β n N = k = 0 n c k , n , N x k
t ˜ 0 x N = 1
t ˜ 1 x N = 1 N N + 2 N x
t ˜ 2 x N = N 2 3 N + 2 N 2 + 6 ( 1 N ) N 2 x + 6 N 2 x 2
t ˜ 3 x N = N 3 + 6 N 2 11 N + 6 N 3 + 12 N 2 30 N + 22 N 3 x + 30 ( 1 N ) N 3 x 2 + 20 N 3 x 3
c 0 , 0 , N = 1
c 0,1 , N = 1 N N c 1,1 , N = 2 N
c 0,2 , N = N 2 3 N + 2 N 2 c 1,2 . N = 6 ( 1 N ) N 2 c 2,2 , N = 6 N 2
c 0,3 , N = N 3 + 6 N 2 11 N + 6 N 3 c 1,3 , N = 12 N 2 30 N + 22 N 3 c 2,3 , N = 30 ( 1 N ) N 3 c 3,3 , N = 20 N N 3
T pq = 1 ρ ˜ p N ρ ˜ q N x = 0 N 1 y = 0 N 1 ( i = 0 p c i , p , N x i ) ( j = 0 q c j , q , N x j ) f ( x , y )
m pq = x = 0 N 1 y = 0 N 1 x p y q f x y
T pq = 1 ρ ˜ p N ρ ˜ q N i = 0 p j = 0 q c j , q , N c i , p , N m i , j
T 00 = m 00 N 2 T 10 = 6 m 10 + 3 ( 1 N ) m 00 N ( N 2 1 ) T 01 = 6 m 01 + 3 ( 1 N ) m 00 N ( N 2 1 )
T 20 = 30 m 20 + 30 ( 1 N ) m 10 + 5 ( 1 N ) ( 2 N ) m 00 ( N 2 1 ) ( N 2 2 ) T 02 = 30 m 02 + 30 ( 1 N ) m 01 + 5 ( 1 N ) ( 2 N ) m 00 ( N 2 1 ) ( N 2 2 )
T 11 = 36 m 11 + 18 ( 1 N ) ( m 10 + m 01 ) + 9 ( 1 N 2 ) m 00 ( N 2 1 ) 2
μ nm = ( x x ¯ ) n ( x y ¯ ) m f ( x , y ) dxdy
g x + N 2 y + N 2 = f x ¯ + x cos θ y sin θ α y ¯ + y cos θ x sin θ α
m ̅ nm = x = 0 N 1 x = 0 N 1 α 2 f x y { α [ ( x x ¯ ) cos θ + ( y y ¯ ) sin θ ] + N 2 } n × { α [ ( y y ¯ ) cos θ ( x x ¯ ) sin θ ] + N 2 } m
T ̅ pq = 1 ρ ˜ p N ρ ˜ q N i = 0 p j = 0 q c j , q , N c i , P , N m ¯ i , j
T ¯ pq = x = 0 N 1 y = 0 N 1 N 2 2 m 00 f x y { [ ( x x ¯ ) cos θ + ( y y ¯ ) sin θ ] N 2 2 m 00 + N 2 } p { [ ( y y ¯ ) cos θ ( x x ¯ ) sin θ ] N 2 2 m 00 + N 2 } q
T ̂ m i , n i = T om i , n i + a m i n i T wm i , n i
x j = a j 1 s 1 + a j 2 s 2 + + a jn s n for all j = 1 . . n
t ˜ n ( N 1 x ) = ( 1 ) n t ˜ n ( x )
T pq = 1 ρ ˜ p N ρ ˜ q N x = 0 N 2 1 y = 0 N 2 1 t ˜ p ( x ) t ˜ q ( y ) { f x y + ( 1 ) p f ( N 1 x , y ) + ( 1 ) q f ( x , N 1 y ) + ( 1 ) p + q f ( N 1 x , N 1 y ) }
f x y = { m = 0 N 1 n = 0 N 1 T mn t ˜ m ( x ) t ˜ n ( y ) x , y < ( N 2 ) m = 0 N 1 n = 0 N 1 ( 1 ) m T mn t ˜ m ( N 1 x ) t ˜ n ( y ) y < ( N 2 ) ; x ( N 2 ) m = 0 N 1 n = 0 N 1 ( 1 ) m T mn t ˜ m ( x ) t ˜ n ( N 1 y ) x < ( N 2 ) ; y ( N 2 ) m = 0 N 1 n = 0 N 1 ( 1 ) m + n T mn t ˜ m ( N 1 x ) t ˜ n ( N 1 y ) x , y ( N 2 )
T = CFC T
F = C T TC
PSNR = 10 log ( f ( x , y ) max 2 / ( N 2 x = 0 N 1 y = 0 N 1 ( f ( x , y ) z ( x , y ) ) 2 ) )
NC ( w , w ) = i w ( i ) w ´ ( i ) / ( i w 2 ( i ) i w 2 ( i ) )
f ̂ ( x , y ) = f x y + v ( x , y )
p ( V = v ) = 1 2 π σ exp ( ( v μ ´ ) 2 2 σ 2 )
v x y = λ f x y

Metrics