Abstract

Generally, conventional transform (DWT and DFT, etc.) -based watermarking techniques provide only one spectrum plane for embedding the watermark, thus the embedding watermark information can be easily removed. To solve this problem, we propose an efficient cellular automata (CA) based watermarking method that CA transform (CAT) with various gateway values can provide many transform planes for watermark embedding according to various CA rules. In this paper, multiple ownership watermarks are first recorded in the form of an elemental image array (EIA), simultaneously, and then the recorded EIA as the watermark data is embedded into the CAT coefficient. An additional advantage of this proposed method is that EIA is composed of many elemental images and each elemental image has its own property of watermarks. Even though most data of elemental images are lost, the watermarks can be reconstructed from the remaining elemental images successfully. Experimental results show that the proposed technique provides good image quality and is robust in varying degree to some image processing attacks.

© 2015 Optical Society of America

Full Article  |  PDF Article
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References

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2014 (4)

X. Li, S. Cho, and S. Kim, “Combined use of BP neural network and computational integral imaging reconstruction for optical multiple-image security,” Opt. Commun. 315, 147–158 (2014).
[Crossref]

X. Li, S. Cho, and S. Kim, “High security and robust optical image encryption approach based on computer-generated integral imaging pickup and iterative back-projection techniques,” Opt. Lasers Eng. 55, 162–182 (2014).
[Crossref]

Y. Oh, D. H. Shin, B. G. Lee, S. I. Jeong, and H. J. Choi, “Resolution-enhanced integral imaging in focal mode with a time-multiplexed electrical mask array,” Opt. Express 22(15), 17620–17629 (2014).
[Crossref] [PubMed]

X. Li, S. Cho, and S. Kim, “Computational integral imaging-based 3D digital watermarking scheme using cellular automata transform and maximum length cellular automata,” Multidim. Syst. Sign. P. 25(3), 405–424 (2014).
[Crossref]

2013 (5)

2012 (1)

2011 (2)

C. C. Lai, “A digital watermarking scheme based on singular value decomposition and tiny genetic algorithm,” Digit. Signal Process. 21(4), 522–527 (2011).
[Crossref]

L. Cui and W. Li, “Adaptive multiwavelet-based watermarking through JPW masking,” IEEE Trans. Image Process. 20(4), 1047–1060 (2011).
[Crossref] [PubMed]

2010 (2)

B. Surekha, G. Swamy, and K. S. Rao, “A multiple watermarking technique for images based on visual cryptography,” Comput. Appl. Eng. Educ. 1, 77–81 (2010).

B. Lee, H. H. Kang, and E. S. Kim, “Occlusion removal method of partially occluded object using variance in computational integral imaging,” 3D Res. 1, 6–10 (2010).

2009 (1)

C. Li, S. Li, G. Chen, and W. Halang, “Cryptanalysis of an image encryption scheme based on a compound chaotic sequence,” Image Vis. Comput. 27(8), 1035–1039 (2009).
[Crossref]

2008 (3)

2007 (1)

2006 (3)

B. Javidi, R. Ponce-Díaz, and S. H. Hong, “Three-dimensional recognition of occluded objects by using computational integral imaging,” Opt. Lett. 31(8), 1106–1108 (2006).
[Crossref] [PubMed]

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

X. W. Wang and H. Zhao, “A novel synchronization invariant audio watermarking scheme based on DWT and DCT,” IEEE T. Signal Process. 54(12), 4835–4840 (2006).

2005 (2)

P. Bao and X. Ma, “Image adaptive watermarking using wavelet domain singular value decomposition,” IEEE T. Circ. Syst. Vid. 15(1), 96–102 (2005).
[Crossref]

D. H. Shin, E. S. Kim, and B. Lee, “Computational reconstruction of three-dimensional objects in integral imaging using lenslet array,” Jpn. J. Appl. Phys. 44(11), 8016–8018 (2005).
[Crossref]

2004 (2)

S. H. Hong, J. S. Jang, and B. Javidi, “Three-dimensional volumetric object reconstruction using computational integral imaging,” Opt. Express 12(3), 483–491 (2004).
[Crossref] [PubMed]

D. P. Mukherjee, S. Maitra, and S. T. Acton, “Spatial domain digital watermarking of multimedia objects for buyer authentication,” IEEE Trans. Multimed. 6(1), 1–15 (2004).
[Crossref]

2003 (2)

X. Kang, J. Huang, Y. Shi, and Y. Lin, “A DWT-DFT composite watermarking scheme robust to both affine transform and JPEG compression,” IEEE T. Circ. Syst. Vid. 13(8), 776–786 (2003).
[Crossref]

A. Stern and B. Javidi, “3-D computational synthetic aperture integral imaging (COMPSAII),” Opt. Express 11(19), 2446–2451 (2003).
[Crossref] [PubMed]

2002 (2)

Y. Frauel and B. Javidi, “Digital three-dimensional image correlation by use of computer-reconstructed integral imaging,” Appl. Opt. 41(26), 5488–5496 (2002).
[Crossref] [PubMed]

R. Liu and T. Tan, “An SVD-based watermarking scheme for protecting rightful ownership,” IEEE Trans. Multimed. 4(1), 121–128 (2002).
[Crossref]

Acton, S. T.

D. P. Mukherjee, S. Maitra, and S. T. Acton, “Spatial domain digital watermarking of multimedia objects for buyer authentication,” IEEE Trans. Multimed. 6(1), 1–15 (2004).
[Crossref]

Ailing, T.

Atrey, P. K.

G. Bhatnagar, Q. M. Wu, and P. K. Atrey, “Secure randomized image watermarking based on singular value decomposition,” ACM T. Multim. Comput. 10, 41–421 (2013).

Bao, P.

P. Bao and X. Ma, “Image adaptive watermarking using wavelet domain singular value decomposition,” IEEE T. Circ. Syst. Vid. 15(1), 96–102 (2005).
[Crossref]

Baskurt, A.

K. Wang, G. Lavoué, F. Denis, and A. Baskurt, “Hierarchical watermarking of semiregular meshes based on wavelet transform,” IEEE T. Inf. Foren. Sec. 3(4), 620–634 (2008).
[Crossref]

Bhatnagar, G.

G. Bhatnagar, Q. M. Wu, and P. K. Atrey, “Secure randomized image watermarking based on singular value decomposition,” ACM T. Multim. Comput. 10, 41–421 (2013).

Chen, G.

C. Li, S. Li, G. Chen, and W. Halang, “Cryptanalysis of an image encryption scheme based on a compound chaotic sequence,” Image Vis. Comput. 27(8), 1035–1039 (2009).
[Crossref]

Cho, S.

X. Li, S. Cho, and S. Kim, “Combined use of BP neural network and computational integral imaging reconstruction for optical multiple-image security,” Opt. Commun. 315, 147–158 (2014).
[Crossref]

X. Li, S. Cho, and S. Kim, “High security and robust optical image encryption approach based on computer-generated integral imaging pickup and iterative back-projection techniques,” Opt. Lasers Eng. 55, 162–182 (2014).
[Crossref]

X. Li, S. Cho, and S. Kim, “Computational integral imaging-based 3D digital watermarking scheme using cellular automata transform and maximum length cellular automata,” Multidim. Syst. Sign. P. 25(3), 405–424 (2014).
[Crossref]

Choi, H. J.

Choi, J. H.

Cui, L.

L. Cui and W. Li, “Adaptive multiwavelet-based watermarking through JPW masking,” IEEE Trans. Image Process. 20(4), 1047–1060 (2011).
[Crossref] [PubMed]

Denis, F.

K. Wang, G. Lavoué, F. Denis, and A. Baskurt, “Hierarchical watermarking of semiregular meshes based on wavelet transform,” IEEE T. Inf. Foren. Sec. 3(4), 620–634 (2008).
[Crossref]

Erdenebat, M. U.

Frauel, Y.

Halang, W.

C. Li, S. Li, G. Chen, and W. Halang, “Cryptanalysis of an image encryption scheme based on a compound chaotic sequence,” Image Vis. Comput. 27(8), 1035–1039 (2009).
[Crossref]

Hong, S. H.

Huang, J.

X. Kang, J. Huang, Y. Shi, and Y. Lin, “A DWT-DFT composite watermarking scheme robust to both affine transform and JPEG compression,” IEEE T. Circ. Syst. Vid. 13(8), 776–786 (2003).
[Crossref]

Hwang, D. C.

Jang, J. S.

Javidi, B.

Jeong, J. S.

Jeong, S. I.

Kang, H. H.

B. Lee, H. H. Kang, and E. S. Kim, “Occlusion removal method of partially occluded object using variance in computational integral imaging,” 3D Res. 1, 6–10 (2010).

Kang, X.

X. Kang, J. Huang, Y. Shi, and Y. Lin, “A DWT-DFT composite watermarking scheme robust to both affine transform and JPEG compression,” IEEE T. Circ. Syst. Vid. 13(8), 776–786 (2003).
[Crossref]

Kim, E. S.

B. Lee, H. H. Kang, and E. S. Kim, “Occlusion removal method of partially occluded object using variance in computational integral imaging,” 3D Res. 1, 6–10 (2010).

D. C. Hwang, D. H. Shin, S. C. Kim, and E. S. Kim, “Depth extraction of three-dimensional objects in space by the computational integral imaging reconstruction technique,” Appl. Opt. 47(19), D128–D135 (2008).
[Crossref] [PubMed]

D. H. Shin, E. S. Kim, and B. Lee, “Computational reconstruction of three-dimensional objects in integral imaging using lenslet array,” Jpn. J. Appl. Phys. 44(11), 8016–8018 (2005).
[Crossref]

Kim, N.

Kim, S.

X. Li, S. Cho, and S. Kim, “High security and robust optical image encryption approach based on computer-generated integral imaging pickup and iterative back-projection techniques,” Opt. Lasers Eng. 55, 162–182 (2014).
[Crossref]

X. Li, S. Cho, and S. Kim, “Combined use of BP neural network and computational integral imaging reconstruction for optical multiple-image security,” Opt. Commun. 315, 147–158 (2014).
[Crossref]

X. Li, S. Cho, and S. Kim, “Computational integral imaging-based 3D digital watermarking scheme using cellular automata transform and maximum length cellular automata,” Multidim. Syst. Sign. P. 25(3), 405–424 (2014).
[Crossref]

Kim, S. C.

Kwon, K. C.

Lai, C. C.

C. C. Lai, “A digital watermarking scheme based on singular value decomposition and tiny genetic algorithm,” Digit. Signal Process. 21(4), 522–527 (2011).
[Crossref]

Lavoué, G.

K. Wang, G. Lavoué, F. Denis, and A. Baskurt, “Hierarchical watermarking of semiregular meshes based on wavelet transform,” IEEE T. Inf. Foren. Sec. 3(4), 620–634 (2008).
[Crossref]

Lee, B.

B. Lee, H. H. Kang, and E. S. Kim, “Occlusion removal method of partially occluded object using variance in computational integral imaging,” 3D Res. 1, 6–10 (2010).

D. H. Shin, E. S. Kim, and B. Lee, “Computational reconstruction of three-dimensional objects in integral imaging using lenslet array,” Jpn. J. Appl. Phys. 44(11), 8016–8018 (2005).
[Crossref]

Lee, B. G.

Li, C.

C. Li, S. Li, G. Chen, and W. Halang, “Cryptanalysis of an image encryption scheme based on a compound chaotic sequence,” Image Vis. Comput. 27(8), 1035–1039 (2009).
[Crossref]

Li, S.

C. Li, S. Li, G. Chen, and W. Halang, “Cryptanalysis of an image encryption scheme based on a compound chaotic sequence,” Image Vis. Comput. 27(8), 1035–1039 (2009).
[Crossref]

Li, W.

L. Cui and W. Li, “Adaptive multiwavelet-based watermarking through JPW masking,” IEEE Trans. Image Process. 20(4), 1047–1060 (2011).
[Crossref] [PubMed]

Li, X.

X. Li, S. Cho, and S. Kim, “Combined use of BP neural network and computational integral imaging reconstruction for optical multiple-image security,” Opt. Commun. 315, 147–158 (2014).
[Crossref]

X. Li, S. Cho, and S. Kim, “High security and robust optical image encryption approach based on computer-generated integral imaging pickup and iterative back-projection techniques,” Opt. Lasers Eng. 55, 162–182 (2014).
[Crossref]

X. Li, S. Cho, and S. Kim, “Computational integral imaging-based 3D digital watermarking scheme using cellular automata transform and maximum length cellular automata,” Multidim. Syst. Sign. P. 25(3), 405–424 (2014).
[Crossref]

Liansheng, S.

Lim, Y. T.

Lin, Y.

X. Kang, J. Huang, Y. Shi, and Y. Lin, “A DWT-DFT composite watermarking scheme robust to both affine transform and JPEG compression,” IEEE T. Circ. Syst. Vid. 13(8), 776–786 (2003).
[Crossref]

Liu, R.

R. Liu and T. Tan, “An SVD-based watermarking scheme for protecting rightful ownership,” IEEE Trans. Multimed. 4(1), 121–128 (2002).
[Crossref]

Ma, X.

P. Bao and X. Ma, “Image adaptive watermarking using wavelet domain singular value decomposition,” IEEE T. Circ. Syst. Vid. 15(1), 96–102 (2005).
[Crossref]

Maitra, S.

D. P. Mukherjee, S. Maitra, and S. T. Acton, “Spatial domain digital watermarking of multimedia objects for buyer authentication,” IEEE Trans. Multimed. 6(1), 1–15 (2004).
[Crossref]

Martinez-Corral, B. M.

Meiting, X.

Mukherjee, D. P.

D. P. Mukherjee, S. Maitra, and S. T. Acton, “Spatial domain digital watermarking of multimedia objects for buyer authentication,” IEEE Trans. Multimed. 6(1), 1–15 (2004).
[Crossref]

Oh, Y.

Park, C.

Park, J. H.

Ponce-Díaz, R.

Rao, K. S.

B. Surekha, G. Swamy, and K. S. Rao, “A multiple watermarking technique for images based on visual cryptography,” Comput. Appl. Eng. Educ. 1, 77–81 (2010).

Shi, Y.

X. Kang, J. Huang, Y. Shi, and Y. Lin, “A DWT-DFT composite watermarking scheme robust to both affine transform and JPEG compression,” IEEE T. Circ. Syst. Vid. 13(8), 776–786 (2003).
[Crossref]

Shin, D. H.

Stern, A.

Surekha, B.

B. Surekha, G. Swamy, and K. S. Rao, “A multiple watermarking technique for images based on visual cryptography,” Comput. Appl. Eng. Educ. 1, 77–81 (2010).

Swamy, G.

B. Surekha, G. Swamy, and K. S. Rao, “A multiple watermarking technique for images based on visual cryptography,” Comput. Appl. Eng. Educ. 1, 77–81 (2010).

Tan, T.

R. Liu and T. Tan, “An SVD-based watermarking scheme for protecting rightful ownership,” IEEE Trans. Multimed. 4(1), 121–128 (2002).
[Crossref]

Telatar, Z.

E. Yavuz and Z. Telatar, “Comments on “A digital watermarking scheme based on singular value decomposition and tiny genetic algorithm,”,” Digit. Signal Process. 23(4), 1335–1336 (2013).
[Crossref]

Wang, K.

K. Wang, G. Lavoué, F. Denis, and A. Baskurt, “Hierarchical watermarking of semiregular meshes based on wavelet transform,” IEEE T. Inf. Foren. Sec. 3(4), 620–634 (2008).
[Crossref]

Wang, X. W.

X. W. Wang and H. Zhao, “A novel synchronization invariant audio watermarking scheme based on DWT and DCT,” IEEE T. Signal Process. 54(12), 4835–4840 (2006).

Wu, Q. M.

G. Bhatnagar, Q. M. Wu, and P. K. Atrey, “Secure randomized image watermarking based on singular value decomposition,” ACM T. Multim. Comput. 10, 41–421 (2013).

Xiao, X.

Yavuz, E.

E. Yavuz and Z. Telatar, “Comments on “A digital watermarking scheme based on singular value decomposition and tiny genetic algorithm,”,” Digit. Signal Process. 23(4), 1335–1336 (2013).
[Crossref]

Yoo, H.

Yoo, K. H.

Zhao, H.

X. W. Wang and H. Zhao, “A novel synchronization invariant audio watermarking scheme based on DWT and DCT,” IEEE T. Signal Process. 54(12), 4835–4840 (2006).

3D Res. (1)

B. Lee, H. H. Kang, and E. S. Kim, “Occlusion removal method of partially occluded object using variance in computational integral imaging,” 3D Res. 1, 6–10 (2010).

ACM T. Multim. Comput. (1)

G. Bhatnagar, Q. M. Wu, and P. K. Atrey, “Secure randomized image watermarking based on singular value decomposition,” ACM T. Multim. Comput. 10, 41–421 (2013).

Appl. Opt. (3)

Comput. Appl. Eng. Educ. (1)

B. Surekha, G. Swamy, and K. S. Rao, “A multiple watermarking technique for images based on visual cryptography,” Comput. Appl. Eng. Educ. 1, 77–81 (2010).

Digit. Signal Process. (2)

E. Yavuz and Z. Telatar, “Comments on “A digital watermarking scheme based on singular value decomposition and tiny genetic algorithm,”,” Digit. Signal Process. 23(4), 1335–1336 (2013).
[Crossref]

C. C. Lai, “A digital watermarking scheme based on singular value decomposition and tiny genetic algorithm,” Digit. Signal Process. 21(4), 522–527 (2011).
[Crossref]

IEEE T. Circ. Syst. Vid. (2)

P. Bao and X. Ma, “Image adaptive watermarking using wavelet domain singular value decomposition,” IEEE T. Circ. Syst. Vid. 15(1), 96–102 (2005).
[Crossref]

X. Kang, J. Huang, Y. Shi, and Y. Lin, “A DWT-DFT composite watermarking scheme robust to both affine transform and JPEG compression,” IEEE T. Circ. Syst. Vid. 13(8), 776–786 (2003).
[Crossref]

IEEE T. Inf. Foren. Sec. (1)

K. Wang, G. Lavoué, F. Denis, and A. Baskurt, “Hierarchical watermarking of semiregular meshes based on wavelet transform,” IEEE T. Inf. Foren. Sec. 3(4), 620–634 (2008).
[Crossref]

IEEE T. Signal Process. (1)

X. W. Wang and H. Zhao, “A novel synchronization invariant audio watermarking scheme based on DWT and DCT,” IEEE T. Signal Process. 54(12), 4835–4840 (2006).

IEEE Trans. Image Process. (1)

L. Cui and W. Li, “Adaptive multiwavelet-based watermarking through JPW masking,” IEEE Trans. Image Process. 20(4), 1047–1060 (2011).
[Crossref] [PubMed]

IEEE Trans. Multimed. (2)

D. P. Mukherjee, S. Maitra, and S. T. Acton, “Spatial domain digital watermarking of multimedia objects for buyer authentication,” IEEE Trans. Multimed. 6(1), 1–15 (2004).
[Crossref]

R. Liu and T. Tan, “An SVD-based watermarking scheme for protecting rightful ownership,” IEEE Trans. Multimed. 4(1), 121–128 (2002).
[Crossref]

Image Vis. Comput. (1)

C. Li, S. Li, G. Chen, and W. Halang, “Cryptanalysis of an image encryption scheme based on a compound chaotic sequence,” Image Vis. Comput. 27(8), 1035–1039 (2009).
[Crossref]

Jpn. J. Appl. Phys. (1)

D. H. Shin, E. S. Kim, and B. Lee, “Computational reconstruction of three-dimensional objects in integral imaging using lenslet array,” Jpn. J. Appl. Phys. 44(11), 8016–8018 (2005).
[Crossref]

Multidim. Syst. Sign. P. (1)

X. Li, S. Cho, and S. Kim, “Computational integral imaging-based 3D digital watermarking scheme using cellular automata transform and maximum length cellular automata,” Multidim. Syst. Sign. P. 25(3), 405–424 (2014).
[Crossref]

Opt. Commun. (1)

X. Li, S. Cho, and S. Kim, “Combined use of BP neural network and computational integral imaging reconstruction for optical multiple-image security,” Opt. Commun. 315, 147–158 (2014).
[Crossref]

Opt. Express (7)

Opt. Lasers Eng. (1)

X. Li, S. Cho, and S. Kim, “High security and robust optical image encryption approach based on computer-generated integral imaging pickup and iterative back-projection techniques,” Opt. Lasers Eng. 55, 162–182 (2014).
[Crossref]

Opt. Lett. (2)

Proc. IEEE (1)

A. Stern and B. Javidi, “Three-dimensional image sensing, visualization, and processing using integral imaging,” Proc. IEEE 94(3), 591–607 (2006).
[Crossref]

Other (3)

O. Lafe, Cellular Automata Transforms: Theory and Applications in Multimedia Compression, Encryption and Modeling, (Kluwer Academic Pub., 2000).

M. Radulescu, F. Ionescu, and R. Dogaru, “Improving multimedia techniques for watermarking color images using cellular automata,” in IEEE International Symposium on Signals, Circuits and Systems (IEEE, 2007) pp. 1–4 (2007).
[Crossref]

R. Shiba, S. Kang, and Y. Aoki, “An image watermarking technique using cellular automata transform,” in IEEE Region 10 Conference (IEEE 2004), pp. 303–306.
[Crossref]

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Figures (9)

Fig. 1
Fig. 1 Principle of the multiple watermarks pickup and reconstruction: (a) pickup process, (b) reconstruction process.
Fig. 2
Fig. 2 Plane images reconstruction process from an EIA with the occlusion attack.
Fig. 3
Fig. 3 Block diagram of the proposed watermark embedding process.
Fig. 4
Fig. 4 (a) and (b) original images ‘Malan flower’ and ‘Baboon’, (c) and (d) watermarked images based on the proposed method.
Fig. 5
Fig. 5 Simulation results: (a) 2D basis functions Aijkl (8 × 8 × 64 cells): A00kl is the block at the extreme upper left corner. The top row represents 0 j < 8 , i = 0. The left column is j = 0, 0 i < 8 . Aij00 is the upper left corner of each block. The white rectangular dots represent ‘1’ (addition) while the black ones are ‘-1’ (subtraction); (b) level-1 CAT coefficient of the test image ‘Malan flower’; (c) energy distribution of four classes of CAT coefficient; (d) recorded EIA from the multiple watermarks; (e) scrambled watermark; (f) extracted EIA from the watermarked image ‘Malan flower’; (g)-(i) reconstructed multiple watermarks by using the CIIR technique.
Fig. 6
Fig. 6 Statistical analysis: (a)-(c) autocorrelation of the multiple watermarks, (d) autocorrelation of the picked-up EIA, (e) autocorrelation of the scrambled EIA.
Fig. 7
Fig. 7 Proposed extracted multiple watermarks against attacks: (a) speckle (variance: V = 0.6), (b) sharpening (alpha: α = 0.5), (c) blur noise (radius R = 2) (d) salt & pepper (density: D = 0.2).
Fig. 8
Fig. 8 Proposed extracted multiple watermarks against some image processing attacks: (a) Gaussian noise (V = 0.5), (b) median filter (3 × 3), (c) occlusion attack (size S = 30%), (d) JPEG compression attack (Q = 0.3).
Fig. 9
Fig. 9 Extracted multiple watermarks based on the conventional watermarking method: (a) Gaussian noise (V = 0.5), (b) median filter (3 × 3), (c) occlusion attack (occlusion size S = 30%), (d) JPEG compression attack (Q = 0.3).

Tables (2)

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Table 1 The multiple watermarks embedding process.

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Table 2 The multiple watermarks extraction process

Equations (13)

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E i j ( x , y ) = O ( x g z k + i ρ , y g z k + j ρ )
R k ( x , y ) = i = 0 M 1 j = 0 N 1 E ( x z k g + i ρ , y z k g + j ρ ) i j
a ( p ) ( t + 1 ) = ( j = 0 2 m 2 j a j + 2 m 1 ) 2 m mod K
a 1 ( t + 1 ) = ( 0 a 0 t + 1 a 1 t + 2 a 2 t + 3 a 0 t a 1 t + 4 a 1 t a 2 t + 5 a 2 t a 0 t + 6 a 0 t a 1 t a 2 t + 7 ) 8 mod ( 2 )
7 = x 0 ; 6 = x 1 7 ; 5 = x 2 7 ; 4 = x 3 5 6 7 ; 3 = x 4 7 ; 2 = x 5 5 6 7 ; 1 = x 6 3 5 7 ; 0 = x 7 1 1 3 4 5 6 7 ;
ξ i k = α + β a i k
ξ i k = α + β a i k a k i
ξ i j k l = ξ i k ξ j l
f i j = k = 0 N 1 l = 0 N 1 σ k l ξ i j k l
σ k l = i = 0 N 1 j = 0 N 1 f i j ξ ' i j k l
A = U S V T = i = 1 r s i U i V i T
P S N R ( O , O ' ) = 10 log 10 ( 255 2 M S E ( O , O ' ) )
M S E ( O , O ' ) = 1 M N x = 0 M 1 y = 0 N 1 [ O ( x , y ) O ' ( x , y ) ] 2

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