## Novel Bayesian deringing method in image interpolation and compression using a SGLI prior

Optics Express, Vol. 18, Issue 7, pp. 7138-7149 (2010)

http://dx.doi.org/10.1364/OE.18.007138

Acrobat PDF (707 KB)

### 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

## 1. Introduction

5. 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]

13. 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]

5. 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]

14. 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]

22. 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]

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

24. 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]

## 2. Methods

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

24. 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]

*X*(

*i*,

*j*) at a pixel (

*i*,

*j*) is defined as the first partial derivatives of its image intensity with respect to coordinates

*x*and

*y*:where

*G*and

_{x}*G*represent the horizontal and vertical first partial derivatives, respectively.

_{y}*G*and

_{x}*G*are expressed as: Then, the magnitude of the gradient vector in Eq. (1) is expressed as:The local inhomogeneity is another measure of discontinuity to show the degree of uniformity/dis-uniformity between the center pixel and its neighboring pixels. The average of intensity difference between the center pixel (

_{y}*i, j*) and its neighboring pixels can be expressed as:where Ω represents a local neighborhood of the pixel (

*i, j*), and (

*m, n*) indicates the locations of pixels in the neighborhood Ω. Here, we only consider the 3x3 neighborhood of the pixel (

*i, j*). Then,

*L*(

*i, j*) is normalized as follows:where

*L*

_{max}and

*L*

_{min}represent the maximum and minimum value of

*L*(

*i, j*) in the entire image, respectively. To emphasize the higher value of

*L*(

*i, j*), a nonlinear transformation is applied as follows.By combining the two discontinuity measures into a prior value, the prior energy

*U*(

*X*) is defined as follows.where the regularization parameters

*γ*

_{1}and

*γ*

_{2}control the influence of the two values.

*U*(

*X*). Let

*Y*be an

*N × M*observed image corrupted by ringing artifacts from an unknown image

*X*, the optimal solution

*X** is determined by the maximum a posteriori (MAP) estimation as follows [15

15. 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]

*p*(

*X*) and

*p*(

*Y|X*) denote the prior distribution for the unknown image

*X*and the conditional probability of

*Y*given

*X*respectively. In addition, a general model for the prior distribution

*p*(

*X*) is a Markov random field (MRF) which is characterized by its Gibbs distribution given bywhere

*Q*is the partition function,

*λ*is a constant known as the temperature in the terminology of physical systems, and

*U*(

*X*) is energy function of

*X*[14

14. 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]

*λ*,

*p*(

*X*) becomes flat, and for small

*λ*,

*p*(

*X*) has sharp modes. Consequently, the probability function is converted into energy function by Eq. (10).

26. 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]

*K*is a normalizing positive constant and

*σ*

^{2}is the noise variance. If

*α*is

*λ*

^{−1}, the MAP estimation in Eq. (9) can be expressed as:As a result, total energy

*U*using the SGLI prior in the Bayesian framework can be expressed as:Energy of each pixel can be computed using Eq. (13), and thus we can reconstruct the original image through energy optimization techniques. In our method, the termination of iteration is determined automatically based on the energy difference between iteration

_{T}*t*and

*t*-1 as follows:where

*M*and

*N*represent the height and width of the estimated image, respectively [27

27. 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]

*U*

_{T}^{(}

^{t}^{)}(

*x,y*) denotes the total energy of the image at iteration

*t*. Figure 2 shows the evolution of

*U*

_{T}^{(}

*for the*

^{t)}*Cameraman*image as the iteration proceeds. As can be seen, energy function is monotonically decreasing as the number of iterations increases. In addition, the observed corrupted image gets smoothed as the iteration proceeds. We find that after a relatively small number of iterations,

*U*

_{T}^{(}

*changes slightly in value from various images. Thus, we determine the optimal termination time of iterations from the energy difference*

^{t)}*Ψ*(t). Our method is iterated until

*Ψ*(t) is lower than 8% of the first energy

*U*

_{T}^{(0)}.

## 3. Results

*Lena*,

*Cameraman*,

*Man*,

*Woman*,

*Airfield*, and

*House*, whose sizes are 256x256 pixels as shown in Fig. 3 . The LR images were generated by low-pass filtering and down-sampling the HR images. The down-sampling factor was 4 and the down-sampled images were interpolated to the same size of the original HR image. The weight

*α*of Eq. (12) was initially set as 0<

*α*≤0.3. The optimal weight was selected through exhausting experiments. The optimal weight was 0.2 in the interpolation artifact reduction and 0.03 in the JPEG compression artifact reduction. The regularization parameters were chosen heuristically and the values of them were

*γ*

_{1}=

*γ*

_{2}= 1.0.

*X*is the uncorrupted original image and

*X**is either the estimated image or the observed image corrupted by noise or artifacts. In addition,

*MSE*,

*SNR*, and

*PSNR*are mean squared error, signal to noise ratio, and peak signal to noise ratio, respectively.

### 3.1 Performance evaluation in the interpolation artifact reduction

*Cameraman*image, the ringing artifacts in non-edge regions at object boundaries are removed effectively.

*MSE*,

*SNR*, and

*PSNR*values are measured over the 6 test images. Table 1 lists performance evaluation results of our proposed method compared with the bicubic interpolation, bilateral filtering [28], adaptive smoothing [24

24. 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]

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

13. 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]

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

22. 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]

*Cameraman*,

*House*, and

*Airfield*. Although the FoE method produces the best evaluation results in the other three images, the proposed method performs better than any other methods in average performance. Our method achieves an average PSNR gain of 0.09 dB as compared to the bicubic interpolation method. The results show that our method reduces the interpolation ringing artifacts efficiently and improve picture quality successfully.

### 3.2 Performance evaluation in the JPEG compression artifact reduction

*MSE*,

*SNR*, and

*PSNR*values are also measured over the 6 test images. Table 2 lists performance evaluation results of the proposed method compared with JPEG compression, bilateral filtering [28], adaptive smoothing [24

**88**(8), 1929–1945 (2008). [CrossRef]

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

13. 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]

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

## 4. Conclusion

## Acknowledgements

## References and links

1. | J. D. Ouwerkerk, “Image super-resolution survey,” Image Vis. Comput. |

2. | J. Sun, J. Sun, Z. Xu, and H. Y. Shum, “Image super-resolution using gradient prior,” in: Proceedings of IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8. |

3. | F. Pan and L. Zhang, “New image super-resolution scheme based on residual error restoration by neural networks,” Opt. Eng. |

4. | J. S. Chitode, Digital Signal Processing (Technical Publications, Pune-India, 2008). |

5. | 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. |

6. | Y. Wu, O. C. Au, E. Luo, D. Tu, and L. Yeung, “A novel deringing method based on MAP image restoration,” in: Proceedings of IEEE International Conference on Multimedia and Exposition (IEEE, 2009), pp. 217–220. |

7. | K. T. Block, M. Uecker, and J. Frahm, “Suppression of MRI truncation artifacts using total variation constrained data extrapolation,” Int. J. Biomed. Imaging |

8. | B. Münch, P. Trtik, F. Marone, and M. Stampanoni, “Stripe and ring artifact removal with combined wavelet--Fourier filtering,” Opt. Express |

9. | V. B. S. Prasath, and A. Singh, “Ringing artifact reduction in blind image deblurring and denoising problems by regularization methods,” in: Proceedings of International Conference on Advances in Pattern Recognition (IEEE, 2009), pp. 333–336. |

10. | K. Lee, D. S. Kim, and T. Kim, “Regression-based prediction for blocking artifact reduction in JPEG-compressed images,” IEEE Trans. Image Process. |

11. | C. A. Segall, A. K. Katsaggelos, R. Molina, and J. Mateos, “Bayesian resolution enhancement of compressed video,” IEEE Trans. Image Process. |

12. | G. Wang, T. T. Wong, and P. A. Heng, “Deringing cartoons by image analogies,” ACM Trans. Graph. |

13. | 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. |

14. | S. Geman and D. Geman, “Stochastic relaxation, Gibbs distributions, and the Bayesian restoration of images,” IEEE Trans. Pattern Anal. Mach. Intell. |

15. | J. M. Sanches, J. C. Nascimento, and J. S. Marques, “Medical image noise reduction using the Sylvester-Lyapunov equation,” IEEE Trans. Image Process. |

16. | T. A. Stephenson and T. Chen, “Adaptive Markov random fields for example-based super-resolution of faces,” EURASIP J. Appl. Signal Process. |

17. | D. Rajan and S. Chaudhuri, “An MRF-based approach to generation of super-resolution images from blurred observations,” J. Math. Imaging Vis. |

18. | 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. |

19. | S. Tan and L. Jiao, “A unified iterative denoising algorithm based on natural image statistical models: derivation and examples,” Opt. Express |

20. | S. Roth, and M. J. Black, “Field of experts: a framework for learning image priors,” in: Proceedings of IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 860–867. |

21. | S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. |

22. | 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. |

23. | K. Chen, “Adaptive smoothing via contextual and local discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. |

24. | Y. K. Park, S. L. Park, and J. K. Kim, “Retinex method based on adaptive smoothing for illumination invariant face recognition,” Signal Process. |

25. | J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. [Ser A] |

26. | A. B. Hamza and H. Krim, “A variational approach to maximum a posteriori estimation for image denoising,” Lect. Notes Comput. Sci. |

27. | 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. |

28. | C. Tomasi, and R. Manduch, “Bilateral filtering for gray and color images,” in: Proceedings of IEEE International Conference on Computer Vision (IEEE, 1998), pp. 839–846. |

**OCIS Codes**

(100.0100) Image processing : Image processing

(100.3020) Image processing : Image reconstruction-restoration

**ToC Category:**

Image Processing

**History**

Original Manuscript: January 20, 2010

Revised Manuscript: March 12, 2010

Manuscript Accepted: March 14, 2010

Published: March 23, 2010

**Citation**

Cheolkon Jung and Licheng Jiao, "Novel Bayesian deringing method
in image interpolation and compression
using a SGLI prior," Opt. Express **18**, 7138-7149 (2010)

http://www.opticsinfobase.org/oe/abstract.cfm?URI=oe-18-7-7138

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### References

- J. D. Ouwerkerk, “Image super-resolution survey,” Image Vis. Comput. 24(10), 1039–1052 (2006). [CrossRef]
- J. Sun, J. Sun, Z. Xu, and H. Y. Shum, “Image super-resolution using gradient prior,” in: Proceedings of IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2008), pp. 1–8.
- 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]
- J. S. Chitode, Digital Signal Processing (Technical Publications, Pune-India, 2008).
- 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]
- Y. Wu, O. C. Au, E. Luo, D. Tu, and L. Yeung, “A novel deringing method based on MAP image restoration,” in: Proceedings of IEEE International Conference on Multimedia and Exposition (IEEE, 2009), pp. 217–220.
- 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]
- 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]
- V. B. S. Prasath and A. Singh, “Ringing artifact reduction in blind image deblurring and denoising problems by regularization methods,” in: Proceedings of International Conference on Advances in Pattern Recognition (IEEE, 2009), pp. 333–336.
- 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]
- G. Wang, T. T. Wong, and P. A. Heng, “Deringing cartoons by image analogies,” ACM Trans. Graph. 25(4), 1360–1379 (2006). [CrossRef]
- 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]
- 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]
- 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]
- 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).
- 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]
- 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. 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]
- S. Roth and M. J. Black, “Field of experts: a framework for learning image priors,” in: Proceedings of IEEE International Conference on Computer Vision and Pattern Recognition (IEEE, 2005), pp. 860–867.
- S. Roth and M. J. Black, “Fields of experts,” Int. J. Comput. Vis. 82(2), 205–229 (2009). [CrossRef]
- 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]
- K. Chen, “Adaptive smoothing via contextual and local discontinuities,” IEEE Trans. Pattern Anal. Mach. Intell. 27(10), 1552–1567 (2005). [CrossRef] [PubMed]
- 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]
- J. Besag, “On the statistical analysis of dirty pictures,” J. R. Stat. Soc. [Ser A] 48, 259–302 (1986).
- 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]
- 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]
- C. Tomasi and R. Manduch, “Bilateral filtering for gray and color images,” in: Proceedings of IEEE International Conference on Computer Vision (IEEE, 1998), pp. 839–846.

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