RWRM: Residual Wasserstein regularization model for image restoration

Ruiqiang He; Xiangchu Feng; Xiaolong Zhu; Hua Huang; Bingzhe Wei

doi:10.3934/ipi.2020069

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December 2021, 15(6): 1307-1332. doi: 10.3934/ipi.2020069

RWRM: Residual Wasserstein regularization model for image restoration

Ruiqiang He ^a,b, , Xiangchu Feng ^a,, , Xiaolong Zhu ^a, , Hua Huang ^a, and Bingzhe Wei ^a,

a.	School of Mathematics and Statistics, Xidian University, Xi'an 710126, China
b.	Department of Mathematics, Xinzhou Teachers University, Xinzhou 034000, China

^* Corresponding Author

Received October 2019 Revised August 2020 Published December 2021 Early access November 2020

Existing image restoration methods mostly make full use of various image prior information. However, they rarely exploit the potential of residual histograms, especially their role as ensemble regularization constraint. In this paper, we propose a residual Wasserstein regularization model (RWRM), in which a residual histogram constraint is subtly embedded into a type of variational minimization problems. Specifically, utilizing the Wasserstein distance from the optimal transport theory, this scheme is achieved by enforcing the observed image residual histogram as close as possible to the reference residual histogram. Furthermore, the RWRM unifies the residual Wasserstein regularization and image prior regularization to improve image restoration performance. The robustness of parameter selection in the RWRM makes the proposed algorithms easier to implement. Finally, extensive experiments have confirmed that our RWRM applied to Gaussian denoising and non-blind deconvolution is effective.

Keywords: Residual histogram, Wasserstein distance, Gaussian denoising, non-blind deconvolution.

Mathematics Subject Classification: 68U10.

Citation: Ruiqiang He, Xiangchu Feng, Xiaolong Zhu, Hua Huang, Bingzhe Wei. RWRM: Residual Wasserstein regularization model for image restoration. Inverse Problems and Imaging, 2021, 15 (6) : 1307-1332. doi: 10.3934/ipi.2020069

References:

[1]	E. J Candes and T Tao, Near-optimal signal recovery from random projections: Universal encoding strategies?, IEEE Transactions on Information Theory, 52 (2006), 5406-5425. doi: 10.1109/TIT.2006.885507.
[2]	A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.
[3]	T. S. Cho, C. L. Zitnick, N. Joshi, S. B. Kang, R. Szeliski and W. T. Freeman, Image restoration by matching gradient distributions, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 683-694. doi: 10.1109/TPAMI.2011.166.
[4]	P. L Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling and Simulation, 4 (2005), 1168-1200. doi: 10.1137/050626090.
[5]	K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Transactions on image processing, 16 (2007), 2080-2095. doi: 10.1109/TIP.2007.901238.
[6]	W. Dong, L. Zhang, G. Shi and X. Li, Nonlocally centralized sparse representation for image restoration, IEEE Transactions on Image Processing, 22 (2013), 1620-1630. doi: 10.1109/TIP.2012.2235847.
[7]	D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.
[8]	M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Transactions on Image processing, 15 (2006), 3736-3745. doi: 10.1109/TIP.2006.881969.
[9]	M. El Gheche, J.-F. Aujol, Y. Berthoumieu and C.-A. Deledalle, Texture reconstruction guided by the histogram of a high-resolution patch, IEEE Trans. Image Process, 26 (2017), 549-560. doi: 10.1109/TIP.2016.2627812.
[10]	W. Feller, An Introduction to Probability Theory and Its Applications Ⅱ, John Wiley & Sons, 1968.
[11]	A. L. Gibbs, Convergence in the wasserstein metric for markov chain monte carlo algorithms with applications to image restoration, Stochastic Models, 20 (2004), 473-492. doi: 10.1081/STM-200033117.
[12]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592.
[13]	R. C. Gonzalez, R. E Woods and S. L Eddins, Digital Image Processing Using MATLAB, Prentice Hall Press, 2007.
[14]	S. Harmeling, C. J. Schuler and H. C. Burger, Image denoising: Can plain neural networks compete with bm3d?, In IEEE Conference on Computer Vision and Pattern Recognition, (2012), 2392-2399.
[15]	R. He, X. Feng, W. Wang, X. Zhu and Ch unyu Yang, W-ldmm: A wasserstein driven low-dimensional manifold model for noisy image restoration, Neurocomputing, 371 (2020), 108-123. doi: 10.1016/j.neucom.2019.08.088.
[16]	D. J. Heeger and J. R. Bergen, Pyramid-based texture analysis/synthesis, In International Conference on Image Processing, 1995. Proceedings, (1995), 229-238.
[17]	V. Jain and H. Sebastian Seung, Natural image denoising with convolutional networks, In International Conference on Neural Information Processing Systems, (2008), 769-776.
[18]	D. Krishnan and R. Fergus, Fast image deconvolution using hyper-laplacian priors, In International Conference on Neural Information Processing Systems, (2009), 1033-1041.
[19]	X. Lan, S. Roth, D. Huttenlocher and M. J Black, Efficient belief propagation with learned higher-order markov random fields, In European Conference on Computer Vision, pages 269-282. Springer, 2006. doi: 10.1007/11744047_21.
[20]	S. Z. Li, Markov Random Field Modeling in Image Analysis, Springer-Verlag London, Ltd., London, 2009.
[21]	Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, Journal of Scientific Computing, 42 (2010), 185-197. doi: 10.1007/s10915-009-9320-2.
[22]	J. Mairal, M. Elad and G. Sapiro, Sparse representation for color image restoration, IEEE Transactions on Image Processing, 17 (2008), 53-69. doi: 10.1109/TIP.2007.911828.
[23]	S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.
[24]	X. Mei, W. Dong, B. G. Hu and S. Lyu, Unihist: A unified framework for image restoration with marginal histogram constraints, In Computer Vision and Pattern Recognition, pages 3753-3761, 2015.
[25]	S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling and Simulation, 4 (2005), 460-489. doi: 10.1137/040605412.
[26]	O. Pele and M. Werman, Fast and robust earth mover's distances, In IEEE International Conference on Computer Vision, (2010), 460-467. doi: 10.1109/ICCV.2009.5459199.
[27]	G. Peyré, J. Fadili and J. Rabin, Wasserstein active contours, In IEEE International Conference on Image Processing, (2013), 2541-2544.
[28]	J. Portilla, V. Strela, M. J. Wainwright and E. P. Simoncelli, Image denoising using scale mixtures of gaussians in the wavelet domain, IEEE Transactions on Image Processing, 12 (2003), 1338-1351. doi: 10.1109/TIP.2003.818640.
[29]	J. Rabin and G. Peyré, Wasserstein regularization of imaging problem, In IEEE International Conference on Image Processing, (2011), 1541-1544, . doi: 10.1109/ICIP.2011.6115740.
[30]	A. Rajwade, A. Rangarajan and A. Banerjee, Image denoising using the higher order singular value decomposition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2012), 849-862.
[31]	W. H. Richardson, Bayesian-based iterative method of image restoration, Journal of the Optical Society of America, 62 (1972), 55-59. doi: 10.1364/JOSA.62.000055.
[32]	Y. Romano, M. Protter and M. Elad, Single image interpolation via adaptive nonlocal sparsity-based modeling, IEEE Transactions on Image Processing, 23 (2014), 3085-3098. doi: 10.1109/TIP.2014.2325774.
[33]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Eleventh International Conference of the Center for Nonlinear Studies on Experimental Mathematics: Computational Issues in Nonlinear Science. Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[34]	O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, 2009.
[35]	U. Schmidt, Q. Gao and S. Roth, A generative perspective on mrfs in low-level vision, In Computer Vision and Pattern Recognition, 2010, pages 1751-1758. doi: 10.1109/CVPR.2010.5539844.
[36]	O. Stanley, Z. Shi and W. Zhu, Low dimensional manifold model for image processing, SIAM Journal on Imaging Sciences, 10 (2017), 1669-1690. doi: 10.1137/16M1058686.
[37]	D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), 165-187. doi: 10.1088/0266-5611/19/6/059.
[38]	K. Suzuki, I. Horiba and N. Sugie, Efficient approximation of neural filters for removing quantum noise from images, IEEE Transactions on Signal Processing, 50 (2002), 1787-1799. doi: 10.1109/TSP.2002.1011218.
[39]	P. Swoboda and C. Schnorr, Convex variational image restoration with histogram priors, SIAM Journal on Imaging Sciences, 6 (2013), 1719-1735. doi: 10.1137/120897535.
[40]	G. Tartavel, G. Peyré and Y. Gousseau, Wasserstein loss for image synthesis and restoration, SIAM Journal on Imaging Sciences, 9 (2016), 1726-1755. doi: 10.1137/16M1067494.
[41]	F. Thaler, K. Hammernik, C. Payer, M. Urschler and D. Stern, Sparse-view ct reconstruction using wasserstein gans, 2018, pages 75-82. doi: 10.1007/978-3-030-00129-2_9.
[42]	M. Vauhkonen, D. Vadasz, P. A. Karjalainen, E. Somersalo and J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Transactions on Medical Imaging, 17 (1998), 285-93. doi: 10.1109/42.700740.
[43]	C. Villani, Optimal Transport: Old and New, volume 338, Springer-Verlag, Berlin, 2009 doi: 10.1007/978-3-540-71050-9.
[44]	Y. Weiss and W. T. Freeman, What makes a good model of natural images?, In 2007 IEEE Conference on Computer Vision and Pattern Recognition, (2007) pages 1-8. doi: 10.1109/CVPR.2007.383092.
[45]	O. J. Woodford, C. Rother and V. Kolmogorov, A global perspective on map inference for low-level vision, In IEEE International Conference on Computer Vision, 2009, pages 2319-2326. doi: 10.1109/ICCV.2009.5459434.
[46]	F. Wu, B. Wang, D. Cui and L. Li, Single image super-resolution based on wasserstein gans, Chinese Control Conference (CCC), 2018. doi: 10.23919/ChiCC.2018.8484039.
[47]	Q. Yang, P. Yan, Y. Zhang, H. Yu, Y. Shi, X. Mou, M. K. Kalra, Y. Zhang, L. Sun and G. Wang, Low-dose ct image denoising using a generative adversarial network with wasserstein distance and perceptual loss, IEEE Transactions on Medical Imaging, 37 (2018), 1348-1357. doi: 10.1109/TMI.2018.2827462.
[48]	K. Zhang, W. Zuo, S. Gu and L. Zhang, Learning deep cnn denoiser prior for image restoration, 2017, pages 2808-2817. doi: 10.1109/CVPR.2017.300.
[49]	W. Zuo, L. Zhang, C. Song, D. Zhang and H. Gao, Gradient histogram estimation and preservation for texture enhanced image denoising, IEEE Transactions on Image Processing, 23 (2014), 2459-2472. doi: 10.1109/TIP.2014.2316423.

show all references

References:

[1]	E. J Candes and T Tao, Near-optimal signal recovery from random projections: Universal encoding strategies?, IEEE Transactions on Information Theory, 52 (2006), 5406-5425. doi: 10.1109/TIT.2006.885507.
[2]	A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.
[3]	T. S. Cho, C. L. Zitnick, N. Joshi, S. B. Kang, R. Szeliski and W. T. Freeman, Image restoration by matching gradient distributions, IEEE Transactions on Pattern Analysis and Machine Intelligence, 34 (2012), 683-694. doi: 10.1109/TPAMI.2011.166.
[4]	P. L Combettes and V. Wajs, Signal recovery by proximal forward-backward splitting, Multiscale Modeling and Simulation, 4 (2005), 1168-1200. doi: 10.1137/050626090.
[5]	K. Dabov, A. Foi, V. Katkovnik and K. Egiazarian, Image denoising by sparse 3-d transform-domain collaborative filtering, IEEE Transactions on image processing, 16 (2007), 2080-2095. doi: 10.1109/TIP.2007.901238.
[6]	W. Dong, L. Zhang, G. Shi and X. Li, Nonlocally centralized sparse representation for image restoration, IEEE Transactions on Image Processing, 22 (2013), 1620-1630. doi: 10.1109/TIP.2012.2235847.
[7]	D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306. doi: 10.1109/TIT.2006.871582.
[8]	M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Transactions on Image processing, 15 (2006), 3736-3745. doi: 10.1109/TIP.2006.881969.
[9]	M. El Gheche, J.-F. Aujol, Y. Berthoumieu and C.-A. Deledalle, Texture reconstruction guided by the histogram of a high-resolution patch, IEEE Trans. Image Process, 26 (2017), 549-560. doi: 10.1109/TIP.2016.2627812.
[10]	W. Feller, An Introduction to Probability Theory and Its Applications Ⅱ, John Wiley & Sons, 1968.
[11]	A. L. Gibbs, Convergence in the wasserstein metric for markov chain monte carlo algorithms with applications to image restoration, Stochastic Models, 20 (2004), 473-492. doi: 10.1081/STM-200033117.
[12]	G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Modeling and Simulation, 7 (2008), 1005-1028. doi: 10.1137/070698592.
[13]	R. C. Gonzalez, R. E Woods and S. L Eddins, Digital Image Processing Using MATLAB, Prentice Hall Press, 2007.
[14]	S. Harmeling, C. J. Schuler and H. C. Burger, Image denoising: Can plain neural networks compete with bm3d?, In IEEE Conference on Computer Vision and Pattern Recognition, (2012), 2392-2399.
[15]	R. He, X. Feng, W. Wang, X. Zhu and Ch unyu Yang, W-ldmm: A wasserstein driven low-dimensional manifold model for noisy image restoration, Neurocomputing, 371 (2020), 108-123. doi: 10.1016/j.neucom.2019.08.088.
[16]	D. J. Heeger and J. R. Bergen, Pyramid-based texture analysis/synthesis, In International Conference on Image Processing, 1995. Proceedings, (1995), 229-238.
[17]	V. Jain and H. Sebastian Seung, Natural image denoising with convolutional networks, In International Conference on Neural Information Processing Systems, (2008), 769-776.
[18]	D. Krishnan and R. Fergus, Fast image deconvolution using hyper-laplacian priors, In International Conference on Neural Information Processing Systems, (2009), 1033-1041.
[19]	X. Lan, S. Roth, D. Huttenlocher and M. J Black, Efficient belief propagation with learned higher-order markov random fields, In European Conference on Computer Vision, pages 269-282. Springer, 2006. doi: 10.1007/11744047_21.
[20]	S. Z. Li, Markov Random Field Modeling in Image Analysis, Springer-Verlag London, Ltd., London, 2009.
[21]	Y. Lou, X. Zhang, S. Osher and A. Bertozzi, Image recovery via nonlocal operators, Journal of Scientific Computing, 42 (2010), 185-197. doi: 10.1007/s10915-009-9320-2.
[22]	J. Mairal, M. Elad and G. Sapiro, Sparse representation for color image restoration, IEEE Transactions on Image Processing, 17 (2008), 53-69. doi: 10.1109/TIP.2007.911828.
[23]	S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.
[24]	X. Mei, W. Dong, B. G. Hu and S. Lyu, Unihist: A unified framework for image restoration with marginal histogram constraints, In Computer Vision and Pattern Recognition, pages 3753-3761, 2015.
[25]	S. Osher, M. Burger, D. Goldfarb, J. Xu and W. Yin, An iterative regularization method for total variation-based image restoration, Multiscale Modeling and Simulation, 4 (2005), 460-489. doi: 10.1137/040605412.
[26]	O. Pele and M. Werman, Fast and robust earth mover's distances, In IEEE International Conference on Computer Vision, (2010), 460-467. doi: 10.1109/ICCV.2009.5459199.
[27]	G. Peyré, J. Fadili and J. Rabin, Wasserstein active contours, In IEEE International Conference on Image Processing, (2013), 2541-2544.
[28]	J. Portilla, V. Strela, M. J. Wainwright and E. P. Simoncelli, Image denoising using scale mixtures of gaussians in the wavelet domain, IEEE Transactions on Image Processing, 12 (2003), 1338-1351. doi: 10.1109/TIP.2003.818640.
[29]	J. Rabin and G. Peyré, Wasserstein regularization of imaging problem, In IEEE International Conference on Image Processing, (2011), 1541-1544, . doi: 10.1109/ICIP.2011.6115740.
[30]	A. Rajwade, A. Rangarajan and A. Banerjee, Image denoising using the higher order singular value decomposition, IEEE Transactions on Pattern Analysis and Machine Intelligence, 35 (2012), 849-862.
[31]	W. H. Richardson, Bayesian-based iterative method of image restoration, Journal of the Optical Society of America, 62 (1972), 55-59. doi: 10.1364/JOSA.62.000055.
[32]	Y. Romano, M. Protter and M. Elad, Single image interpolation via adaptive nonlocal sparsity-based modeling, IEEE Transactions on Image Processing, 23 (2014), 3085-3098. doi: 10.1109/TIP.2014.2325774.
[33]	L. I. Rudin, S. Osher and E. Fatemi, Nonlinear total variation based noise removal algorithms, Eleventh International Conference of the Center for Nonlinear Studies on Experimental Mathematics: Computational Issues in Nonlinear Science. Phys. D, 60 (1992), 259-268. doi: 10.1016/0167-2789(92)90242-F.
[34]	O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, 2009.
[35]	U. Schmidt, Q. Gao and S. Roth, A generative perspective on mrfs in low-level vision, In Computer Vision and Pattern Recognition, 2010, pages 1751-1758. doi: 10.1109/CVPR.2010.5539844.
[36]	O. Stanley, Z. Shi and W. Zhu, Low dimensional manifold model for image processing, SIAM Journal on Imaging Sciences, 10 (2017), 1669-1690. doi: 10.1137/16M1058686.
[37]	D. Strong and T. Chan, Edge-preserving and scale-dependent properties of total variation regularization, Inverse Problems, 19 (2003), 165-187. doi: 10.1088/0266-5611/19/6/059.
[38]	K. Suzuki, I. Horiba and N. Sugie, Efficient approximation of neural filters for removing quantum noise from images, IEEE Transactions on Signal Processing, 50 (2002), 1787-1799. doi: 10.1109/TSP.2002.1011218.
[39]	P. Swoboda and C. Schnorr, Convex variational image restoration with histogram priors, SIAM Journal on Imaging Sciences, 6 (2013), 1719-1735. doi: 10.1137/120897535.
[40]	G. Tartavel, G. Peyré and Y. Gousseau, Wasserstein loss for image synthesis and restoration, SIAM Journal on Imaging Sciences, 9 (2016), 1726-1755. doi: 10.1137/16M1067494.
[41]	F. Thaler, K. Hammernik, C. Payer, M. Urschler and D. Stern, Sparse-view ct reconstruction using wasserstein gans, 2018, pages 75-82. doi: 10.1007/978-3-030-00129-2_9.
[42]	M. Vauhkonen, D. Vadasz, P. A. Karjalainen, E. Somersalo and J. P. Kaipio, Tikhonov regularization and prior information in electrical impedance tomography, IEEE Transactions on Medical Imaging, 17 (1998), 285-93. doi: 10.1109/42.700740.
[43]	C. Villani, Optimal Transport: Old and New, volume 338, Springer-Verlag, Berlin, 2009 doi: 10.1007/978-3-540-71050-9.
[44]	Y. Weiss and W. T. Freeman, What makes a good model of natural images?, In 2007 IEEE Conference on Computer Vision and Pattern Recognition, (2007) pages 1-8. doi: 10.1109/CVPR.2007.383092.
[45]	O. J. Woodford, C. Rother and V. Kolmogorov, A global perspective on map inference for low-level vision, In IEEE International Conference on Computer Vision, 2009, pages 2319-2326. doi: 10.1109/ICCV.2009.5459434.
[46]	F. Wu, B. Wang, D. Cui and L. Li, Single image super-resolution based on wasserstein gans, Chinese Control Conference (CCC), 2018. doi: 10.23919/ChiCC.2018.8484039.
[47]	Q. Yang, P. Yan, Y. Zhang, H. Yu, Y. Shi, X. Mou, M. K. Kalra, Y. Zhang, L. Sun and G. Wang, Low-dose ct image denoising using a generative adversarial network with wasserstein distance and perceptual loss, IEEE Transactions on Medical Imaging, 37 (2018), 1348-1357. doi: 10.1109/TMI.2018.2827462.
[48]	K. Zhang, W. Zuo, S. Gu and L. Zhang, Learning deep cnn denoiser prior for image restoration, 2017, pages 2808-2817. doi: 10.1109/CVPR.2017.300.
[49]	W. Zuo, L. Zhang, C. Song, D. Zhang and H. Gao, Gradient histogram estimation and preservation for texture enhanced image denoising, IEEE Transactions on Image Processing, 23 (2014), 2459-2472. doi: 10.1109/TIP.2014.2316423.

Figure 11. The curve graphs of parameter robustness. One can see that $ \lambda $ has a strong influence on PSNR in TV-L2 ($ \beta = 0 $), while in the RWRM ($ \beta \ne 0 $) PSNR is little affected by $ \lambda $. Thus, the parameters in our RWRM are robust

Images	Airfield	Lena	Peppers	Plane	Couple	Girl	Lake	Boat	Lolo	Martha	Avg.
TV-L2	25.55	29.23	28.70	28.02	31.49	29.91	27.19	27.92	25.52	29.44	28.30
TV-L2	0.6689	0.7410	0.7233	0.7198	0.8300	0.7502	0.7129	0.7136	0.6212	0.7458	0.7227
RWRM	26.55	30.16	30.47	29.60	32.28	31.02	27.90	28.71	26.90	30.74	29.43
RWRM	0.7008	0.8183	0.8070	0.8361	0.8515	0.8244	0.7635	0.7990	0.7500	0.8195	0.7970

$ \sigma $	Methods	1	2	3	4	5	6	7	8	Avg.
$ \sigma=60 $	BM3D	24.77	23.33	29.95	23.36	27.43	24.80	23.27	27.81	25.59
	BM3D	0.488	0.472	0.793	0.455	0.604	0.549	0.519	0.687	0.571
	NCSR	24.62	23.25	29.57	23.23	27.05	24.66	23.04	27.61	25.38
	NCSR	0.466	0.460	0.804	0.440	0.589	0.539	0.494	0.690	0.560
	GHP	24.74	23.32	29.49	23.32	27.18	24.71	23.17	27.59	25.44
	GHP	0.486	0.479	0.790	0.458	0.598	0.548	0.512	0.683	0.569
	RWRM	24.75	23.25	29.58	23.27	27.48	24.75	23.00	27.88	25.50
	RWRM	0.490	0.481	0.793	0.463	0.602	0.535	0.505	0.691	0.570
$ \sigma=70 $	BM3D	24.37	22.93	29.26	23.03	26.97	24.43	22.84	27.34	25.15
	BM3D	0.461	0.444	0.777	0.432	0.586	0.530	0.493	0.670	0.549
	NCSR	24.24	22.83	28.99	22.85	26.60	24.25	22.56	27.13	24.93
	NCSR	0.439	0.426	0.794	0.410	0.572	0.517	0.464	0.678	0.538
	GHP	24.35	22.89	28.93	22.93	26.75	24.31	22.65	27.12	24.99
	GHP	0.461	0.446	0.775	0.428	0.582	0.524	0.480	0.667	0.545
	RWRM	24.36	22.90	29.07	22.94	27.10	24.42	22.55	27.51	25.11
	RWRM	0.464	0.451	0.788	0.437	0.588	0.518	0.481	0.681	0.551
$ \sigma=80 $	BM3D	24.04	22.59	28.66	22.75	26.61	24.11	22.47	26.92	24.77
	BM3D	0.441	0.421	0.763	0.414	0.572	0.513	0.472	0.655	0.531
	NCSR	23.92	22.47	28.52	22.55	26.22	23.91	22.17	26.70	24.56
	NCSR	0.419	0.400	0.785	0.388	0.559	0.501	0.441	0.667	0.520
	GHP	24.02	22.52	28.46	22.61	26.36	23.96	22.23	26.66	24.60
	GHP	0.440	0.421	0.763	0.408	0.567	0.508	0.457	0.651	0.527
	RWRM	24.10	22.53	28.66	22.67	26.80	24.14	22.17	27.18	24.78
	RWRM	0.445	0.428	0.782	0.416	0.576	0.504	0.458	0.672	0.535
$ \sigma=90 $	BM3D	23.75	22.32	28.12	22.49	26.30	23.81	22.15	26.48	24.43
	BM3D	0.423	0.403	0.751	0.397	0.559	0.498	0.454	0.640	0.516
	NCSR	23.65	22.16	28.10	22.29	25.88	23.62	21.82	26.30	24.23
	NCSR	0.404	0.379	0.778	0.372	0.548	0.489	0.422	0.658	0.506
	GHP	23.68	22.15	27.99	22.29	25.97	23.65	21.72	26.21	24.21
	GHP	0.424	0.398	0.750	0.390	0.554	0.493	0.430	0.636	0.509
	RWRM	23.84	22.21	28.27	22.40	26.53	23.90	21.87	26.87	24.49
	RWRM	0.435	0.410	0.775	0.402	0.567	0.493	0.437	0.666	0.523
$ \sigma=100 $	BM3D	23.48	22.07	27.69	22.27	26.00	23.55	21.85	26.16	24.13
	BM3D	0.409	0.388	0.738	0.387	0.548	0.487	0.437	0.627	0.503
	NCSR	23.43	21.89	27.71	22.05	25.58	23.35	21.50	25.92	23.93
	NCSR	0.392	0.362	0.771	0.360	0.539	0.479	0.406	0.649	0.495
	GHP	23.35	21.71	27.47	21.93	25.56	23.26	21.22	25.69	23.77
	GHP	0.408	0.374	0.735	0.374	0.537	0.476	0.407	0.616	0.491
	RWRM	23.64	22.00	28.00	22.20	26.32	23.67	21.60	26.61	24.26
	RWRM	0.421	0.392	0.766	0.385	0.560	0.484	0.419	0.660	0.511

$ \sigma $	Methods	1	2	3	4	5	6	7	8
$ \sigma=40 $	RWRM	25.76	24.38	30.74	24.30	28.50	25.71	24.28	28.88
	RWRM	0.5612	0.5609	0.8095	0.5510	0.6468	0.5914	0.5947	0.7204
	LDCNN	26.14	24.83	31.58	24.81	28.66	26.05	24.92	29.24
	LDCNN	0.6104	0.6141	0.8342	0.6043	0.6625	0.6419	0.6457	0.7424
$ \sigma=50 $	RWRM	25.06	23.66	30.11	23.64	27.92	25.15	23.51	28.31
	RWRM	0.4914	0.4893	0.8013	0.4781	0.6176	0.5570	0.5402	0.7012
	LDCNN	25.47	24.09	30.90	24.12	28.07	25.45	24.17	28.55
	LDCNN	0.5522	0.5520	0.8178	0.5415	0.6330	0.5994	0.5894	0.7132

Images	Plane	Goldhill	Couple	Peppers	Martha	Boat	Girl	Lena	Bacteria	Brain	Avg.
TV-L2	24.07	25.89	27.81	25.70	24.89	24.43	27.12	24.19	27.77	28.20	26.01
RWRM	24.80	26.36	28.30	26.57	25.89	24.94	27.89	24.82	28.37	28.69	26.66

Images	1	2	3	4	5	6	7	8	Avg.
NL-$ {{\rm{H}}^{\rm{1}}} $	25.64	30.17	26.48	27.11	26.03	26.36	26.47	25.30	26.70
NL-$ {{\rm{H}}^{\rm{1}}} $	0.7800	0.8218	0.7982	0.7600	0.8481	0.8044	0.8785	0.7999	0.8114
NL-TV	25.83	29.91	26.42	26.95	26.74	26.41	27.40	25.50	26.90
NL-TV	0.7752	0.8209	0.7944	0.7649	0.8504	0.8048	0.8631	0.7972	0.8089
RWRM	26.24	30.95	26.83	27.88	27.98	26.90	28.24	26.32	27.67
RWRM	0.7993	0.8358	0.8096	0.7800	0.8707	0.8148	0.8599	0.8217	0.8240

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American Institute of Mathematical Sciences

RWRM: Residual Wasserstein regularization model for image restoration

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