doi: 10.3934/ipi.2021065
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Nonconvex regularization for blurred images with Cauchy noise

1. 

School of Mathematics and Statistics, Northeast Normal University, Changchun 130024, China

2. 

LCP, Institute of Applied Physics and Computational Mathematics, Beijing 100088, China

3. 

Department of Mathematics, The Chinese University of Hong Kong, Hong Kong 999077

* Corresponding author: Guoxi Ni (nijiusuo09@163.com)

Received  January 2021 Revised  August 2021 Early access October 2021

In this paper, we propose a nonconvex regularization model for images damaged by Cauchy noise and blur. This model is based on the method of the total variational proposed by Federica, Dong and Zeng [SIAM J. Imaging Sci.(2015)], where a variational approach for restoring blurred images with Cauchy noise is used. Here we consider the nonconvex regularization, namely a weighted difference of $ l_1 $-norm and $ l_2 $-norm coupled with wavelet frame, the alternating direction method of multiplier is carried out for this minimization problem, we describe the details of the algorithm and prove its convergence. Numerical experiments are tested by adding different levels of noise and blur, results show that our method can denoise and deblur the image better.

Citation: Xiao Ai, Guoxi Ni, Tieyong Zeng. Nonconvex regularization for blurred images with Cauchy noise. Inverse Problems & Imaging, doi: 10.3934/ipi.2021065
References:
[1]

A. Achim and E. Kuruoglu, Image denoising using bivariate -stable distributions in the complex wavelet domain, IEEE Signal Process Letters, 12 (2005), 17-20.  doi: 10.1109/LSP.2004.839692.  Google Scholar

[2]

H. C. Andrews and B. R. Hunt, Digital Image Restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977. Google Scholar

[3]

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2$^nd$ edition, Appl. Math. Sci., 147. Springer, New York, 2006.  Google Scholar

[4] A. Bovik, Handbook of Image and Video Processing, Academic Press, New York, 2000.   Google Scholar
[5]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1-122.  doi: 10.1561/9781601984616.  Google Scholar

[6]

J. F. CaiR. ChanL. Shen and Z. Shen, Convergence analysis of tight framelet approach for missing data recovery, Adv. Comput. Math., 31 (2009), 87-113.  doi: 10.1007/s10444-008-9084-5.  Google Scholar

[7]

J. F. CaiR. Chan and Z. Shen, Simultaneous cartoon and texture inpainting, Inverse Probl. Imaging, 4 (2010), 379-395.  doi: 10.3934/ipi.2010.4.379.  Google Scholar

[8]

M. C. Cai and X. Q. Jin, BCCB preconditioners for solving linear systems from delay differential equations, Comput. Math. Appl., 50 (2005), 281-288.  doi: 10.1016/j.camwa.2004.03.019.  Google Scholar

[9]

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

[10]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[11]

R. ChanY. Dong and M. Hintermuller, An effcient two-phase $L_1$-TV method for restoring blurred images with impulse noise, IEEE Trans. Image Process., 19 (2010), 1731-1739.  doi: 10.1109/TIP.2010.2045148.  Google Scholar

[12]

T. F. ChanG. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar

[13]

R. ChanH. Yang and T. Zeng, A two-stage image segmentation method for blurry images with Poisson or multiplicative gamma noise, SIAM J. Imaging Sci., 7 (2014), 98-127.  doi: 10.1137/130920241.  Google Scholar

[14]

R. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), 427-482.  doi: 10.1137/S0036144594276474.  Google Scholar

[15]

R. Chan and X. Q. Jin, An Introduction to Iterative Toeplitz Solvers. Fundamentals of Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898718850.  Google Scholar

[16]

Y. ChangS. KadabaP. Doerschuk and S. Gelfand, Image restoration using recursive Markov random field models driven by Cauchy distributed noise, IEEE Signal Process. Lett., 8 (2001), 65-66.   Google Scholar

[17]

L. Condat, A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479.  doi: 10.1007/s10957-012-0245-9.  Google Scholar

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I. DaubechiesG. Teschke and L. Vese, Iteratively solving linear inverse problems under general convex constraints, Inverse Probl. Imag., 1 (2007), 29-46.  doi: 10.3934/ipi.2007.1.29.  Google Scholar

[19]

N. DeyL. Blanc-FeraudC. ZimmerP. RouxZ. KamJ. Olivo-Marin and J. Zerubia, Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution, Microsc. Res. Tech., 69 (2006), 260-266.  doi: 10.1002/jemt.20294.  Google Scholar

[20]

B. DongH. JiZ. W. Shen and Y. H. Xu, Wavelet frame based blind image inpainting, Appl. Comput. Harmon. Anal., 32 (2012), 268-279.  doi: 10.1016/j.acha.2011.06.001.  Google Scholar

[21]

Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imaging Sci., 6 (2013), 1598-1625.  doi: 10.1137/120870621.  Google Scholar

[22]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), 3736-3745.  doi: 10.1109/TIP.2006.881969.  Google Scholar

[23]

M. EladJ. StarckP. Querre and D. Donoho, Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Appl. Comput Harmon Anal., 19 (2005), 340-358.  doi: 10.1016/j.acha.2005.03.005.  Google Scholar

[24]

F. SciacchitanoY. Q. Dong and T. Y. Zeng, Variational approach for restoring blurred images with cauchy noise, SIAM J. Imaging Sci., 8 (2015), 1894-1922.  doi: 10.1137/140997816.  Google Scholar

[25]

M. Figueiredo and J. Bioucas-Dias, Restoration of poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

[26]

M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255.  Google Scholar

[27]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[28]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[29]

R. Gonzalez and R. Woods, Digital Image Processing, 3rd edition, Pearson, London, 2008. Google Scholar

[30]

G. Grimmett and D. Welsh, Oxford Science Publications, Oxford Science Publications, London, 1986. Google Scholar

[31]

Y.-M. HuangM. K. Ng and Y.-W. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593.  Google Scholar

[32]

M. Idan and J. Speyer, Cauchy estimation for linear scalar systems, IEEE Trans. Automat. Control, 55 (2010), 1329-1342.  doi: 10.1109/TAC.2010.2042009.  Google Scholar

[33]

E. KuruogluW. Fitzgerald and P. Rayner, Near optimal detection of signals in impulsive noise modeled with asymmetric alpha-stable distribution, IEEE Commun. Lett., 2 (1998), 282-284.   Google Scholar

[34]

T. LeR. Chartrand and T. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[35]

J. Liu, Y. Lou, G. Ni and T. Zeng, An image sharpening operator combined with framelet for image deblurring, Inverse Problems, 36 (2020), 29pp. doi: 10.1088/1361-6420/ab6df0.  Google Scholar

[36]

J. Liu, A. Ni and G. Ni, A nonconvex $l_1(l_1 - l_2)$ model for image restoration with impulse noise, J. Comput. Appl. Math., 378 (2020), 16pp. doi: 10.1016/j.cam.2020.112934.  Google Scholar

[37]

Y. F. Lou and M. Yan, Fast $L1-L2$ minimization via a proximal operator, J. Sci. Comput., 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.  Google Scholar

[38]

Y. F. Lou, S. Osher and J. Xin, Computational aspects of constrained L1-L2 minimization for compressive sensing, J. Infect. Dis., (2015), 169–180. Google Scholar

[39]

Y. LouT. ZengS. Osher and J. Xin, A weighted difference of anisotropic and isotropic total variation model for image processing, SIAM J. Imaging Sci., 8 (2015), 1798-1823.  doi: 10.1137/14098435X.  Google Scholar

[40]

J. MeiY. DongT. Huang and W. Yin, Cauchy noise removal by nonconvex ADMM with convergence guarantees, J. Sci Comput., 74 (2018), 743-766.  doi: 10.1007/s10915-017-0460-5.  Google Scholar

[41]

M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 90-120.   Google Scholar

[42]

Y. PengJ. ChenX. Xu and F. Pu, SAR images statistical modeling and classification based on the mixture of alpha-stable distributions, Remote Sens., 5 (2013), 2145-2163.  doi: 10.3390/rs5052145.  Google Scholar

[43]

N. PustelnikC. Chaux and J. Pesquet, Parallel proximal algorithm for image restoration using hybrid regularization, IEEE Trans. Image Process., 20 (2011), 2450-2462.  doi: 10.1109/TIP.2011.2128335.  Google Scholar

[44]

P. Reeves, A non-gaussian turbulence simulation, Technical Report AFFDL-TR-, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, (1969), 69–67. Google Scholar

[45]

L. RudinS. 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.  Google Scholar

[46]

S. SetzerG. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques, J. Visual Commun. Image Represent., 21 (2010), 193-199.  doi: 10.1016/j.jvcir.2009.10.006.  Google Scholar

[47]

J. StarckM. Elad and D. Donoho, Image decomposition via the combination of sparse representations and a variational approach, IEEE Trans. Image Process., 14 (2005), 1570-1582.  doi: 10.1109/TIP.2005.852206.  Google Scholar

[48]

T. WanN. Canagarajah and A. Achim, Segmentation of noisy colour images using Cauchy distribution in the complex wavelet domain, IET Image Process., 5 (2011), 159-170.   Google Scholar

[49]

C. Wu and X.-C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[50]

J. YangY. Zhang and W. Yin, An effcient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894.  Google Scholar

[51]

P. H. YinY. F. Lou and J. Xin, Minimization of $l_{1-2}$ for compressed sensing, SIAM J. Sci. Comput., 37 (2015), 536-563.  doi: 10.1137/140952363.  Google Scholar

[52]

W. ZhouA. BovikH. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.   Google Scholar

show all references

References:
[1]

A. Achim and E. Kuruoglu, Image denoising using bivariate -stable distributions in the complex wavelet domain, IEEE Signal Process Letters, 12 (2005), 17-20.  doi: 10.1109/LSP.2004.839692.  Google Scholar

[2]

H. C. Andrews and B. R. Hunt, Digital Image Restoration, Prentice-Hall, Englewood Cliffs, NJ, 1977. Google Scholar

[3]

G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing: Partial Differential Equations and the Calculus of Variations, 2$^nd$ edition, Appl. Math. Sci., 147. Springer, New York, 2006.  Google Scholar

[4] A. Bovik, Handbook of Image and Video Processing, Academic Press, New York, 2000.   Google Scholar
[5]

S. BoydN. ParikhE. ChuB. Peleato and J. Eckstein, Distributed optimization and statistical learning via the alternating direction method of multipliers, Found. Trends Mach. Learn., 3 (2010), 1-122.  doi: 10.1561/9781601984616.  Google Scholar

[6]

J. F. CaiR. ChanL. Shen and Z. Shen, Convergence analysis of tight framelet approach for missing data recovery, Adv. Comput. Math., 31 (2009), 87-113.  doi: 10.1007/s10444-008-9084-5.  Google Scholar

[7]

J. F. CaiR. Chan and Z. Shen, Simultaneous cartoon and texture inpainting, Inverse Probl. Imaging, 4 (2010), 379-395.  doi: 10.3934/ipi.2010.4.379.  Google Scholar

[8]

M. C. Cai and X. Q. Jin, BCCB preconditioners for solving linear systems from delay differential equations, Comput. Math. Appl., 50 (2005), 281-288.  doi: 10.1016/j.camwa.2004.03.019.  Google Scholar

[9]

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

[10]

A. Chambolle and T. Pock, A first-order primal-dual algorithm for convex problems with applications to imaging, J. Math. Imaging Vision, 40 (2011), 120-145.  doi: 10.1007/s10851-010-0251-1.  Google Scholar

[11]

R. ChanY. Dong and M. Hintermuller, An effcient two-phase $L_1$-TV method for restoring blurred images with impulse noise, IEEE Trans. Image Process., 19 (2010), 1731-1739.  doi: 10.1109/TIP.2010.2045148.  Google Scholar

[12]

T. F. ChanG. H. Golub and P. Mulet, A nonlinear primal-dual method for total variation-based image restoration, SIAM J. Sci. Comput., 20 (1999), 1964-1977.  doi: 10.1137/S1064827596299767.  Google Scholar

[13]

R. ChanH. Yang and T. Zeng, A two-stage image segmentation method for blurry images with Poisson or multiplicative gamma noise, SIAM J. Imaging Sci., 7 (2014), 98-127.  doi: 10.1137/130920241.  Google Scholar

[14]

R. Chan and M. K. Ng, Conjugate gradient methods for Toeplitz systems, SIAM Rev., 38 (1996), 427-482.  doi: 10.1137/S0036144594276474.  Google Scholar

[15]

R. Chan and X. Q. Jin, An Introduction to Iterative Toeplitz Solvers. Fundamentals of Algorithms, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2007. doi: 10.1137/1.9780898718850.  Google Scholar

[16]

Y. ChangS. KadabaP. Doerschuk and S. Gelfand, Image restoration using recursive Markov random field models driven by Cauchy distributed noise, IEEE Signal Process. Lett., 8 (2001), 65-66.   Google Scholar

[17]

L. Condat, A primal-dual splitting method for convex optimization involving Lipschitzian, proximable and linear composite terms, J. Optim. Theory Appl., 158 (2013), 460-479.  doi: 10.1007/s10957-012-0245-9.  Google Scholar

[18]

I. DaubechiesG. Teschke and L. Vese, Iteratively solving linear inverse problems under general convex constraints, Inverse Probl. Imag., 1 (2007), 29-46.  doi: 10.3934/ipi.2007.1.29.  Google Scholar

[19]

N. DeyL. Blanc-FeraudC. ZimmerP. RouxZ. KamJ. Olivo-Marin and J. Zerubia, Richardson-Lucy algorithm with total variation regularization for 3D confocal microscope deconvolution, Microsc. Res. Tech., 69 (2006), 260-266.  doi: 10.1002/jemt.20294.  Google Scholar

[20]

B. DongH. JiZ. W. Shen and Y. H. Xu, Wavelet frame based blind image inpainting, Appl. Comput. Harmon. Anal., 32 (2012), 268-279.  doi: 10.1016/j.acha.2011.06.001.  Google Scholar

[21]

Y. Dong and T. Zeng, A convex variational model for restoring blurred images with multiplicative noise, SIAM J. Imaging Sci., 6 (2013), 1598-1625.  doi: 10.1137/120870621.  Google Scholar

[22]

M. Elad and M. Aharon, Image denoising via sparse and redundant representations over learned dictionaries, IEEE Trans. Image Process., 15 (2006), 3736-3745.  doi: 10.1109/TIP.2006.881969.  Google Scholar

[23]

M. EladJ. StarckP. Querre and D. Donoho, Simultaneous cartoon and texture image inpainting using morphological component analysis (MCA), Appl. Comput Harmon Anal., 19 (2005), 340-358.  doi: 10.1016/j.acha.2005.03.005.  Google Scholar

[24]

F. SciacchitanoY. Q. Dong and T. Y. Zeng, Variational approach for restoring blurred images with cauchy noise, SIAM J. Imaging Sci., 8 (2015), 1894-1922.  doi: 10.1137/140997816.  Google Scholar

[25]

M. Figueiredo and J. Bioucas-Dias, Restoration of poissonian images using alternating direction optimization, IEEE Trans. Image Process., 19 (2010), 3133-3145.  doi: 10.1109/TIP.2010.2053941.  Google Scholar

[26]

M. Figueiredo and R. Nowak, An EM algorithm for wavelet-based image restoration, IEEE Trans. Image Process., 12 (2003), 906-916.  doi: 10.1109/TIP.2003.814255.  Google Scholar

[27]

G. Gilboa and S. Osher, Nonlocal operators with applications to image processing, Multiscale Model. Simul., 7 (2008), 1005-1028.  doi: 10.1137/070698592.  Google Scholar

[28]

T. Goldstein and S. Osher, The split Bregman method for L1-regularized problems, SIAM J. Imaging Sci., 2 (2009), 323-343.  doi: 10.1137/080725891.  Google Scholar

[29]

R. Gonzalez and R. Woods, Digital Image Processing, 3rd edition, Pearson, London, 2008. Google Scholar

[30]

G. Grimmett and D. Welsh, Oxford Science Publications, Oxford Science Publications, London, 1986. Google Scholar

[31]

Y.-M. HuangM. K. Ng and Y.-W. Wen, A new total variation method for multiplicative noise removal, SIAM J. Imaging Sci., 2 (2009), 20-40.  doi: 10.1137/080712593.  Google Scholar

[32]

M. Idan and J. Speyer, Cauchy estimation for linear scalar systems, IEEE Trans. Automat. Control, 55 (2010), 1329-1342.  doi: 10.1109/TAC.2010.2042009.  Google Scholar

[33]

E. KuruogluW. Fitzgerald and P. Rayner, Near optimal detection of signals in impulsive noise modeled with asymmetric alpha-stable distribution, IEEE Commun. Lett., 2 (1998), 282-284.   Google Scholar

[34]

T. LeR. Chartrand and T. Asaki, A variational approach to reconstructing images corrupted by Poisson noise, J. Math. Imaging Vision, 27 (2007), 257-263.  doi: 10.1007/s10851-007-0652-y.  Google Scholar

[35]

J. Liu, Y. Lou, G. Ni and T. Zeng, An image sharpening operator combined with framelet for image deblurring, Inverse Problems, 36 (2020), 29pp. doi: 10.1088/1361-6420/ab6df0.  Google Scholar

[36]

J. Liu, A. Ni and G. Ni, A nonconvex $l_1(l_1 - l_2)$ model for image restoration with impulse noise, J. Comput. Appl. Math., 378 (2020), 16pp. doi: 10.1016/j.cam.2020.112934.  Google Scholar

[37]

Y. F. Lou and M. Yan, Fast $L1-L2$ minimization via a proximal operator, J. Sci. Comput., 74 (2018), 767-785.  doi: 10.1007/s10915-017-0463-2.  Google Scholar

[38]

Y. F. Lou, S. Osher and J. Xin, Computational aspects of constrained L1-L2 minimization for compressive sensing, J. Infect. Dis., (2015), 169–180. Google Scholar

[39]

Y. LouT. ZengS. Osher and J. Xin, A weighted difference of anisotropic and isotropic total variation model for image processing, SIAM J. Imaging Sci., 8 (2015), 1798-1823.  doi: 10.1137/14098435X.  Google Scholar

[40]

J. MeiY. DongT. Huang and W. Yin, Cauchy noise removal by nonconvex ADMM with convergence guarantees, J. Sci Comput., 74 (2018), 743-766.  doi: 10.1007/s10915-017-0460-5.  Google Scholar

[41]

M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 90-120.   Google Scholar

[42]

Y. PengJ. ChenX. Xu and F. Pu, SAR images statistical modeling and classification based on the mixture of alpha-stable distributions, Remote Sens., 5 (2013), 2145-2163.  doi: 10.3390/rs5052145.  Google Scholar

[43]

N. PustelnikC. Chaux and J. Pesquet, Parallel proximal algorithm for image restoration using hybrid regularization, IEEE Trans. Image Process., 20 (2011), 2450-2462.  doi: 10.1109/TIP.2011.2128335.  Google Scholar

[44]

P. Reeves, A non-gaussian turbulence simulation, Technical Report AFFDL-TR-, Air Force Flight Dynamics Laboratory, Wright-Patterson Air Force Base, (1969), 69–67. Google Scholar

[45]

L. RudinS. 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.  Google Scholar

[46]

S. SetzerG. Steidl and T. Teuber, Deblurring Poissonian images by split Bregman techniques, J. Visual Commun. Image Represent., 21 (2010), 193-199.  doi: 10.1016/j.jvcir.2009.10.006.  Google Scholar

[47]

J. StarckM. Elad and D. Donoho, Image decomposition via the combination of sparse representations and a variational approach, IEEE Trans. Image Process., 14 (2005), 1570-1582.  doi: 10.1109/TIP.2005.852206.  Google Scholar

[48]

T. WanN. Canagarajah and A. Achim, Segmentation of noisy colour images using Cauchy distribution in the complex wavelet domain, IET Image Process., 5 (2011), 159-170.   Google Scholar

[49]

C. Wu and X.-C. Tai, Augmented Lagrangian method, dual methods, and split Bregman iteration for ROF, vectorial TV, and high order models, SIAM J. Imaging Sci., 3 (2010), 300-339.  doi: 10.1137/090767558.  Google Scholar

[50]

J. YangY. Zhang and W. Yin, An effcient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865.  doi: 10.1137/080732894.  Google Scholar

[51]

P. H. YinY. F. Lou and J. Xin, Minimization of $l_{1-2}$ for compressed sensing, SIAM J. Sci. Comput., 37 (2015), 536-563.  doi: 10.1137/140952363.  Google Scholar

[52]

W. ZhouA. BovikH. Sheikh and E. Simoncelli, Image quality assessment: From error visibility to structural similarity, IEEE Trans. Image Process., 13 (2004), 600-612.   Google Scholar

Figure 1.  Testing images
Figure 2.  Restored images from blurred and noised images by different methods. (a):Images with Cauchy noise ($ \xi = 0.02 $) and Gaussian blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method
Figure 3.  Restored images from blurred and noised images by different methods. (a):Images with Cauchy noise ($ \xi = 0.04 $) and Gaussian blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method
Figure 4.  Restored images from blurred and noised images by different methods. (a):Images destroyed by Cauchy noise($ \xi = 0.02 $) and motion blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method
Figure 5.  Restored images from blurred and noised images by different methods. (a):Images destroyed by Cauchy noise($ \xi = 0.02 $) and average blur; (b): Results by median filter (MD); (c): Results by convex TV method; (d): Results by our method
Table 1.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods ($ \xi = 0.02 $, Gaussian blur)
Image Value Corrupted MD TV Ours
Lena SNR 5.76 12.43 14.31 14.93
PSNR 18.47 24.39 27.64 28.21
SSIM 0.1951 0.7884 0.8168 0.8389
Cameraman SNR 6.16 11.72 13.69 14.57
PSNR 18.19 24.39 26.08 26.84
SSIM 0.1566 0.7483 0.8061 0.8083
Parrot SNR 6.65 12.15 14.85 15.72
PSNR 18.27 24.42 26.72 27.50
SSIM 0.1909 0.7833 0.8239 0.8394
Boat SNR 6.80 13.85 15.40 16.52
PSNR 18.02 25.70 27.02 28.07
SSIM 0.1981 0.7935 0.8286 0.8559
Kitten SNR 6.15 10.13 11.86 13.01
PSNR 17.98 22.67 24.03 25.06
SSIM 0.2496 0.7195 0.7696 0.8161
House SNR 5.03 13.70 16.11 17.05
PSNR 18.38 28.48 30.52 31.44
SSIM 0.1425 0.7908 0.8304 0.8494
Image Value Corrupted MD TV Ours
Lena SNR 5.76 12.43 14.31 14.93
PSNR 18.47 24.39 27.64 28.21
SSIM 0.1951 0.7884 0.8168 0.8389
Cameraman SNR 6.16 11.72 13.69 14.57
PSNR 18.19 24.39 26.08 26.84
SSIM 0.1566 0.7483 0.8061 0.8083
Parrot SNR 6.65 12.15 14.85 15.72
PSNR 18.27 24.42 26.72 27.50
SSIM 0.1909 0.7833 0.8239 0.8394
Boat SNR 6.80 13.85 15.40 16.52
PSNR 18.02 25.70 27.02 28.07
SSIM 0.1981 0.7935 0.8286 0.8559
Kitten SNR 6.15 10.13 11.86 13.01
PSNR 17.98 22.67 24.03 25.06
SSIM 0.2496 0.7195 0.7696 0.8161
House SNR 5.03 13.70 16.11 17.05
PSNR 18.38 28.48 30.52 31.44
SSIM 0.1425 0.7908 0.8304 0.8494
Table 2.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods ($ \xi = 0.04 $, Gaussian blur)
Image Value Corrupted MD TV Ours
Lena SNR 3.8 11.24 12.94 13.35
PSNR 15.87 23.53 26.31 26.65
SSIM 0.0928 0.6944 0.7675 0.7928
Cameraman SNR 4.16 10.88 12.39 13.09
PSNR 15.67 23.53 24.94 25.44
SSIM 0.0761 0.6241 0.7532 0.7602
Parrot SNR 4.58 11.19 13.46 14.12
PSNR 15.75 23.55 25.40 25.91
SSIM 0.0964 0.6752 0.7763 0.7847
Boat SNR 4.65 12.59 14.13 14.81
PSNR 15.33 24.50 25.83 26.41
SSIM 0.0932 0.6936 0.7843 0.8074
Kitten SNR 4.24 9.52 10.48 11.37
PSNR 15.46 22.02 22.76 23.48
SSIM 0.1235 0.6611 0.6967 0.7413
House SNR 3.19 11.83 14.61 15.29
PSNR 15.53 26.49 29.16 29.70
SSIM 0.0636 0.6628 0.8159 0.8216
Image Value Corrupted MD TV Ours
Lena SNR 3.8 11.24 12.94 13.35
PSNR 15.87 23.53 26.31 26.65
SSIM 0.0928 0.6944 0.7675 0.7928
Cameraman SNR 4.16 10.88 12.39 13.09
PSNR 15.67 23.53 24.94 25.44
SSIM 0.0761 0.6241 0.7532 0.7602
Parrot SNR 4.58 11.19 13.46 14.12
PSNR 15.75 23.55 25.40 25.91
SSIM 0.0964 0.6752 0.7763 0.7847
Boat SNR 4.65 12.59 14.13 14.81
PSNR 15.33 24.50 25.83 26.41
SSIM 0.0932 0.6936 0.7843 0.8074
Kitten SNR 4.24 9.52 10.48 11.37
PSNR 15.46 22.02 22.76 23.48
SSIM 0.1235 0.6611 0.6967 0.7413
House SNR 3.19 11.83 14.61 15.29
PSNR 15.53 26.49 29.16 29.70
SSIM 0.0636 0.6628 0.8159 0.8216
Table 3.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods($ \xi = 0.02 $, motion blur)
Image Value Corrupted MD TV Ours
Lena SNR 4.15 7.78 11.66 12.40
PSNR 17.37 22.05 25.16 25.73
SSIM 0.1294 0.6209 0.7328 0.7969
Cameraman SNR 4.98 8.60 11.60 12.42
PSNR 17.19 21.53 24.13 24.60
SSIM 0.1149 0.6478 0.7777 0.7797
Parrot SNR 5.73 9.61 12.16 13.28
PSNR 17.46 22.00 24.15 24.73
SSIM 0.1574 0.7011 0.7782 0.7848
Boat SNR 5.93 10.80 13.02 13.68
PSNR 17.23 22.77 24.77 25.27
SSIM 0.1458 0.6731 0.7652 0.7675
Kitten SNR 4.93 7.63 10.15 10.76
PSNR 16.92 20.37 22.44 22.87
SSIM 0.1802 0.5619 0.6901 0.7216
House SNR 4.54 10.72 14.85 15.65
PSNR 18.01 25.60 29.34 30.03
SSIM 0.1272 0.7211 0.8192 0.8264
Image Value Corrupted MD TV Ours
Lena SNR 4.15 7.78 11.66 12.40
PSNR 17.37 22.05 25.16 25.73
SSIM 0.1294 0.6209 0.7328 0.7969
Cameraman SNR 4.98 8.60 11.60 12.42
PSNR 17.19 21.53 24.13 24.60
SSIM 0.1149 0.6478 0.7777 0.7797
Parrot SNR 5.73 9.61 12.16 13.28
PSNR 17.46 22.00 24.15 24.73
SSIM 0.1574 0.7011 0.7782 0.7848
Boat SNR 5.93 10.80 13.02 13.68
PSNR 17.23 22.77 24.77 25.27
SSIM 0.1458 0.6731 0.7652 0.7675
Kitten SNR 4.93 7.63 10.15 10.76
PSNR 16.92 20.37 22.44 22.87
SSIM 0.1802 0.5619 0.6901 0.7216
House SNR 4.54 10.72 14.85 15.65
PSNR 18.01 25.60 29.34 30.03
SSIM 0.1272 0.7211 0.8192 0.8264
Table 4.  The SNR, PSNR and SSIM values for corrupted images and recovered images given by different methods($ \xi = 0.02 $, average blur)
Image Value Corrupted MD TV Ours
Lena SNR 5.87 12.88 14.56 15.35
PSNR 18.55 26.47 27.87 28.64
SSIM 0.1991 0.8031 0.8203 0.8534
Cameraman SNR 6.20 12.04 14.17 15.16
PSNR 18.15 24.65 26.54 27.40
SSIM 0.1571 0.7623 0.8077 0.8309
Parrot SNR 6.78 12.64 15.35 16.11
PSNR 18.34 24.83 27.20 27.88
SSIM 0.1938 0.7974 0.8228 0.8542
Boat SNR 6.86 14.24 15.79 16.90
PSNR 18.00 26.04 27.38 28.45
SSIM 0.2014 0.8075 0.8345 0.8701
Kitten SNR 6.31 10.62 12.46 13.39
PSNR 18.06 23.08 24.56 25.38
SSIM 0.2598 0.7467 0.7956 0.8327
House SNR 5.09 14.08 16.54 17.41
PSNR 18.39 28.70 30.98 31.81
SSIM 0.1436 0.8013 0.8448 0.8581
Image Value Corrupted MD TV Ours
Lena SNR 5.87 12.88 14.56 15.35
PSNR 18.55 26.47 27.87 28.64
SSIM 0.1991 0.8031 0.8203 0.8534
Cameraman SNR 6.20 12.04 14.17 15.16
PSNR 18.15 24.65 26.54 27.40
SSIM 0.1571 0.7623 0.8077 0.8309
Parrot SNR 6.78 12.64 15.35 16.11
PSNR 18.34 24.83 27.20 27.88
SSIM 0.1938 0.7974 0.8228 0.8542
Boat SNR 6.86 14.24 15.79 16.90
PSNR 18.00 26.04 27.38 28.45
SSIM 0.2014 0.8075 0.8345 0.8701
Kitten SNR 6.31 10.62 12.46 13.39
PSNR 18.06 23.08 24.56 25.38
SSIM 0.2598 0.7467 0.7956 0.8327
House SNR 5.09 14.08 16.54 17.41
PSNR 18.39 28.70 30.98 31.81
SSIM 0.1436 0.8013 0.8448 0.8581
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