# American Institute of Mathematical Sciences

August  2022, 16(4): 895-924. doi: 10.3934/ipi.2022004

## A new anisotropic fourth-order diffusion equation model based on image features for image denoising

 School of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

* Corresponding author: Zhichang Guo, mathgzc@hit.edu.cn

Received  March 2021 Revised  December 2021 Published  August 2022 Early access  February 2022

Fund Project: The authors would like to thank all anonymous referees for their valuable comments and suggestions

Image denoising has always been a challenging task. For performing this task, one of the most effective methods is based on variational PDE. Inspired by the LLT model, we first propose a new adaptive LLT model by adding a weighted function, and then we propose a class of fourth-order diffusion equations based on the new functional. Owing to the adaptive function, the new functional is better than the LLT model and other fourth-order models in terms of edge preservation. While generalizing the Euler-Lagrange equation of the new functional, we discuss a new fourth-order diffusion framework for image denoising. Different from those of other fourth-order diffusion models, the new diffusion coefficients depend on the first-order and second-order derivatives, which can preserve edges and smooth images, respectively. Regarding numerical implementations, we first design an explicit scheme for the proposed model. However, fourth-order diffusion equations require strict stability conditions, and the number of iterations needed is considerable. Consequently, we apply the fast explicit diffusion algorithm (FED) to the explicit scheme to reduce the time consumption of the proposed approach. Furthermore, the additive operator splitting (AOS) scheme is applied for the numerical implementation, and it is the most efficient among all of our algorithms. Finally, compared with other models, the new model exhibits superior effectiveness and efficiency.

Citation: Ying Wen, Jiebao Sun, Zhichang Guo. A new anisotropic fourth-order diffusion equation model based on image features for image denoising. Inverse Problems and Imaging, 2022, 16 (4) : 895-924. doi: 10.3934/ipi.2022004
##### References:
 [1] A. Bertozzi and J. Greer, Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Commun. Pur. Appl. Math., 57 (2004), 764-790.  doi: 10.1002/cpa.20019. [2] A. Bertozzi, J. Greer, S. Osher and K. Vixie, Nonlinear regularizations of TV based PDEs for image processing, Contemp. Math, 371 (2005), 29-40.  doi: 10.1090/conm/371/06846. [3] C. Brito-Loeza and K. Chen, Multigrid algorithm for high order denoising, SIAM J. Imaging Sci., 3 (2010), 363-389.  doi: 10.1137/080737903. [4] D. Calvetti and L. Reichel, Adaptive Richardson iteration based on Leja points, J. Comput. Appl. Math., 71 (1996), 267-286.  doi: 10.1016/0377-0427(96)87162-7. [5] F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193.  doi: 10.1137/0729012. [6] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169. [7] P. Chen and Y. Wang, Fourth-order partial differential equations for image inpainting, In 2008 International Conference on Audio, Language and Image Processing, (2008), 1713–1717. [8] P. Chen and Y. Wang, A new fourth-order equation model for image inpainting, 2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery, 5 (2009), 320-324.  doi: 10.1109/FSKD.2009.201. [9] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238. [10] S. Durand, J. Fadili and M. Nikolova, Multiplicative noise removal using L1 fidelity on frame coefficients, J. Math. Imaging Vision, 36 (2010), 201-226.  doi: 10.1007/s10851-009-0180-z. [11] S. Grewenig, J. Weickert and A. Bruhn, From box filtering to fast explicit diffusion, Pattern Recognit., 6376 (2010), 533-542.  doi: 10.1007/978-3-642-15986-2_54. [12] M. R. Hajiaboli, A self-governing hybrid model for noise removal, Pacific-Rim Symposium on Image and Video Technology, 5414 (2009), 295-305.  doi: 10.1007/978-3-540-92957-4_26. [13] M. R. Hajiaboli, An anisotropic fourth-order diffusion filter for image noise removal, Int. J. Comput. Vis., 92 (2011), 177-191.  doi: 10.1007/s11263-010-0330-1. [14] S. Kim and H. Lim, Fourth-order partial differential equations for effective image denoising, Electron. J. Differ. Equ. Conf., 17 (2009), 107-121. [15] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint, arXiv: 1412.6980. [16] F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image Represent., 18 (2007), 322-330.  doi: 10.1016/j.jvcir.2007.04.005. [17] P. Li, S.-J. Li, Z.-A. Yao and Z.-J. Zhang, Two anisotropic fourth-order partial differential equations for image inpainting, IET Image Process., 7 (2013), 260-269.  doi: 10.1049/iet-ipr.2012.0592. [18] P. Li, Y. Zou and Z. Yao, Fourth-order anisotropic diffusion equations for image zooming, Journal of Image and Graphics, 18 (2013), 1261-1269. [19] S. Li and X. Yang, Novel image inpainting algorithm based on adaptive fourth-order partial differential equation, IET Image Process., 11 (2017), 870-879.  doi: 10.1049/iet-ipr.2016.0898. [20] F. Liu and J. Liu, Anisotropic diffusion for image denoising based on diffusion tensors, Journal of Visual Communication and Image Representation, 23 (2012), 516-521.  doi: 10.1016/j.jvcir.2012.01.012. [21] X. Liu, L. Huang and Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising, Appl. Math. Lett., 24 (2011), 1282-1288.  doi: 10.1016/j.aml.2011.01.028. [22] X. Y. Liu, C.-H. Lai and K. A. Pericleous, A fourth-order partial differential equation denoising model with an adaptive relaxation method, Int. J. Comput. Math., 92 (2015), 608-622.  doi: 10.1080/00207160.2014.904854. [23] B. Lu and Q. Liu, Image restoration with surface-based fourth-order partial differential equation, In Visual Communications and Image Processing 2010, International Society for Optics and Photonics, 7744 (2010), 774424. [24] T. Lu, P. Neittaanmaki and X.-C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, Math. Anal. Numér., 26 (1992), 673-708.  doi: 10.1051/m2an/1992260606731. [25] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590.  doi: 10.1109/TIP.2003.819229. [26] M. Lysaker, S. Osher and X.-C. Tai, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13 (2004), 1345-1357.  doi: 10.1109/TIP.2004.834662. [27] M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vis., 66 (2006), 5-18.  doi: 10.1007/s11263-005-3219-7. [28] D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, Proceedings Eighth IEEE International Conference on Computer Vision, 2 (2001), 416-423.  doi: 10.1109/ICCV.2001.937655. [29] V. Murali and P. Sudeep, Image denoising using DnCNN: An exploration study, In Advances in Communication Systems and Networks, Springer, 656 (2020), 847–859. doi: 10.1007/978-981-15-3992-3_72. [30] K. Papafitsoros and C.-B. Schönlieb, A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vision, 48 (2014), 308-338.  doi: 10.1007/s10851-013-0445-4. [31] Y. Pathak, K. Arya and S. Tiwari, Fourth-order partial differential equations based anisotropic diffusion model for low-dose CT images, Modern Phys. Lett. B, 32 (2018), 1850300, 26 pp. doi: 10.1142/S0217984918503001. [32] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205. [33] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F. [34] K. Shi, D. Zhang, Z. Guo and B. Wu, A linear reaction-diffusion system with interior degeneration for color image compression, SIAM J. Imaging Sci., 11 (2018), 442-472.  doi: 10.1137/17M1137991. [35] R. S. Varga, Matrix Iterative Analysis, vol. 27, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2. [36] Z. Wang, A. Bovik, H. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861. [37] J. Weickert, Applications of nonlinear diffusion in image processing and computer vision, Acta Math. Univ. Comenian., 70 (2001), 33-50. [38] J. Weickert, B. Romeny and M. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Process., 7 (1998), 398-410.  doi: 10.1109/83.661190. [39] J. Weickert, S. Grewenig, C. Schroers and A. Bruhn, Cyclic schemes for PDE-based image analysis, Int. J. Comput. Vis., 118 (2016), 275-299.  doi: 10.1007/s11263-015-0874-1. [40] W. Yao, Z. Guo, J. Sun, B. Wu and H. Gao, Multiplicative noise removal for texture images based on adaptive anisotropic fractional diffusion equations, SIAM J. Imaging Sci., 12 (2019), 839-873.  doi: 10.1137/18M1187192. [41] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730.  doi: 10.1109/83.869184. [42] W. Zeng, X. Lu and X. Tan, A class of fourth-order telegraph-diffusion equations for image restoration, J. Appl. Math., 2011 (2011), Art. ID 240370, 20 pp. doi: 10.1155/2011/240370. [43] K. Zhang, W. Zuo, Y. Chen, D. Meng and L. Zhang, Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising, IEEE Trans. Image Process., 26 (2017), 3142-3155.  doi: 10.1109/TIP.2017.2662206. [44] Z. Zhou, Z. Guo, G. Dong, J. Sun, D. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Trans. Image Process., 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185. [45] W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5 (2012), 1-32.  doi: 10.1137/110822268.

show all references

##### References:
 [1] A. Bertozzi and J. Greer, Low-curvature image simplifiers: Global regularity of smooth solutions and Laplacian limiting schemes, Commun. Pur. Appl. Math., 57 (2004), 764-790.  doi: 10.1002/cpa.20019. [2] A. Bertozzi, J. Greer, S. Osher and K. Vixie, Nonlinear regularizations of TV based PDEs for image processing, Contemp. Math, 371 (2005), 29-40.  doi: 10.1090/conm/371/06846. [3] C. Brito-Loeza and K. Chen, Multigrid algorithm for high order denoising, SIAM J. Imaging Sci., 3 (2010), 363-389.  doi: 10.1137/080737903. [4] D. Calvetti and L. Reichel, Adaptive Richardson iteration based on Leja points, J. Comput. Appl. Math., 71 (1996), 267-286.  doi: 10.1016/0377-0427(96)87162-7. [5] F. Catté, P.-L. Lions, J.-M. Morel and T. Coll, Image selective smoothing and edge detection by nonlinear diffusion, SIAM J. Numer. Anal., 29 (1992), 182-193.  doi: 10.1137/0729012. [6] T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM J. Sci. Comput., 22 (2000), 503-516.  doi: 10.1137/S1064827598344169. [7] P. Chen and Y. Wang, Fourth-order partial differential equations for image inpainting, In 2008 International Conference on Audio, Language and Image Processing, (2008), 1713–1717. [8] P. Chen and Y. Wang, A new fourth-order equation model for image inpainting, 2009 Sixth International Conference on Fuzzy Systems and Knowledge Discovery, 5 (2009), 320-324.  doi: 10.1109/FSKD.2009.201. [9] K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-D transform-domain collaborative filtering, IEEE Trans. Image Process., 16 (2007), 2080-2095.  doi: 10.1109/TIP.2007.901238. [10] S. Durand, J. Fadili and M. Nikolova, Multiplicative noise removal using L1 fidelity on frame coefficients, J. Math. Imaging Vision, 36 (2010), 201-226.  doi: 10.1007/s10851-009-0180-z. [11] S. Grewenig, J. Weickert and A. Bruhn, From box filtering to fast explicit diffusion, Pattern Recognit., 6376 (2010), 533-542.  doi: 10.1007/978-3-642-15986-2_54. [12] M. R. Hajiaboli, A self-governing hybrid model for noise removal, Pacific-Rim Symposium on Image and Video Technology, 5414 (2009), 295-305.  doi: 10.1007/978-3-540-92957-4_26. [13] M. R. Hajiaboli, An anisotropic fourth-order diffusion filter for image noise removal, Int. J. Comput. Vis., 92 (2011), 177-191.  doi: 10.1007/s11263-010-0330-1. [14] S. Kim and H. Lim, Fourth-order partial differential equations for effective image denoising, Electron. J. Differ. Equ. Conf., 17 (2009), 107-121. [15] D. P. Kingma and J. Ba, Adam: A method for stochastic optimization, arXiv preprint, arXiv: 1412.6980. [16] F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, J. Vis. Commun. Image Represent., 18 (2007), 322-330.  doi: 10.1016/j.jvcir.2007.04.005. [17] P. Li, S.-J. Li, Z.-A. Yao and Z.-J. Zhang, Two anisotropic fourth-order partial differential equations for image inpainting, IET Image Process., 7 (2013), 260-269.  doi: 10.1049/iet-ipr.2012.0592. [18] P. Li, Y. Zou and Z. Yao, Fourth-order anisotropic diffusion equations for image zooming, Journal of Image and Graphics, 18 (2013), 1261-1269. [19] S. Li and X. Yang, Novel image inpainting algorithm based on adaptive fourth-order partial differential equation, IET Image Process., 11 (2017), 870-879.  doi: 10.1049/iet-ipr.2016.0898. [20] F. Liu and J. Liu, Anisotropic diffusion for image denoising based on diffusion tensors, Journal of Visual Communication and Image Representation, 23 (2012), 516-521.  doi: 10.1016/j.jvcir.2012.01.012. [21] X. Liu, L. Huang and Z. Guo, Adaptive fourth-order partial differential equation filter for image denoising, Appl. Math. Lett., 24 (2011), 1282-1288.  doi: 10.1016/j.aml.2011.01.028. [22] X. Y. Liu, C.-H. Lai and K. A. Pericleous, A fourth-order partial differential equation denoising model with an adaptive relaxation method, Int. J. Comput. Math., 92 (2015), 608-622.  doi: 10.1080/00207160.2014.904854. [23] B. Lu and Q. Liu, Image restoration with surface-based fourth-order partial differential equation, In Visual Communications and Image Processing 2010, International Society for Optics and Photonics, 7744 (2010), 774424. [24] T. Lu, P. Neittaanmaki and X.-C. Tai, A parallel splitting-up method for partial differential equations and its applications to Navier-Stokes equations, Math. Anal. Numér., 26 (1992), 673-708.  doi: 10.1051/m2an/1992260606731. [25] M. Lysaker, A. Lundervold and X.-C. Tai, Noise removal using fourth-order partial differential equation with applications to medical magnetic resonance images in space and time, IEEE Trans. Image Process., 12 (2003), 1579-1590.  doi: 10.1109/TIP.2003.819229. [26] M. Lysaker, S. Osher and X.-C. Tai, Noise removal using smoothed normals and surface fitting, IEEE Trans. Image Process., 13 (2004), 1345-1357.  doi: 10.1109/TIP.2004.834662. [27] M. Lysaker and X.-C. Tai, Iterative image restoration combining total variation minimization and a second-order functional, Int. J. Comput. Vis., 66 (2006), 5-18.  doi: 10.1007/s11263-005-3219-7. [28] D. Martin, C. Fowlkes, D. Tal and J. Malik, A database of human segmented natural images and its application to evaluating segmentation algorithms and measuring ecological statistics, Proceedings Eighth IEEE International Conference on Computer Vision, 2 (2001), 416-423.  doi: 10.1109/ICCV.2001.937655. [29] V. Murali and P. Sudeep, Image denoising using DnCNN: An exploration study, In Advances in Communication Systems and Networks, Springer, 656 (2020), 847–859. doi: 10.1007/978-981-15-3992-3_72. [30] K. Papafitsoros and C.-B. Schönlieb, A combined first and second order variational approach for image reconstruction, J. Math. Imaging Vision, 48 (2014), 308-338.  doi: 10.1007/s10851-013-0445-4. [31] Y. Pathak, K. Arya and S. Tiwari, Fourth-order partial differential equations based anisotropic diffusion model for low-dose CT images, Modern Phys. Lett. B, 32 (2018), 1850300, 26 pp. doi: 10.1142/S0217984918503001. [32] P. Perona and J. Malik, Scale-space and edge detection using anisotropic diffusion, IEEE Trans. Pattern Anal. Mach. Intell., 12 (1990), 629-639.  doi: 10.1109/34.56205. [33] L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Phys. D, 60 (1992), 259-268.  doi: 10.1016/0167-2789(92)90242-F. [34] K. Shi, D. Zhang, Z. Guo and B. Wu, A linear reaction-diffusion system with interior degeneration for color image compression, SIAM J. Imaging Sci., 11 (2018), 442-472.  doi: 10.1137/17M1137991. [35] R. S. Varga, Matrix Iterative Analysis, vol. 27, Springer-Verlag, Berlin, 2000. doi: 10.1007/978-3-642-05156-2. [36] Z. Wang, A. Bovik, H. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.  doi: 10.1109/TIP.2003.819861. [37] J. Weickert, Applications of nonlinear diffusion in image processing and computer vision, Acta Math. Univ. Comenian., 70 (2001), 33-50. [38] J. Weickert, B. Romeny and M. Viergever, Efficient and reliable schemes for nonlinear diffusion filtering, IEEE Trans. Image Process., 7 (1998), 398-410.  doi: 10.1109/83.661190. [39] J. Weickert, S. Grewenig, C. Schroers and A. Bruhn, Cyclic schemes for PDE-based image analysis, Int. J. Comput. Vis., 118 (2016), 275-299.  doi: 10.1007/s11263-015-0874-1. [40] W. Yao, Z. Guo, J. Sun, B. Wu and H. Gao, Multiplicative noise removal for texture images based on adaptive anisotropic fractional diffusion equations, SIAM J. Imaging Sci., 12 (2019), 839-873.  doi: 10.1137/18M1187192. [41] Y.-L. You and M. Kaveh, Fourth-order partial differential equations for noise removal, IEEE Trans. Image Process., 9 (2000), 1723-1730.  doi: 10.1109/83.869184. [42] W. Zeng, X. Lu and X. Tan, A class of fourth-order telegraph-diffusion equations for image restoration, J. Appl. Math., 2011 (2011), Art. ID 240370, 20 pp. doi: 10.1155/2011/240370. [43] K. Zhang, W. Zuo, Y. Chen, D. Meng and L. Zhang, Beyond a Gaussian denoiser: Residual learning of deep CNN for image denoising, IEEE Trans. Image Process., 26 (2017), 3142-3155.  doi: 10.1109/TIP.2017.2662206. [44] Z. Zhou, Z. Guo, G. Dong, J. Sun, D. Zhang and B. Wu, A doubly degenerate diffusion model based on the gray level indicator for multiplicative noise removal, IEEE Trans. Image Process., 24 (2015), 249-260.  doi: 10.1109/TIP.2014.2376185. [45] W. Zhu and T. Chan, Image denoising using mean curvature of image surface, SIAM J. Imaging Sci., 5 (2012), 1-32.  doi: 10.1137/110822268.
The noise-free image, noisy image $f$ (the standard deviation of noise $\sigma_{\text{n}} = 10$), the map of $\alpha(f)$, and the corresponding $\Omega_{h}$ (white regions) and $\varGamma$ (black regions) with parameters $K = 10, \sigma = 1, \gamma = 0.55$
Comparison between the adaptive LLT and the original LLT in the 1-D case
Partially enlarged results of Fig. 2
Analysis of $\Phi_a$ and $\Phi_b$. (a) Mapping of $\Phi_a$ and $\Phi_b$ with $K$ = 1. (b) Mapping of $\Phi_a$ with different values of $K$
Comparison of edge detection operators in synthetic images. (a) Synthetic image. (b) Magnitude of the gradient. (c) Absolute value of the Laplacian
Comparison of edge detection operators in natural images. First row: images. Second row: magnitudes of gradients. Third row: absolute values of Laplacian
Comparison between the diffusion coefficients of the proposed model and the LLT model ($\Phi = \Phi_b$, $K = 1$). First row: diffusion coefficients of the LLT model. Second row: diffusion coefficients of the proposed model
Noise-free images for the evaluations
Denoising results of geometry ($300 \times 300$), $\sigma_{\text{n}} = 40$
Plots of column signals. (a) Location of the signals. (b) Plots of the signals for the original image and the images denoised by the LLT, FED and AOS approaches
Denoising results of slope-1 ($128 \times 128$), $\sigma_{\text{n}} = 20$
Differences between the noisy slope-1 ($128 \times 128$), $\sigma_{\text{n}} = 20$ and obtained denoised images
Denoising results of slope-2 ($128 \times 128$), $\sigma_{\text{n}} = 15$
The plots between the variance and number of iterations or cycles of the Expct, FED and AOS methods for the noisy image castle with $\sigma_{\text{n}} = 30$. For the PSNR values, only focus on its tendency and ignore the exact values
Robustness of parameters. (a) Scatter plot of PSNRs obtained from parametric experiments sorted from large to small. (b) Heat map of PSNR values with different settings of $\sigma$ and $K$. (c) Heat map of the number of iterations with different settings of $\sigma$ and $K$
Denoising results of starfish($256 \times 256$), $\sigma_{\text{n}} = 30$. (f)-(l) Expct with different settings of $\sigma$ and $K$
Denoising results of piggy ($341 \times 199$), $\sigma_{\text{n}} = 30$
Denoising results of butterfly ($256 \times 256$), $\sigma_{\text{n}} = 30$
Denoising results of Lena ($371 \times 302$), $\sigma_{\text{n}} = 40$
Denoising results of parrot ($256 \times 256$), $\sigma_{\text{n}} = 50$
Denoising results of the noisy image Fig. 1(b), and the PSNR values of the LLT, adaptive LLT, and functional $\widetilde{E}$ approaches are $32.757$, $41.407$, and $37.239$, respectively
Plots of column signals. (a) Location of the signals. (b) Plots of the signals for the images denoised by the LLT, adaptive LLT and functional $\widetilde{E}$ (the ideal case) approaches. (c)-(e) Partially enlarged results
PSNR (dB) and SSIM values of the synthetic images denoising results generated by different methods
 $\sigma_\text{n}$ Images TV LLT AFD MC TDM BM3D DnCNN FED AOS 40 geometry $32.873$ $27.400$ 32.343 $31.800$ 35.768 28.691 29.557 $35.006$ $34.207$ $0.188$ $0.145$ 0.190 $0.197$ 0.266 0.186 0.923 $0.202$ $0.575$ 20 slope-1 $37.707$ $33.053$ $38.895$ $37.696$ $40.264$ $39.834$ $40.119$ $41.928$ $42.866$ $0.727$ $0.646$ $0.763$ $0.739$ $0.767$ $0.711$ $0.991$ $0.774$ $0.826$ 15 slope-2 $37.023$ $32.372$ $36.156$ $37.043$ $43.642$ $40.553$ $39.417$ $42.125$ $42.307$ $0.872$ $0.604$ $0.871$ $0.891$ $0.981$ $0.878$ $0.993$ $0.991$ $0.992$
 $\sigma_\text{n}$ Images TV LLT AFD MC TDM BM3D DnCNN FED AOS 40 geometry $32.873$ $27.400$ 32.343 $31.800$ 35.768 28.691 29.557 $35.006$ $34.207$ $0.188$ $0.145$ 0.190 $0.197$ 0.266 0.186 0.923 $0.202$ $0.575$ 20 slope-1 $37.707$ $33.053$ $38.895$ $37.696$ $40.264$ $39.834$ $40.119$ $41.928$ $42.866$ $0.727$ $0.646$ $0.763$ $0.739$ $0.767$ $0.711$ $0.991$ $0.774$ $0.826$ 15 slope-2 $37.023$ $32.372$ $36.156$ $37.043$ $43.642$ $40.553$ $39.417$ $42.125$ $42.307$ $0.872$ $0.604$ $0.871$ $0.891$ $0.981$ $0.878$ $0.993$ $0.991$ $0.992$
PSNR (dB) and SSIM values of the natural images denoising results generated by different methods
 $\sigma_\text{n}$ Images TV LLT AFD MC TDM BM3D DnCNN Expct FED AOS 10 piggy 38.328 37.305 38.418 37.222 38.432 40.535 41.522 39.505 39.571 39.763 0.614 0.555 0.629 0.505 0.619 0.627 0.993 0.657 0.649 0.670 butterfly 32.716 32.194 32.231 32.473 33.151 33.257 34.712 33.158 33.074 33.143 0.812 0.795 0.812 0.797 0.824 0.830 0.984 0.821 0.818 0.823 castle 32.705 32.140 31.834 32.131 32.740 34.089 34.616 32.940 32.787 32.803 0.563 0.554 0.548 0.550 0.579 0.626 0.967 0.561 0.559 0.562 Lena 32.926 33.061 32.535 33.019 33.238 34.513 34.892 33.370 33.230 33.386 0.719 0.726 0.706 0.731 0.727 0.743 0.965 0.724 0.715 0.730 starfish 32.224 32.368 31.651 32.190 32.768 33.305 34.305 32.805 32.720 32.684 0.827 0.835 0.825 0.835 0.842 0.844 0.970 0.839 0.835 0.839 parrot 32.362 31.787 31.592 31.766 32.550 33.365 33.940 32.505 32.376 32.434 0.693 0.686 0.693 0.674 0.709 0.745 0.968 0.706 0.705 0.695 20 piggy 34.776 33.147 34.775 33.691 35.002 36.572 37.803 35.896 35.942 36.101 0.541 0.460 0.560 0.432 0.600 0.531 0.988 0.606 0.595 0.617 butterfly 28.628 27.698 28.486 28.647 29.267 29.575 30.948 29.033 29.054 29.196 0.741 0.700 0.748 0.721 0.764 0.775 0.969 0.754 0.758 0.758 castle 28.986 28.038 28.369 28.287 29.023 30.482 31.214 29.080 29.018 29.047 0.437 0.420 0.440 0.431 0.480 0.506 0.943 0.452 0.443 0.446 Lena 29.567 29.582 29.416 29.592 29.852 31.314 31.682 29.990 29.941 30.070 0.589 0.604 0.589 0.606 0.613 0.643 0.938 0.607 0.603 0.612 starfish 28.342 28.307 28.065 28.344 28.787 29.655 30.498 28.777 28.740 28.782 0.725 0.737 0.725 0.742 0.746 0.752 0.939 0.744 0.741 0.745 parrot 28.725 27.863 28.240 28.257 28.992 29.749 30.553 28.906 28.864 28.937 0.579 0.567 0.595 0.576 0.614 0.611 0.940 0.605 0.600 0.603 30 piggy 32.805 31.089 33.046 31.812 32.968 34.099 35.553 33.896 33.923 34.104 0.498 0.427 0.528 0.405 0.562 0.490 0.982 0.553 0.554 0.574 butterfly 26.345 25.334 26.424 26.591 26.997 27.648 28.819 26.856 26.826 27.017 0.691 0.646 0.697 0.679 0.712 0.730 0.955 0.710 0.704 0.710 castle 26.979 25.834 26.548 26.177 27.031 28.454 29.280 26.996 26.973 27.032 0.376 0.351 0.390 0.367 0.423 0.441 0.923 0.399 0.414 0.417 Lena 27.841 27.685 27.735 27.740 28.047 29.489 29.863 28.189 28.171 28.183 0.514 0.531 0.519 0.535 0.548 0.574 0.915 0.536 0.534 0.540 starfish 26.305 26.111 26.152 26.149 26.649 27.525 28.368 26.563 26.540 26.622 0.639 0.652 0.643 0.657 0.665 0.684 0.908 0.659 0.659 0.662 parrot 26.677 25.633 26.432 26.271 26.995 27.616 28.679 26.876 26.858 26.947 0.509 0.494 0.526 0.509 0.550 0.531 0.918 0.539 0.537 0.539 40 piggy 31.204 29.491 31.406 30.134 31.335 31.773 33.642 32.193 32.207 32.151 0.466 0.413 0.506 0.367 0.541 0.452 0.975 0.527 0.522 0.523 butterfly 24.837 23.764 24.995 25.077 25.383 26.097 27.347 25.396 25.423 25.435 0.646 0.601 0.656 0.644 0.664 0.688 0.938 0.668 0.667 0.668 castle 25.672 24.460 25.379 24.780 25.619 26.923 27.868 25.619 25.671 25.699 0.325 0.299 0.341 0.316 0.376 0.380 0.902 0.343 0.360 0.368 Lena 26.657 26.381 26.528 26.479 26.741 28.139 28.575 26.938 26.925 26.975 0.452 0.469 0.460 0.472 0.495 0.515 0.893 0.479 0.478 0.481 starfish 24.962 24.734 24.785 24.697 25.206 25.907 26.849 25.189 25.171 25.237 0.585 0.600 0.589 0.600 0.616 0.627 0.883 0.609 0.608 0.612 parrot 25.318 24.286 25.270 24.966 25.666 25.904 27.387 25.640 25.622 25.739 0.459 0.452 0.479 0.469 0.516 0.476 0.899 0.494 0.489 0.489 50 piggy 30.071 28.470 30.225 28.882 30.005 29.625 32.247 30.977 31.053 30.944 0.446 0.399 0.472 0.342 0.525 0.439 0.965 0.490 0.492 0.497 butterfly 23.581 22.625 23.872 23.926 24.048 24.465 26.321 24.189 24.169 24.258 0.608 0.563 0.626 0.610 0.625 0.643 0.926 0.634 0.633 0.637 castle 24.870 23.713 24.662 23.967 24.755 25.678 27.002 24.903 24.883 24.916 0.285 0.262 0.302 0.276 0.342 0.330 0.885 0.313 0.308 0.307 Lena 25.720 25.308 25.505 25.389 25.694 26.866 27.466 25.873 25.864 25.847 0.407 0.422 0.409 0.421 0.455 0.456 0.869 0.426 0.424 0.433 starfish 23.966 23.781 23.783 23.779 24.101 24.348 25.621 24.144 24.128 24.191 0.532 0.553 0.535 0.557 0.569 0.554 0.856 0.555 0.554 0.559 parrot 24.302 23.222 24.385 23.931 24.652 24.401 26.434 24.629 24.639 24.715 0.412 0.412 0.429 0.426 0.474 0.424 0.883 0.443 0.442 0.444
 $\sigma_\text{n}$ Images TV LLT AFD MC TDM BM3D DnCNN Expct FED AOS 10 piggy 38.328 37.305 38.418 37.222 38.432 40.535 41.522 39.505 39.571 39.763 0.614 0.555 0.629 0.505 0.619 0.627 0.993 0.657 0.649 0.670 butterfly 32.716 32.194 32.231 32.473 33.151 33.257 34.712 33.158 33.074 33.143 0.812 0.795 0.812 0.797 0.824 0.830 0.984 0.821 0.818 0.823 castle 32.705 32.140 31.834 32.131 32.740 34.089 34.616 32.940 32.787 32.803 0.563 0.554 0.548 0.550 0.579 0.626 0.967 0.561 0.559 0.562 Lena 32.926 33.061 32.535 33.019 33.238 34.513 34.892 33.370 33.230 33.386 0.719 0.726 0.706 0.731 0.727 0.743 0.965 0.724 0.715 0.730 starfish 32.224 32.368 31.651 32.190 32.768 33.305 34.305 32.805 32.720 32.684 0.827 0.835 0.825 0.835 0.842 0.844 0.970 0.839 0.835 0.839 parrot 32.362 31.787 31.592 31.766 32.550 33.365 33.940 32.505 32.376 32.434 0.693 0.686 0.693 0.674 0.709 0.745 0.968 0.706 0.705 0.695 20 piggy 34.776 33.147 34.775 33.691 35.002 36.572 37.803 35.896 35.942 36.101 0.541 0.460 0.560 0.432 0.600 0.531 0.988 0.606 0.595 0.617 butterfly 28.628 27.698 28.486 28.647 29.267 29.575 30.948 29.033 29.054 29.196 0.741 0.700 0.748 0.721 0.764 0.775 0.969 0.754 0.758 0.758 castle 28.986 28.038 28.369 28.287 29.023 30.482 31.214 29.080 29.018 29.047 0.437 0.420 0.440 0.431 0.480 0.506 0.943 0.452 0.443 0.446 Lena 29.567 29.582 29.416 29.592 29.852 31.314 31.682 29.990 29.941 30.070 0.589 0.604 0.589 0.606 0.613 0.643 0.938 0.607 0.603 0.612 starfish 28.342 28.307 28.065 28.344 28.787 29.655 30.498 28.777 28.740 28.782 0.725 0.737 0.725 0.742 0.746 0.752 0.939 0.744 0.741 0.745 parrot 28.725 27.863 28.240 28.257 28.992 29.749 30.553 28.906 28.864 28.937 0.579 0.567 0.595 0.576 0.614 0.611 0.940 0.605 0.600 0.603 30 piggy 32.805 31.089 33.046 31.812 32.968 34.099 35.553 33.896 33.923 34.104 0.498 0.427 0.528 0.405 0.562 0.490 0.982 0.553 0.554 0.574 butterfly 26.345 25.334 26.424 26.591 26.997 27.648 28.819 26.856 26.826 27.017 0.691 0.646 0.697 0.679 0.712 0.730 0.955 0.710 0.704 0.710 castle 26.979 25.834 26.548 26.177 27.031 28.454 29.280 26.996 26.973 27.032 0.376 0.351 0.390 0.367 0.423 0.441 0.923 0.399 0.414 0.417 Lena 27.841 27.685 27.735 27.740 28.047 29.489 29.863 28.189 28.171 28.183 0.514 0.531 0.519 0.535 0.548 0.574 0.915 0.536 0.534 0.540 starfish 26.305 26.111 26.152 26.149 26.649 27.525 28.368 26.563 26.540 26.622 0.639 0.652 0.643 0.657 0.665 0.684 0.908 0.659 0.659 0.662 parrot 26.677 25.633 26.432 26.271 26.995 27.616 28.679 26.876 26.858 26.947 0.509 0.494 0.526 0.509 0.550 0.531 0.918 0.539 0.537 0.539 40 piggy 31.204 29.491 31.406 30.134 31.335 31.773 33.642 32.193 32.207 32.151 0.466 0.413 0.506 0.367 0.541 0.452 0.975 0.527 0.522 0.523 butterfly 24.837 23.764 24.995 25.077 25.383 26.097 27.347 25.396 25.423 25.435 0.646 0.601 0.656 0.644 0.664 0.688 0.938 0.668 0.667 0.668 castle 25.672 24.460 25.379 24.780 25.619 26.923 27.868 25.619 25.671 25.699 0.325 0.299 0.341 0.316 0.376 0.380 0.902 0.343 0.360 0.368 Lena 26.657 26.381 26.528 26.479 26.741 28.139 28.575 26.938 26.925 26.975 0.452 0.469 0.460 0.472 0.495 0.515 0.893 0.479 0.478 0.481 starfish 24.962 24.734 24.785 24.697 25.206 25.907 26.849 25.189 25.171 25.237 0.585 0.600 0.589 0.600 0.616 0.627 0.883 0.609 0.608 0.612 parrot 25.318 24.286 25.270 24.966 25.666 25.904 27.387 25.640 25.622 25.739 0.459 0.452 0.479 0.469 0.516 0.476 0.899 0.494 0.489 0.489 50 piggy 30.071 28.470 30.225 28.882 30.005 29.625 32.247 30.977 31.053 30.944 0.446 0.399 0.472 0.342 0.525 0.439 0.965 0.490 0.492 0.497 butterfly 23.581 22.625 23.872 23.926 24.048 24.465 26.321 24.189 24.169 24.258 0.608 0.563 0.626 0.610 0.625 0.643 0.926 0.634 0.633 0.637 castle 24.870 23.713 24.662 23.967 24.755 25.678 27.002 24.903 24.883 24.916 0.285 0.262 0.302 0.276 0.342 0.330 0.885 0.313 0.308 0.307 Lena 25.720 25.308 25.505 25.389 25.694 26.866 27.466 25.873 25.864 25.847 0.407 0.422 0.409 0.421 0.455 0.456 0.869 0.426 0.424 0.433 starfish 23.966 23.781 23.783 23.779 24.101 24.348 25.621 24.144 24.128 24.191 0.532 0.553 0.535 0.557 0.569 0.554 0.856 0.555 0.554 0.559 parrot 24.302 23.222 24.385 23.931 24.652 24.401 26.434 24.629 24.639 24.715 0.412 0.412 0.429 0.426 0.474 0.424 0.883 0.443 0.442 0.444
Average PSNR (dB) and SSIM values for 6 natural images
 $\sigma_\text{n}$ TV LLT AFD MC TDM BM3D DnCNN Expct FED AOS 10 33.544 33.142 33.044 33.134 33.813 34.844 35.665 34.047 33.960 34.036 0.705 0.692 0.702 0.682 0.717 0.736 0.975 0.718 0.714 0.720 20 29.837 29.106 29.559 29.470 30.154 31.224 32.116 30.280 30.260 30.355 0.602 0.581 0.610 0.585 0.636 0.637 0.953 0.628 0.623 0.630 30 27.825 26.948 27.723 23.534 28.115 29.138 30.094 28.229 28.215 28.317 0.538 0.517 0.551 0.450 0.577 0.575 0.933 0.566 0.567 0.574 40 26.442 25.519 26.394 26.022 26.658 27.457 28.611 26.829 26.836 26.873 0.489 0.472 0.505 0.478 0.535 0.523 0.915 0.520 0.521 0.524 50 25.419 24.520 25.405 24.979 25.542 25.897 27.515 25.786 25.789 25.812 0.448 0.435 0.462 0.439 0.498 0.474 0.897 0.477 0.475 0.479 Average 28.613 27.847 28.425 27.428 28.856 29.712 30.800 29.034 29.012 29.079 0.556 0.539 0.566 0.527 0.592 0.589 0.935 0.582 0.580 0.585
 $\sigma_\text{n}$ TV LLT AFD MC TDM BM3D DnCNN Expct FED AOS 10 33.544 33.142 33.044 33.134 33.813 34.844 35.665 34.047 33.960 34.036 0.705 0.692 0.702 0.682 0.717 0.736 0.975 0.718 0.714 0.720 20 29.837 29.106 29.559 29.470 30.154 31.224 32.116 30.280 30.260 30.355 0.602 0.581 0.610 0.585 0.636 0.637 0.953 0.628 0.623 0.630 30 27.825 26.948 27.723 23.534 28.115 29.138 30.094 28.229 28.215 28.317 0.538 0.517 0.551 0.450 0.577 0.575 0.933 0.566 0.567 0.574 40 26.442 25.519 26.394 26.022 26.658 27.457 28.611 26.829 26.836 26.873 0.489 0.472 0.505 0.478 0.535 0.523 0.915 0.520 0.521 0.524 50 25.419 24.520 25.405 24.979 25.542 25.897 27.515 25.786 25.789 25.812 0.448 0.435 0.462 0.439 0.498 0.474 0.897 0.477 0.475 0.479 Average 28.613 27.847 28.425 27.428 28.856 29.712 30.800 29.034 29.012 29.079 0.556 0.539 0.566 0.527 0.592 0.589 0.935 0.582 0.580 0.585
Comparison of CPU Times
 $\sigma_{\rm{n}}$ Expct FED AOS 10 4.083 2.333 1.922 20 13.863 6.509 4.222 30 28.385 21.427 9.419 40 44.120 32.749 19.449 50 63.855 32.065 15.770
 $\sigma_{\rm{n}}$ Expct FED AOS 10 4.083 2.333 1.922 20 13.863 6.509 4.222 30 28.385 21.427 9.419 40 44.120 32.749 19.449 50 63.855 32.065 15.770
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