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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 & 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.  Google Scholar

[2]

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306.  doi: 10.1109/TIT.2006.871582.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

W. Feller, An Introduction to Probability Theory and Its Applications Ⅱ, John Wiley & Sons, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13] R. C. Gonzalez, R. E Woods and S. L Eddins, Digital Image Processing Using MATLAB, Prentice Hall Press, 2007.   Google Scholar
[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. Google Scholar

[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.  Google Scholar

[16]

D. J. Heeger and J. R. Bergen, Pyramid-based texture analysis/synthesis, In International Conference on Image Processing, 1995. Proceedings, (1995), 229-238. Google Scholar

[17]

V. Jain and H. Sebastian Seung, Natural image denoising with convolutional networks, In International Conference on Neural Information Processing Systems, (2008), 769-776. Google Scholar

[18]

D. Krishnan and R. Fergus, Fast image deconvolution using hyper-laplacian priors, In International Conference on Neural Information Processing Systems, (2009), 1033-1041. Google Scholar

[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.  Google Scholar

[20]

S. Z. Li, Markov Random Field Modeling in Image Analysis, Springer-Verlag London, Ltd., London, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.   Google Scholar
[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Peyré, J. Fadili and J. Rabin, Wasserstein active contours, In IEEE International Conference on Image Processing, (2013), 2541-2544. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

C. Villani, Optimal Transport: Old and New, volume 338, Springer-Verlag, Berlin, 2009 doi: 10.1007/978-3-540-71050-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[2]

A. Chambolle, An algorithm for total variation minimization and applications, Journal of Mathematical Imaging and Vision, 20 (2004), 89-97.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

D. L. Donoho, Compressed sensing, IEEE Transactions on Information Theory, 52 (2006), 1289-1306.  doi: 10.1109/TIT.2006.871582.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

W. Feller, An Introduction to Probability Theory and Its Applications Ⅱ, John Wiley & Sons, 1968.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13] R. C. Gonzalez, R. E Woods and S. L Eddins, Digital Image Processing Using MATLAB, Prentice Hall Press, 2007.   Google Scholar
[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. Google Scholar

[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.  Google Scholar

[16]

D. J. Heeger and J. R. Bergen, Pyramid-based texture analysis/synthesis, In International Conference on Image Processing, 1995. Proceedings, (1995), 229-238. Google Scholar

[17]

V. Jain and H. Sebastian Seung, Natural image denoising with convolutional networks, In International Conference on Neural Information Processing Systems, (2008), 769-776. Google Scholar

[18]

D. Krishnan and R. Fergus, Fast image deconvolution using hyper-laplacian priors, In International Conference on Neural Information Processing Systems, (2009), 1033-1041. Google Scholar

[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.  Google Scholar

[20]

S. Z. Li, Markov Random Field Modeling in Image Analysis, Springer-Verlag London, Ltd., London, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23] S. Mallat, A Wavelet Tour of Signal Processing, Academic Press, Inc., San Diego, CA, 1998.   Google Scholar
[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. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Peyré, J. Fadili and J. Rabin, Wasserstein active contours, In IEEE International Conference on Image Processing, (2013), 2541-2544. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[34]

O. Scherzer, M. Grasmair, H. Grossauer, M. Haltmeier and F. Lenzen, Variational Methods in Imaging, Springer, 2009.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

C. Villani, Optimal Transport: Old and New, volume 338, Springer-Verlag, Berlin, 2009 doi: 10.1007/978-3-540-71050-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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 $. Thus, the parameters in our RWRM are robust
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Figure 1.  Ten test images for Gaussian denoising. From left to right, they are Airfield, Lena, Peppers, Plane, Couple, Girl, Lake, Boat, Loco and Martha respectively
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Figure 2.  TV-L2 denoising results on noisy image ($ \sigma = 25 $) with different regularization parameters $ \lambda $. One can find that the TV-L2 model with a moderate $ \lambda \in [1500,1800] $ can optimize the denoising effect
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$ \beta = 0 $), the effect of the parameter $ \lambda $ on PSNR. (b) In our RWRM, the effect of Wasserstein regularization parameter $ \beta $ on PSNR in the cases of three different $ \lambda $">Figure 3.  Relationships between PSNR and model parameters taking the image "Boat". (a) In the TV-L2 model ($ \beta = 0 $), the effect of the parameter $ \lambda $ on PSNR. (b) In our RWRM, the effect of Wasserstein regularization parameter $ \beta $ on PSNR in the cases of three different $ \lambda $
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Figure 4.  PSNR comparison curve on the ten images about the proposed RWRM ($ \beta \ne 0 $) and TV-L2 ($ \beta = 0 $) in Gaussian denoising
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Figure 5.  The distribution comparison for the final residuals of TV-L2 (green), RWRM (red), and the ground truth (blue). One can see that RWRM residual estimate is more accurate than TV-L2
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Figure 6.  Denoising results on the three images Peppers, Plane and Martha. Here (c) and (d) are the result images generated by the TV-L2 and our RWRM respectively. One can see that our approach clearly yields better visual results
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Figure 7.  Local magnification comparison of denoised images. One can see that the RWRM can recover better image edge details
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Figure 8.  Eight test images. From left to right, they are labeled as 1 to 8 respectively
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5], NCSR [6], GHP [49] and our RWRM on the test image 8. From the first to the third row, the Gaussian noise standard deviations of the corresponding noisy images are 60, 80, and 100, respectively">Figure 9.  The denoising output results (PSNR and SSIM) of BM3D [5], NCSR [6], GHP [49] and our RWRM on the test image 8. From the first to the third row, the Gaussian noise standard deviations of the corresponding noisy images are 60, 80, and 100, respectively
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48]">Figure 10.  The denoising effect of RWRM and learning-based method LDCNN [48]
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Figure 12.  Ten test images for non-blind deconvolution. From left to right, they are Plane, Glodhill, Couple, Peppers, Martha, Boat, Girl, Lena, Bacteria and Brain respectively
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Figure 13.  The graph of the PSNR with $ \lambda $ in the RWRM ($ \beta = 0 $) applied to non-blind deconvolution
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Figure 14.  The graph of the PSNR with $ \beta $ ($ \lambda = 1800 $) in the RWRM applied to non-blind deconvolution
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Figure 15.  In non-blind deconvolution, PSNR comparison curves of the TV-L2 ($ \beta = 0 $) and the RWRM ($ \beta \ne 0 $) on the ten test images
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Figure 16.  The results of non-blind deconvolution on the images Couple, Martha and Bacteria. Here (a) is the blurred and noisy images, (b) and (c) are the result images of the TV-L2 and the RWRM, respectively. One can see that the RWRM has obvious advantages in terms of both visual and numerical aspects
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Figure 17.  Local enlarged images on non-blind deconvolution. One can see that the RWRM can recover better edge information
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Figure 18.  Another eight test images for the non-blind deconvolution. From left to right, they are labeled as 1 to 8 respectively
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Figure 19.  Visual effect comparison of non-blind deconvolution by three methods NL-$ {{\rm{H}}^{\rm{1}}} $, NL-TV and the proposed RWRM
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Table 1.  Denoising PSNR comparison on the ten test images in Fig. 1: the TV-L2 model and our RWRM.
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
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
0.7008 0.8183 0.8070 0.8361 0.8515 0.8244 0.7635 0.7990 0.7500 0.8195 0.7970
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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
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
0.7008 0.8183 0.8070 0.8361 0.8515 0.8244 0.7635 0.7990 0.7500 0.8195 0.7970
Table 2.  PSNR (dB) and SSIM values of BM3D [5], NCSR [6], GHP [49] and our RWRM in Gaussian denoising
$ \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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
0.421 0.392 0.766 0.385 0.560 0.484 0.419 0.660 0.511
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$ \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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
0.421 0.392 0.766 0.385 0.560 0.484 0.419 0.660 0.511
Table 3.  Denoising PSNR and SSIM results of RWRM and learning-based method LDCNN [48]
$ \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
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
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
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
0.5522 0.5520 0.8178 0.5415 0.6330 0.5994 0.5894 0.7132
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$ \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
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
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
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
0.5522 0.5520 0.8178 0.5415 0.6330 0.5994 0.5894 0.7132
Table 4.  PSNR (dB) comparison of non-blind deconvolution on the ten test images: the TV-L2 model ($ \beta = 0 $) and the RWRM ($ \beta \ne 0 $).
Images Plane Goldhill Couple Peppers Martha Boat Girl Lena Bacteria Brain Avg.
TV-L2 24.07 25.89 27.81 25.70 24.89 24.43 27.12 24.19 27.77 28.20 26.01
RWRM 24.80 26.36 28.30 26.57 25.89 24.94 27.89 24.82 28.37 28.69 26.66
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Images Plane Goldhill Couple Peppers Martha Boat Girl Lena Bacteria Brain Avg.
TV-L2 24.07 25.89 27.81 25.70 24.89 24.43 27.12 24.19 27.77 28.20 26.01
RWRM 24.80 26.36 28.30 26.57 25.89 24.94 27.89 24.82 28.37 28.69 26.66
Table 5.  The non-blind deconvolution PSNR and SSIM result comparison of the proposed RWRM, NL-$ {{\rm{H}}^{\rm{1}}} $ and NL-TV
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
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
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
0.7993 0.8358 0.8096 0.7800 0.8707 0.8148 0.8599 0.8217 0.8240
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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
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
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
0.7993 0.8358 0.8096 0.7800 0.8707 0.8148 0.8599 0.8217 0.8240
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