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

Ying Wen; Jiebao Sun; Zhichang Guo

doi:10.3934/ipi.2022004

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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

Ying Wen , Jiebao Sun and Zhichang Guo ^,

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.

Keywords: image denoising, adaptive functional, fourth-order diffusion equation.

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.

Figure 1. 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 $

$ \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 $
40	geometry	$ 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 $
20	slope-1	$ 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 $
15	slope-2	$ 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	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
	piggy	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
	butterfly	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
	castle	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
	Lena	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
	starfish	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
	parrot	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
	piggy	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
	butterfly	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
	castle	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
	Lena	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
	starfish	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
	parrot	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
	piggy	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
	butterfly	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
	castle	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
	Lena	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
	starfish	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
	parrot	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
	piggy	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
	butterfly	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
	castle	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
	Lena	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
	starfish	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
	parrot	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
	piggy	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
	butterfly	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
	castle	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
	Lena	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
	starfish	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
	parrot	0.412	0.412	0.429	0.426	0.474	0.424	0.883	0.443	0.442	0.444

$ \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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$ \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
10	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
20	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
30	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
40	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
50	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
Average	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
10	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
20	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
30	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
40	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
50	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
Average	0.556	0.539	0.566	0.527	0.592	0.589	0.935	0.582	0.580	0.585

American Institute of Mathematical Sciences

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

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