Non-convex TV denoising corrupted by impulse noise

Yoon Mo Jung; Taeuk Jeong; Sangwoon Yun

doi:10.3934/ipi.2017032

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August 2017, 11(4): 689-702. doi: 10.3934/ipi.2017032

Non-convex TV denoising corrupted by impulse noise

Yoon Mo Jung ^1, , Taeuk Jeong ^2, and Sangwoon Yun ^3,

1.	Department of Mathematics, Sungkyunkwan University, Suwon 16419, Korea
2.	Department of Computational Science and Engineering, Yonsei University, Seoul 03722, Korea
3.	Department of Mathematics Education, Sungkyunkwan University, Seoul 03063, Korea

Received May 2016 Revised April 2017 Published June 2017

Fund Project: The first author was supported by the National Research Foundation of Korea (NRF) NRF-2014R1A1A2054763 and NRF-2016R1A5A1008055. The third author was supported by the National Research Foundation of Korea (NRF) NRF-2014R1A1A2056038 and NRF-2016R1A5A1008055.

We propose a non-convex type total variation model for impulse noise removal by incorporating TV and the quasi-norm $\ell_q $, $0 < q < 1 $. Since the proposed model is non-convex and non-smooth, an iteratively reweighted algorithm is adapted and combined with a linearized ADMM. The convergence of the proposed algorithm is established and numerical results are given to illustrate the validity and efficiency of the proposed model.

Keywords: Iterative reweighted algorithm, impulse noise, image restoration, non-convex TV.

Mathematics Subject Classification: Primary: 94A08, 90C26.

Citation: Yoon Mo Jung, Taeuk Jeong, Sangwoon Yun. Non-convex TV denoising corrupted by impulse noise. Inverse Problems & Imaging, 2017, 11 (4) : 689-702. doi: 10.3934/ipi.2017032

References:

[1]	A. C. Bovik, Handbook of Image and Video Processing, Academic Press, Inc., Orlando, FL, USA, 2005. Google Scholar
[2]	E. J. Candès, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $l_1 $ minimization, J. Fourier Anal. Appl., 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. Google Scholar
[3]	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
[4]	R. Chan, C.-W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Transactions on Image Processing, 14 (2005), 1479-1485. doi: 10.1109/TIP.2005.852196. Google Scholar
[5]	T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516. doi: 10.1137/S1064827598344169. Google Scholar
[6]	T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Transactions on Circuits and Systems Ⅱ: Analog and Digital Signal Processing, 48 (2001), 784-789. Google Scholar
[7]	X. Chen and W. Zhou, Convergence of the reweighted $\ell_1 $ minimization algorithm for $ \ell_2$-$\ell_p $ minimization, Comput. Optim. Appl., 59 (2014), 47-61. doi: 10.1007/s10589-013-9553-8. Google Scholar
[8]	M. Hintermüller and T. Wu, Nonconvex $ {\rm TV}^q$-models in image restoration: Analysis and a trust-region regularization-based superlinearly convergent solver, SIAM Journal on Imaging Sciences, 6 (2013), 1385-1415. doi: 10.1137/110854746. Google Scholar
[9]	H. Hwang and R. A. Haddad, Adaptive median filters: new algorithms and results, IEEE Transactions on Image Processing, 4 (1995), 499-502. doi: 10.1109/83.370679. Google Scholar
[10]	D. Krishnan and R. Fergus, Fast image deconvolution using hyper-laplacian priors, in Advances in Neural Information Processing Systems 22 (eds. Y. Bengio, D. Schuurmans, J. Lafferty, C. Williams and A. Culotta), Curran Associates, Inc., 2009,1033-1041. Google Scholar
[11]	Z. Lin, R. Liu and Z. Su, Linearized alternating direction method with adaptive penalty for low-rank representation, in Advances in Neural Information Processing Systems (eds. J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira and K. Q. Weinberger), Curran Associates, Inc., 24 (2011), 612-620. Google Scholar
[12]	X. Liu, Alternating minimization method for image restoration corrupted by impulse noise, Multimedia Tools and Applications, 76 (2017), 12505-12516. doi: 10.1007/s11042-016-3631-8. Google Scholar
[13]	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 Transactions on Image Processing, 12 (2003), 1579-1590. doi: 10.1109/TIP.2003.819229. Google Scholar
[14]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 99-120, Special issue on mathematics and image analysis. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar
[15]	S. Oh, H. Woo, S. Yun and M. Kang, Non-convex hybrid total variation for image denoising, Journal of Visual Communication and Image Representation, 24 (2013), 332-344. doi: 10.1016/j.jvcir.2013.01.010. Google Scholar
[16]	R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer-Verlag, 1998. doi: 10.1007/978-3-642-02431-3. Google Scholar
[17]	Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265. Google Scholar
[18]	J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865. doi: 10.1137/080732894. Google Scholar

show all references

References:

[1]	A. C. Bovik, Handbook of Image and Video Processing, Academic Press, Inc., Orlando, FL, USA, 2005. Google Scholar
[2]	E. J. Candès, M. B. Wakin and S. P. Boyd, Enhancing sparsity by reweighted $l_1 $ minimization, J. Fourier Anal. Appl., 14 (2008), 877-905. doi: 10.1007/s00041-008-9045-x. Google Scholar
[3]	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
[4]	R. Chan, C.-W. Ho and M. Nikolova, Salt-and-pepper noise removal by median-type noise detectors and detail-preserving regularization, IEEE Transactions on Image Processing, 14 (2005), 1479-1485. doi: 10.1109/TIP.2005.852196. Google Scholar
[5]	T. Chan, A. Marquina and P. Mulet, High-order total variation-based image restoration, SIAM Journal on Scientific Computing, 22 (2000), 503-516. doi: 10.1137/S1064827598344169. Google Scholar
[6]	T. Chen and H. R. Wu, Space variant median filters for the restoration of impulse noise corrupted images, IEEE Transactions on Circuits and Systems Ⅱ: Analog and Digital Signal Processing, 48 (2001), 784-789. Google Scholar
[7]	X. Chen and W. Zhou, Convergence of the reweighted $\ell_1 $ minimization algorithm for $ \ell_2$-$\ell_p $ minimization, Comput. Optim. Appl., 59 (2014), 47-61. doi: 10.1007/s10589-013-9553-8. Google Scholar
[8]	M. Hintermüller and T. Wu, Nonconvex $ {\rm TV}^q$-models in image restoration: Analysis and a trust-region regularization-based superlinearly convergent solver, SIAM Journal on Imaging Sciences, 6 (2013), 1385-1415. doi: 10.1137/110854746. Google Scholar
[9]	H. Hwang and R. A. Haddad, Adaptive median filters: new algorithms and results, IEEE Transactions on Image Processing, 4 (1995), 499-502. doi: 10.1109/83.370679. Google Scholar
[10]	D. Krishnan and R. Fergus, Fast image deconvolution using hyper-laplacian priors, in Advances in Neural Information Processing Systems 22 (eds. Y. Bengio, D. Schuurmans, J. Lafferty, C. Williams and A. Culotta), Curran Associates, Inc., 2009,1033-1041. Google Scholar
[11]	Z. Lin, R. Liu and Z. Su, Linearized alternating direction method with adaptive penalty for low-rank representation, in Advances in Neural Information Processing Systems (eds. J. Shawe-Taylor, R. S. Zemel, P. L. Bartlett, F. Pereira and K. Q. Weinberger), Curran Associates, Inc., 24 (2011), 612-620. Google Scholar
[12]	X. Liu, Alternating minimization method for image restoration corrupted by impulse noise, Multimedia Tools and Applications, 76 (2017), 12505-12516. doi: 10.1007/s11042-016-3631-8. Google Scholar
[13]	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 Transactions on Image Processing, 12 (2003), 1579-1590. doi: 10.1109/TIP.2003.819229. Google Scholar
[14]	M. Nikolova, A variational approach to remove outliers and impulse noise, J. Math. Imaging Vision, 20 (2004), 99-120, Special issue on mathematics and image analysis. doi: 10.1023/B:JMIV.0000011920.58935.9c. Google Scholar
[15]	S. Oh, H. Woo, S. Yun and M. Kang, Non-convex hybrid total variation for image denoising, Journal of Visual Communication and Image Representation, 24 (2013), 332-344. doi: 10.1016/j.jvcir.2013.01.010. Google Scholar
[16]	R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer-Verlag, 1998. doi: 10.1007/978-3-642-02431-3. Google Scholar
[17]	Y. Wang, J. Yang, W. Yin and Y. Zhang, A new alternating minimization algorithm for total variation image reconstruction, SIAM Journal on Imaging Sciences, 1 (2008), 248-272. doi: 10.1137/080724265. Google Scholar
[18]	J. Yang, Y. Zhang and W. Yin, An efficient TVL1 algorithm for deblurring multichannel images corrupted by impulsive noise, SIAM J. Sci. Comput., 31 (2009), 2842-2865. doi: 10.1137/080732894. Google Scholar

Figure 1. Reconstruction of a piecewise smooth image under salt-and-pepper noise of the noise level $30\%$. SNR: (a) $4.08$dB, (b) $21.70$dB, (c) $23.24$dB, (d) $31.60$dB

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Figure 2. Reconstruction under salt-and-pepper noise of the noise level $50\%$. SNR: (a) $1.89$dB, (b) $10.01$dB, (c) $11.48$dB, (d) $21.85$dB

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$256\times 256$) under the noise level $20\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNRs: (a) $12.00$dB, (b) $29.55$dB, (c) $29.63$dB, (c) $31.24$dB, (e) $13.35$dB, (f) $29.31$dB, (g) $29.59$dB, (h) $30.72$dB">Figure 3. Reconstructed subimages of the "lori" image ($256\times 256$) under the noise level $20\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNRs: (a) $12.00$dB, (b) $29.55$dB, (c) $29.63$dB, (c) $31.24$dB, (e) $13.35$dB, (f) $29.31$dB, (g) $29.59$dB, (h) $30.72$dB

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$512\times 512$) under the noise level $30\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNRs: (a) $10.55$dB, (b) $28.59$dB, (c) $28.84$dB, (d) $31.44$dB, (e) $12.04$dB, (f) $28.71$dB, (g) $28.87$dB, (h) $31.15$dB">Figure 4. Reconstructed subimages of the "boat" image ($512\times 512$) under the noise level $30\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNRs: (a) $10.55$dB, (b) $28.59$dB, (c) $28.84$dB, (d) $31.44$dB, (e) $12.04$dB, (f) $28.71$dB, (g) $28.87$dB, (h) $31.15$dB

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$512\times 512$) under the noise level $40\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNRs: (a) $9.06$dB, (b) $28.44$dB, (c) $28.96$dB, (d) $32.65$dB, (e) $10.43$dB, (f) $28.68$dB, (g) $28.62$dB, (h) $32.09$dB">Figure 5. Reconstructed subimages of the "cameraman" image ($512\times 512$) under the noise level $40\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNRs: (a) $9.06$dB, (b) $28.44$dB, (c) $28.96$dB, (d) $32.65$dB, (e) $10.43$dB, (f) $28.68$dB, (g) $28.62$dB, (h) $32.09$dB

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$512\times 512$) under the noise level $40\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNR: (a) $9.48$dB, (b) $24.62$dB, (c) $24.91$dB, (d) $25.54$dB, (e) $11.07$dB, (f) $24.78$dB, (g) $24.95$dB, (h) $25.47$dB">Figure 6. Reconstructed subimages of the "lighthouse" image ($512\times 512$) under the noise level $40\%$. The first row is salt-and-pepper noise, and the second row is RVIN. PSNR: (a) $9.48$dB, (b) $24.62$dB, (c) $24.91$dB, (d) $25.54$dB, (e) $11.07$dB, (f) $24.78$dB, (g) $24.95$dB, (h) $25.47$dB

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Table1

Algorithm 1 IR$\ell_1$
Update$u^{(k+1)}$ and $\eta^{(k+1)}$ from $u^{(k)}$ and $\eta^{(k)}$:
1. $\begin{array}{rcl} u^{(k+1)} = \arg\min_{u\in U}\; c_1\sum_{i=1}^{n}\nu_1((\nabla u^{(k)})_i)\\| (\nabla u)_i\\|\\ +c_2\sum_{i=1}^{n}\nu_2((\nabla^2 u^{(k)})_i)\\| (\nabla^2 u)_i\\| + \|\|u-b\|\|_1. \end{array}$
2. $\eta^{(k+1)} = \nu\eta^{(k)}$ with $\nu\in(0,1)$.

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Algorithm 1 IR$\ell_1$
Update$u^{(k+1)}$ and $\eta^{(k+1)}$ from $u^{(k)}$ and $\eta^{(k)}$:
1. $\begin{array}{rcl} u^{(k+1)} = \arg\min_{u\in U}\; c_1\sum_{i=1}^{n}\nu_1((\nabla u^{(k)})_i)\\| (\nabla u)_i\\|\\ +c_2\sum_{i=1}^{n}\nu_2((\nabla^2 u^{(k)})_i)\\| (\nabla^2 u)_i\\| + \|\|u-b\|\|_1. \end{array}$
2. $\eta^{(k+1)} = \nu\eta^{(k)}$ with $\nu\in(0,1)$.

Table 1. The performance comparison under the noise level 20%

image size	Noise Type	Noisy PSNR	Results Method Iter(Total inners) time PSNR
lori 256×256	salt-and-pepper (20%)	12.00	TV$\ell_1$	57	0.21	29.55
			AMM	51(255)	0.60	29.63
			NCTV$\ell_1$	5(100)	0.88	31.24
	RVIN (20%)	13.35	TV$\ell_1$	56	0.21	29.31
			AMM	50(250)	0.59	29.59
			NCTV$\ell_1$	5(100)	0.86	30.72
boat 512×512	salt-and-pepper (20%)	12.31	TV$\ell_1$	49	1.27	29.88
			AMM	44(220)	3.75	30.31
			NCTV$\ell_1$	4(74)	4.80	33.74
	RVIN (20%)	13.81	TV$\ell_1$	48	1.17	30.39
			AMM	43(215)	3.54	30.42
			NCTV$\ell_1$	4(76)	4.89	33.65
cameraman 512×512	salt-and-pepper (20%)	12.06	TV$\ell_1$	46	1.14	32.09
			AMM	47(235)	3.90	33.05
			NCTV$\ell_1$	4(70)	4.61	39.02
	RVIN (20%)	13.43	TV$\ell_1$	44	1.08	32.05
			AMM	45(225)	3.76	32.65
			NCTV$\ell_1$	4(73)	4.83	37.99
lighthouse 512×512	salt-and-pepper (20%)	9.48	TV$\ell_1$	55	1.44	27.43
			AMM	52(260)	4.52	27.00
			NCTV$\ell_1$	5(100)	6.63	27.85
	RVIN (20%)	11.07	TV$\ell_1$	54	1.34	27.52
			AMM	50(250)	4.17	26.96
			NCTV$\ell_1$	5(100)	6.56	27.80

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image size	Noise Type	Noisy PSNR	Results Method Iter(Total inners) time PSNR
lori 256×256	salt-and-pepper (20%)	12.00	TV$\ell_1$	57	0.21	29.55
			AMM	51(255)	0.60	29.63
			NCTV$\ell_1$	5(100)	0.88	31.24
	RVIN (20%)	13.35	TV$\ell_1$	56	0.21	29.31
			AMM	50(250)	0.59	29.59
			NCTV$\ell_1$	5(100)	0.86	30.72
boat 512×512	salt-and-pepper (20%)	12.31	TV$\ell_1$	49	1.27	29.88
			AMM	44(220)	3.75	30.31
			NCTV$\ell_1$	4(74)	4.80	33.74
	RVIN (20%)	13.81	TV$\ell_1$	48	1.17	30.39
			AMM	43(215)	3.54	30.42
			NCTV$\ell_1$	4(76)	4.89	33.65
cameraman 512×512	salt-and-pepper (20%)	12.06	TV$\ell_1$	46	1.14	32.09
			AMM	47(235)	3.90	33.05
			NCTV$\ell_1$	4(70)	4.61	39.02
	RVIN (20%)	13.43	TV$\ell_1$	44	1.08	32.05
			AMM	45(225)	3.76	32.65
			NCTV$\ell_1$	4(73)	4.83	37.99
lighthouse 512×512	salt-and-pepper (20%)	9.48	TV$\ell_1$	55	1.44	27.43
			AMM	52(260)	4.52	27.00
			NCTV$\ell_1$	5(100)	6.63	27.85
	RVIN (20%)	11.07	TV$\ell_1$	54	1.34	27.52
			AMM	50(250)	4.17	26.96
			NCTV$\ell_1$	5(100)	6.56	27.80

Table 2. The performance comparison under the noise level 30%

image size	Noise Type	Noisy PSNR	Results Method Iter(Total inners) time PSNR
lori 256×256	salt-and-pepper (30%)	10.23	TV$\ell_1$	65	0.25	27.51
			AMM	58(290)	0.69	28.04
			NCTV$\ell_1$	6(120)	1.05	29.57
	RVIN (30%)	11.57	TV$\ell_1$	65	0.24	28.10
			AMM	57(285)	0.69	28.25
			NCTV$\ell_1$	6(120)	1.07	29.49
boat 512×512	salt-and-pepper (30%)	10.55	TV$\ell_1$	58	1.48	28.59
			AMM	54(270)	4.87	28.84
			NCTV$\ell_1$	5(93)	6.21	31.44
	RVIN (30%)	12.04	TV$\ell_1$	58	1.42	28.71
			AMM	50(250)	4.21	28.87
			NCTV$\ell_1$	5(93)	5.93	31.15
cameraman 512×512	salt-and-pepper (30%)	10.29	TV$\ell_1$	55	1.34	30.41
			AMM	57(285)	4.83	31.00
			NCTV$\ell_1$	5(91)	5.92	35.57
	RVIN (30%)	11.7	TV$\ell_1$	54	1.33	30.65
			AMM	52(260)	4.39	30.72
			NCTV$\ell_1$	5(92)	5.94	35.22
lighthouse 512×512	salt-and-pepper (30%)	10.75	TV$\ell_1$	64	1.65	25.98
			AMM	61(305)	5.33	25.98
			NCTV$\ell_1$	6(120)	7.98	26.64
	RVIN (30%)	12.31	TV$\ell_1$	64	1.6	25.96
			AMM	57(285)	4.75	25.92
			NCTV$\ell_1$	6(120)	7.87	26.53

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image size	Noise Type	Noisy PSNR	Results Method Iter(Total inners) time PSNR
lori 256×256	salt-and-pepper (30%)	10.23	TV$\ell_1$	65	0.25	27.51
			AMM	58(290)	0.69	28.04
			NCTV$\ell_1$	6(120)	1.05	29.57
	RVIN (30%)	11.57	TV$\ell_1$	65	0.24	28.10
			AMM	57(285)	0.69	28.25
			NCTV$\ell_1$	6(120)	1.07	29.49
boat 512×512	salt-and-pepper (30%)	10.55	TV$\ell_1$	58	1.48	28.59
			AMM	54(270)	4.87	28.84
			NCTV$\ell_1$	5(93)	6.21	31.44
	RVIN (30%)	12.04	TV$\ell_1$	58	1.42	28.71
			AMM	50(250)	4.21	28.87
			NCTV$\ell_1$	5(93)	5.93	31.15
cameraman 512×512	salt-and-pepper (30%)	10.29	TV$\ell_1$	55	1.34	30.41
			AMM	57(285)	4.83	31.00
			NCTV$\ell_1$	5(91)	5.92	35.57
	RVIN (30%)	11.7	TV$\ell_1$	54	1.33	30.65
			AMM	52(260)	4.39	30.72
			NCTV$\ell_1$	5(92)	5.94	35.22
lighthouse 512×512	salt-and-pepper (30%)	10.75	TV$\ell_1$	64	1.65	25.98
			AMM	61(305)	5.33	25.98
			NCTV$\ell_1$	6(120)	7.98	26.64
	RVIN (30%)	12.31	TV$\ell_1$	64	1.6	25.96
			AMM	57(285)	4.75	25.92
			NCTV$\ell_1$	6(120)	7.87	26.53

Table 3. The performance comparison under the noise level 40%

image size	Noise Type	Noisy PSNR	Results Method Iter(Total inners) time PSNR
lori 256×256	salt-and-pepper (40%)	8.99	TV$\ell_1$	80	0.30	26.8
			AMM	76(380)	0.92	27.23
			NCTV$\ell_1$	7(140)	1.21	28.67
	RVIN (40%)	10.46	TV$\ell_1$	77	0.29	27.16
			AMM	71(355)	0.85	27.24
			NCTV$\ell_1$	7(140)	1.21	28.40
boat 512×512	salt-and-pepper (40%)	9.33	TV$\ell_1$	70	1.75	27.22
			AMM	68(340)	6.01	27.54
			NCTV$\ell_1$	6(115)	7.34	29.63
	RVIN (40%)	10.76	TV$\ell_1$	70	1.69	27.04
			AMM	64(320)	5.28	27.42
			NCTV$\ell_1$	5(100)	6.45	29.31
cameraman 512×512	salt-and-pepper (40%)	9.06	TV$\ell_1$	67	1.61	28.44
			AMM	69(345)	5.72	28.96
			NCTV$\ell_1$	5(100)	6.48	32.65
	RVIN (40%)	10.43	TV$\ell_1$	67	1.63	28.68
			AMM	63(315)	5.23	28.62
			NCTV$\ell_1$	6(115)	7.51	32.09
lighthouse 512×512	salt-and-pepper (40%)	9.48	TV$\ell_1$	78	2.02	24.62
			AMM	79(395)	6.79	24.91
			NCTV$\ell_1$	7(140)	9.27	25.54
	RVIN (40%)	11.07	TV$\ell_1$	77	1.92	24.78
			AMM	71(355)	6.01	24.95
			NCTV$\ell_1$	7(140)	9.16	25.47

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image size	Noise Type	Noisy PSNR	Results Method Iter(Total inners) time PSNR
lori 256×256	salt-and-pepper (40%)	8.99	TV$\ell_1$	80	0.30	26.8
			AMM	76(380)	0.92	27.23
			NCTV$\ell_1$	7(140)	1.21	28.67
	RVIN (40%)	10.46	TV$\ell_1$	77	0.29	27.16
			AMM	71(355)	0.85	27.24
			NCTV$\ell_1$	7(140)	1.21	28.40
boat 512×512	salt-and-pepper (40%)	9.33	TV$\ell_1$	70	1.75	27.22
			AMM	68(340)	6.01	27.54
			NCTV$\ell_1$	6(115)	7.34	29.63
	RVIN (40%)	10.76	TV$\ell_1$	70	1.69	27.04
			AMM	64(320)	5.28	27.42
			NCTV$\ell_1$	5(100)	6.45	29.31
cameraman 512×512	salt-and-pepper (40%)	9.06	TV$\ell_1$	67	1.61	28.44
			AMM	69(345)	5.72	28.96
			NCTV$\ell_1$	5(100)	6.48	32.65
	RVIN (40%)	10.43	TV$\ell_1$	67	1.63	28.68
			AMM	63(315)	5.23	28.62
			NCTV$\ell_1$	6(115)	7.51	32.09
lighthouse 512×512	salt-and-pepper (40%)	9.48	TV$\ell_1$	78	2.02	24.62
			AMM	79(395)	6.79	24.91
			NCTV$\ell_1$	7(140)	9.27	25.54
	RVIN (40%)	11.07	TV$\ell_1$	77	1.92	24.78
			AMM	71(355)	6.01	24.95
			NCTV$\ell_1$	7(140)	9.16	25.47

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2020 Impact Factor: 1.639

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2020 Impact Factor: 1.639

American Institute of Mathematical Sciences

Non-convex TV denoising corrupted by impulse noise

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Non-convex TV denoising corrupted by impulse noise

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