Abstract

This paper provides a novel Bayesian deringing method to reduce ringing artifacts caused by image interpolation and JPEG compression. To remove the ringing artifacts, the proposed method uses a Bayesian framework based on a SGLI (spatial-gradient-local-inhomogeneity) prior. The SGLI prior employs two complementary discontinuity measures: spatial gradient and local inhomogeniety. The spatial gradient measure effectively detects strong edge components in images. In addition, the local inhomogeniety measure successfully detects locations of the significant discontinuities by taking uniformity of small regions into consideration. The two complementary measures are elaborately combined to create prior probabilities of the Bayesian deringing framework. Thus, the proposed deringing method can effectively preserve the significant discontinuities such as textures of objects as well as the strong edge components in images while reducing the ringing artifacts. Experimental results show that the proposed deringing method achieves average PSNR gains of 0.09 dB in image interpolation artifact reduction and 0.21 dB in JPEG compression artifact reduction.

© 2010 OSA

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2009 (4)

B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express 17(10), 8567–8591 (2009).
[CrossRef] [PubMed]

H. S. Kim, C. Jung, S. Choi, S. Lee, and J. K. Kim, “A novel approach for Bayesian image Denoising using a SGLI Prior,” Lect. Notes Comput. Sci. 5879, 990–1011 (2009).

S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82(2), 205–229 (2009).
[CrossRef]

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

2008 (4)

Y. K. Park, S. L. Park, and J. K. Kim, “Retinex method based on adaptive smoothing for illumination invariant face recognition,” Signal Process. 88(8), 1929–1945 (2008).
[CrossRef]

S. Tan and L. Jiao, “A unified iterative denoising algorithm based on natural image statistical models: derivation and examples,” Opt. Express 16(2), 975–992 (2008).
[CrossRef] [PubMed]

J. M. Sanches, J. C. Nascimento, and J. S. Marques, “Medical image noise reduction using the Sylvester-Lyapunov equation,” IEEE Trans. Image Process. 17(9), 1522–1539 (2008).
[CrossRef] [PubMed]

K. T. Block, M. Uecker, and J. Frahm, “Suppression of MRI truncation artifacts using total variation constrained data extrapolation,” Int. J. Biomed. Imaging 2008, 184123 (2008).
[CrossRef] [PubMed]

2007 (2)

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[CrossRef] [PubMed]

D. Sun and W. K. Cham, “Postprocessing of low bit-rate block DCT coded images based on a fields of experts prior,” IEEE Trans. Image Process. 16(11), 2743–2751 (2007).
[CrossRef] [PubMed]

2006 (3)

T. A. Stephenson and T. Chen, “Adaptive Markov random fields for example-based super-resolution of faces,” EURASIP J. Appl. Signal Process. 2006, 1–12 (2006).

G. Wang, T. T. Wong, and P. A. Heng, “Deringing cartoons by image analogies,” ACM Trans. Graph. 25(4), 1360–1379 (2006).
[CrossRef]

J. D. Ouwerkerk, “Image super-resolution survey,” Image Vis. Comput. 24(10), 1039–1052 (2006).
[CrossRef]

2005 (2)

K. Lee, D. S. Kim, and T. Kim, “Regression-based prediction for blocking artifact reduction in JPEG-compressed images,” IEEE Trans. Image Process. 14(1), 36–48 (2005).
[CrossRef] [PubMed]

K. Chen, “Adaptive smoothing via contextual and local discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1552–1567 (2005).
[CrossRef] [PubMed]

2004 (1)

C. A. Segall, A. K. Katsaggelos, R. Molina, and J. Mateos, “Bayesian resolution enhancement of compressed video,” IEEE Trans. Image Process. 13(7), 898–911 (2004).
[CrossRef]

2003 (1)

F. Pan and L. Zhang, “New image super-resolution scheme based on residual error restoration by neural networks,” Opt. Eng. 42(10), 3038–3046 (2003).
[CrossRef]

2002 (1)

D. Rajan and S. Chaudhuri, “An MRF-based approach to generation of super-resolution images from blurred observations,” J. Math. Imaging Vis. 16(1), 5–15 (2002).
[CrossRef]

2001 (2)

S. Yang, Y. H. Hu, T. Q. Nguyen, and D. L. Tull, “Maximum-likelihood parameter estimation for image ringing-artifact removal,” IEEE Trans. Circ. Syst. Video Tech. 11(8), 963–973 (2001).
[CrossRef]

A. B. Hamza and H. Krim, “A variational approach to maximum a posteriori estimation for image denoising,” Lect. Notes Comput. Sci. 2134, 19–34 (2001).
[CrossRef]

1986 (1)

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. [Ser A] 48, 259–302 (1986).

1984 (1)

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6(6), 721–741 (1984).
[CrossRef]

Besag, J.

J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. [Ser A] 48, 259–302 (1986).

Black, M. J.

S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82(2), 205–229 (2009).
[CrossRef]

Block, K. T.

K. T. Block, M. Uecker, and J. Frahm, “Suppression of MRI truncation artifacts using total variation constrained data extrapolation,” Int. J. Biomed. Imaging 2008, 184123 (2008).
[CrossRef] [PubMed]

Cham, W. K.

D. Sun and W. K. Cham, “Postprocessing of low bit-rate block DCT coded images based on a fields of experts prior,” IEEE Trans. Image Process. 16(11), 2743–2751 (2007).
[CrossRef] [PubMed]

Chaudhuri, S.

D. Rajan and S. Chaudhuri, “An MRF-based approach to generation of super-resolution images from blurred observations,” J. Math. Imaging Vis. 16(1), 5–15 (2002).
[CrossRef]

Chen, K.

K. Chen, “Adaptive smoothing via contextual and local discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1552–1567 (2005).
[CrossRef] [PubMed]

Chen, T.

T. A. Stephenson and T. Chen, “Adaptive Markov random fields for example-based super-resolution of faces,” EURASIP J. Appl. Signal Process. 2006, 1–12 (2006).

Choi, S.

H. S. Kim, C. Jung, S. Choi, S. Lee, and J. K. Kim, “A novel approach for Bayesian image Denoising using a SGLI Prior,” Lect. Notes Comput. Sci. 5879, 990–1011 (2009).

Egiazarian, K.

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[CrossRef] [PubMed]

Foi, A.

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[CrossRef] [PubMed]

Frahm, J.

K. T. Block, M. Uecker, and J. Frahm, “Suppression of MRI truncation artifacts using total variation constrained data extrapolation,” Int. J. Biomed. Imaging 2008, 184123 (2008).
[CrossRef] [PubMed]

Geman, D.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6(6), 721–741 (1984).
[CrossRef]

Geman, S.

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6(6), 721–741 (1984).
[CrossRef]

Hamza, A. B.

A. B. Hamza and H. Krim, “A variational approach to maximum a posteriori estimation for image denoising,” Lect. Notes Comput. Sci. 2134, 19–34 (2001).
[CrossRef]

Heng, P. A.

G. Wang, T. T. Wong, and P. A. Heng, “Deringing cartoons by image analogies,” ACM Trans. Graph. 25(4), 1360–1379 (2006).
[CrossRef]

Hu, Y. H.

S. Yang, Y. H. Hu, T. Q. Nguyen, and D. L. Tull, “Maximum-likelihood parameter estimation for image ringing-artifact removal,” IEEE Trans. Circ. Syst. Video Tech. 11(8), 963–973 (2001).
[CrossRef]

Hur, N.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Jiao, L.

Jung, C.

H. S. Kim, C. Jung, S. Choi, S. Lee, and J. K. Kim, “A novel approach for Bayesian image Denoising using a SGLI Prior,” Lect. Notes Comput. Sci. 5879, 990–1011 (2009).

Jung, K.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Katkovnik, V.

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[CrossRef] [PubMed]

Katsaggelos, A. K.

C. A. Segall, A. K. Katsaggelos, R. Molina, and J. Mateos, “Bayesian resolution enhancement of compressed video,” IEEE Trans. Image Process. 13(7), 898–911 (2004).
[CrossRef]

Kim, D. S.

K. Lee, D. S. Kim, and T. Kim, “Regression-based prediction for blocking artifact reduction in JPEG-compressed images,” IEEE Trans. Image Process. 14(1), 36–48 (2005).
[CrossRef] [PubMed]

Kim, H. S.

H. S. Kim, C. Jung, S. Choi, S. Lee, and J. K. Kim, “A novel approach for Bayesian image Denoising using a SGLI Prior,” Lect. Notes Comput. Sci. 5879, 990–1011 (2009).

Kim, J.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Kim, J. K.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

H. S. Kim, C. Jung, S. Choi, S. Lee, and J. K. Kim, “A novel approach for Bayesian image Denoising using a SGLI Prior,” Lect. Notes Comput. Sci. 5879, 990–1011 (2009).

Y. K. Park, S. L. Park, and J. K. Kim, “Retinex method based on adaptive smoothing for illumination invariant face recognition,” Signal Process. 88(8), 1929–1945 (2008).
[CrossRef]

Kim, T.

K. Lee, D. S. Kim, and T. Kim, “Regression-based prediction for blocking artifact reduction in JPEG-compressed images,” IEEE Trans. Image Process. 14(1), 36–48 (2005).
[CrossRef] [PubMed]

Krim, H.

A. B. Hamza and H. Krim, “A variational approach to maximum a posteriori estimation for image denoising,” Lect. Notes Comput. Sci. 2134, 19–34 (2001).
[CrossRef]

Lee, G.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Lee, H.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Lee, K.

K. Lee, D. S. Kim, and T. Kim, “Regression-based prediction for blocking artifact reduction in JPEG-compressed images,” IEEE Trans. Image Process. 14(1), 36–48 (2005).
[CrossRef] [PubMed]

Lee, S.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

H. S. Kim, C. Jung, S. Choi, S. Lee, and J. K. Kim, “A novel approach for Bayesian image Denoising using a SGLI Prior,” Lect. Notes Comput. Sci. 5879, 990–1011 (2009).

Marone, F.

Marques, J. S.

J. M. Sanches, J. C. Nascimento, and J. S. Marques, “Medical image noise reduction using the Sylvester-Lyapunov equation,” IEEE Trans. Image Process. 17(9), 1522–1539 (2008).
[CrossRef] [PubMed]

Mateos, J.

C. A. Segall, A. K. Katsaggelos, R. Molina, and J. Mateos, “Bayesian resolution enhancement of compressed video,” IEEE Trans. Image Process. 13(7), 898–911 (2004).
[CrossRef]

Molina, R.

C. A. Segall, A. K. Katsaggelos, R. Molina, and J. Mateos, “Bayesian resolution enhancement of compressed video,” IEEE Trans. Image Process. 13(7), 898–911 (2004).
[CrossRef]

Münch, B.

Nascimento, J. C.

J. M. Sanches, J. C. Nascimento, and J. S. Marques, “Medical image noise reduction using the Sylvester-Lyapunov equation,” IEEE Trans. Image Process. 17(9), 1522–1539 (2008).
[CrossRef] [PubMed]

Nguyen, T. Q.

S. Yang, Y. H. Hu, T. Q. Nguyen, and D. L. Tull, “Maximum-likelihood parameter estimation for image ringing-artifact removal,” IEEE Trans. Circ. Syst. Video Tech. 11(8), 963–973 (2001).
[CrossRef]

Oh, Y.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Ouwerkerk, J. D.

J. D. Ouwerkerk, “Image super-resolution survey,” Image Vis. Comput. 24(10), 1039–1052 (2006).
[CrossRef]

Pan, F.

F. Pan and L. Zhang, “New image super-resolution scheme based on residual error restoration by neural networks,” Opt. Eng. 42(10), 3038–3046 (2003).
[CrossRef]

Park, S. L.

Y. K. Park, S. L. Park, and J. K. Kim, “Retinex method based on adaptive smoothing for illumination invariant face recognition,” Signal Process. 88(8), 1929–1945 (2008).
[CrossRef]

Park, Y. K.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Y. K. Park, S. L. Park, and J. K. Kim, “Retinex method based on adaptive smoothing for illumination invariant face recognition,” Signal Process. 88(8), 1929–1945 (2008).
[CrossRef]

Rajan, D.

D. Rajan and S. Chaudhuri, “An MRF-based approach to generation of super-resolution images from blurred observations,” J. Math. Imaging Vis. 16(1), 5–15 (2002).
[CrossRef]

Roth, S.

S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82(2), 205–229 (2009).
[CrossRef]

Sanches, J. M.

J. M. Sanches, J. C. Nascimento, and J. S. Marques, “Medical image noise reduction using the Sylvester-Lyapunov equation,” IEEE Trans. Image Process. 17(9), 1522–1539 (2008).
[CrossRef] [PubMed]

Segall, C. A.

C. A. Segall, A. K. Katsaggelos, R. Molina, and J. Mateos, “Bayesian resolution enhancement of compressed video,” IEEE Trans. Image Process. 13(7), 898–911 (2004).
[CrossRef]

Stampanoni, M.

Stephenson, T. A.

T. A. Stephenson and T. Chen, “Adaptive Markov random fields for example-based super-resolution of faces,” EURASIP J. Appl. Signal Process. 2006, 1–12 (2006).

Sun, D.

D. Sun and W. K. Cham, “Postprocessing of low bit-rate block DCT coded images based on a fields of experts prior,” IEEE Trans. Image Process. 16(11), 2743–2751 (2007).
[CrossRef] [PubMed]

Tan, S.

Trtik, P.

Tull, D. L.

S. Yang, Y. H. Hu, T. Q. Nguyen, and D. L. Tull, “Maximum-likelihood parameter estimation for image ringing-artifact removal,” IEEE Trans. Circ. Syst. Video Tech. 11(8), 963–973 (2001).
[CrossRef]

Uecker, M.

K. T. Block, M. Uecker, and J. Frahm, “Suppression of MRI truncation artifacts using total variation constrained data extrapolation,” Int. J. Biomed. Imaging 2008, 184123 (2008).
[CrossRef] [PubMed]

Wang, G.

G. Wang, T. T. Wong, and P. A. Heng, “Deringing cartoons by image analogies,” ACM Trans. Graph. 25(4), 1360–1379 (2006).
[CrossRef]

Wong, T. T.

G. Wang, T. T. Wong, and P. A. Heng, “Deringing cartoons by image analogies,” ACM Trans. Graph. 25(4), 1360–1379 (2006).
[CrossRef]

Yang, S.

S. Yang, Y. H. Hu, T. Q. Nguyen, and D. L. Tull, “Maximum-likelihood parameter estimation for image ringing-artifact removal,” IEEE Trans. Circ. Syst. Video Tech. 11(8), 963–973 (2001).
[CrossRef]

Yun, K.

Y. K. Park, K. Jung, Y. Oh, S. Lee, J. K. Kim, G. Lee, H. Lee, K. Yun, N. Hur, and J. Kim, “Depth-image-based rendering for 3DTV service over T-DMB,” Signal Process. Image Commun. 24(1-2), 122–136 (2009).
[CrossRef]

Zhang, L.

F. Pan and L. Zhang, “New image super-resolution scheme based on residual error restoration by neural networks,” Opt. Eng. 42(10), 3038–3046 (2003).
[CrossRef]

ACM Trans. Graph. (1)

G. Wang, T. T. Wong, and P. A. Heng, “Deringing cartoons by image analogies,” ACM Trans. Graph. 25(4), 1360–1379 (2006).
[CrossRef]

EURASIP J. Appl. Signal Process. (1)

T. A. Stephenson and T. Chen, “Adaptive Markov random fields for example-based super-resolution of faces,” EURASIP J. Appl. Signal Process. 2006, 1–12 (2006).

IEEE Trans. Circ. Syst. Video Tech. (1)

S. Yang, Y. H. Hu, T. Q. Nguyen, and D. L. Tull, “Maximum-likelihood parameter estimation for image ringing-artifact removal,” IEEE Trans. Circ. Syst. Video Tech. 11(8), 963–973 (2001).
[CrossRef]

IEEE Trans. Image Process. (5)

A. Foi, V. Katkovnik, and K. Egiazarian, “Pointwise shape-adaptive DCT for high-quality denoising and deblocking of grayscale and color images,” IEEE Trans. Image Process. 16(5), 1395–1411 (2007).
[CrossRef] [PubMed]

K. Lee, D. S. Kim, and T. Kim, “Regression-based prediction for blocking artifact reduction in JPEG-compressed images,” IEEE Trans. Image Process. 14(1), 36–48 (2005).
[CrossRef] [PubMed]

C. A. Segall, A. K. Katsaggelos, R. Molina, and J. Mateos, “Bayesian resolution enhancement of compressed video,” IEEE Trans. Image Process. 13(7), 898–911 (2004).
[CrossRef]

J. M. Sanches, J. C. Nascimento, and J. S. Marques, “Medical image noise reduction using the Sylvester-Lyapunov equation,” IEEE Trans. Image Process. 17(9), 1522–1539 (2008).
[CrossRef] [PubMed]

D. Sun and W. K. Cham, “Postprocessing of low bit-rate block DCT coded images based on a fields of experts prior,” IEEE Trans. Image Process. 16(11), 2743–2751 (2007).
[CrossRef] [PubMed]

IEEE Trans. Pattern Anal. Mach. Intell. (2)

K. Chen, “Adaptive smoothing via contextual and local discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1552–1567 (2005).
[CrossRef] [PubMed]

S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. PAMI-6(6), 721–741 (1984).
[CrossRef]

Image Vis. Comput. (1)

J. D. Ouwerkerk, “Image super-resolution survey,” Image Vis. Comput. 24(10), 1039–1052 (2006).
[CrossRef]

Int. J. Biomed. Imaging (1)

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Figures (7)

Fig. 1
Fig. 1

Examples of ringing artifacts. (a) The original Cameraman image. (b) Bicubic interpolated image (the interpolation factor is 4). (c) JPEG compressed image (compression rate is 56.6 Kbps).

Fig. 2
Fig. 2

Evolution of energy UT (t) verse iteration number for the Cameraman image.

Fig. 3
Fig. 3

Test images. (a) Lena, (b) Cameraman, (c) House, (d) Woman, (e) Man, and (f) Airfield.

Fig. 4
Fig. 4

Bicubic interpolated images (the interpolation factor is 4). (a) Lena, (b) Cameraman, (c) House, (d) Woman, (e) Man, and (f) Airfield.

Fig. 5
Fig. 5

Reduction results of the ringing artifacts obtained with the proposed method. (a) Lena, (b) Cameraman, (c) House, (d) Woman, (e) Man, and (f) Airfield.

Fig. 6
Fig. 6

JPEG compressed images (The compression rate is 56.6 Kbps). (a) Lena, (b) Cameraman, (c) House, (d) Woman, (e) Man, and (f) Airfield.

Fig. 7
Fig. 7

Reduction results of the ringing artifacts obtained with the proposed method. (a) Lena, (b) Cameraman, (c) House, (d) Woman, (e) Man, and (f) Airfield.

Tables (2)

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Table 1 Performance evaluation results from test images using the proposed and conventional methods a

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Table 2 Performance evaluation results from test images using the proposed and conventional methods b

Equations (17)

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X ( i , j ) = [ G x , G y ]
G x = X ( i + 1 , j ) X ( i 1 , j )
G y = X ( i , j + 1 ) X ( i , j 1 )
I X ( i , j ) = G x 2 + G y 2
L ( i , j ) = ( m , n ) Ω | X ( i , j ) X ( m , n ) | | Ω | ( i m , j n )
L ^ = L ( i , j ) L min L max L min
L ˜ ( i , j ) = sin ( π 2 L ^ ( i , j ) ) , 0 L ^ ( i , j ) 1
U ( X ) = γ 1 | X ( i , j ) | + γ 2 L ˜ ( i , j )
X = arg max X { log p ( Y | X ) + log p ( X ) }
p ( X ) = 1 Q exp { U ( X ) λ }
p ( Y | X ) = K exp { | X Y | 2 2 σ 2 }
X * = arg min X { | X Y | 2 2 σ 2 + α U ( X ) }
U T ( i , j ) = 1 2 σ 2 [ X ( i , j ) Y ( i , j ) ] 2 + α U ( X )
Ψ ( t ) = x M y N | U T ( t ) ( x , y ) U T ( t 1 ) ( x , y ) | M N
M S E = i = 0 M j = 0 N ( X ( i , j ) X * ( i , j ) ) 2 M N
S N R = 10 log 10 i , j | X * | 2 / M N M S E
P S N R = 10 log 10 255 2 M S E

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