American Institute of Mathematical Sciences

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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 and 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.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer-Verlag, 1998. doi: 10.1007/978-3-642-02431-3.

[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.

[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.

show all references

References:
[1]

A. C. Bovik, Handbook of Image and Video Processing, Academic Press, Inc., Orlando, FL, USA, 2005.

[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.

[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.

[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.

[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.

[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. 

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

R. T. Rockafellar and R. J. -B. Wets, Variational Analysis, Springer-Verlag, 1998. doi: 10.1007/978-3-642-02431-3.

[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.

[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.

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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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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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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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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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
AMM51(255)0.6029.63
NCTV$\ell_1$5(100)0.8831.24
RVIN
(20%)		 13.35 TV$\ell_1$ 56 0.21 29.31
AMM 50(250) 0.59 29.59
NCTV$\ell_1$5(100)0.8630.72
boat
512×512		 salt-and-pepper
(20%)		 12.31 TV$\ell_1$ 49 1.27 29.88
AMM44(220)3.7530.31
NCTV$\ell_1$4(74)4.8033.74
RVIN
(20%)		 13.81 TV$\ell_1$481.1730.39
AMM43(215)3.5430.42
NCTV$\ell_1$4(76)4.8933.65
cameraman
512×512		 salt-and-pepper
(20%)		 12.06 TV$\ell_1$461.1432.09
AMM47(235)3.9033.05
NCTV$\ell_1$4(70)4.6139.02
RVIN
(20%)		 13.43 TV$\ell_1$441.0832.05
AMM45(225)3.7632.65
NCTV$\ell_1$4(73)4.8337.99
lighthouse
512×512		 salt-and-pepper
(20%)		 9.48 TV$\ell_1$551.4427.43
AMM52(260)4.5227.00
NCTV$\ell_1$5(100)6.6327.85
RVIN
(20%)		 11.07 TV$\ell_1$541.3427.52
AMM50(250)4.1726.96
NCTV$\ell_1$5(100)6.5627.80
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image
size		 Noise
Type		 Noisy
PSNR		 Results
Method Iter(Total inners) time PSNR
lori
256×256		 salt-and-pepper
(20%)		 12.00 TV$\ell_1$ 57 0.21 29.55
AMM51(255)0.6029.63
NCTV$\ell_1$5(100)0.8831.24
RVIN
(20%)		 13.35 TV$\ell_1$ 56 0.21 29.31
AMM 50(250) 0.59 29.59
NCTV$\ell_1$5(100)0.8630.72
boat
512×512		 salt-and-pepper
(20%)		 12.31 TV$\ell_1$ 49 1.27 29.88
AMM44(220)3.7530.31
NCTV$\ell_1$4(74)4.8033.74
RVIN
(20%)		 13.81 TV$\ell_1$481.1730.39
AMM43(215)3.5430.42
NCTV$\ell_1$4(76)4.8933.65
cameraman
512×512		 salt-and-pepper
(20%)		 12.06 TV$\ell_1$461.1432.09
AMM47(235)3.9033.05
NCTV$\ell_1$4(70)4.6139.02
RVIN
(20%)		 13.43 TV$\ell_1$441.0832.05
AMM45(225)3.7632.65
NCTV$\ell_1$4(73)4.8337.99
lighthouse
512×512		 salt-and-pepper
(20%)		 9.48 TV$\ell_1$551.4427.43
AMM52(260)4.5227.00
NCTV$\ell_1$5(100)6.6327.85
RVIN
(20%)		 11.07 TV$\ell_1$541.3427.52
AMM50(250)4.1726.96
NCTV$\ell_1$5(100)6.5627.80
Table 2.  The performance comparison under the noise level 30%
image
size		 Noise
Type		 Noisy
PSNR		 Results
Method Iter(Total inners) time PSNR
lori
256×256		 salt-and-pepper
(30%)		 10.23 TV$\ell_1$650.2527.51
AMM58(290)0.6928.04
NCTV$\ell_1$6(120)1.0529.57
RVIN
(30%)		 11.57 TV$\ell_1$650.2428.10
AMM57(285)0.6928.25
NCTV$\ell_1$6(120)1.0729.49
boat
512×512		 salt-and-pepper
(30%)		 10.55 TV$\ell_1$581.4828.59
AMM54(270)4.8728.84
NCTV$\ell_1$5(93)6.2131.44
RVIN
(30%)		 12.04 TV$\ell_1$581.4228.71
AMM50(250)4.2128.87
NCTV$\ell_1$5(93)5.9331.15
cameraman
512×512		 salt-and-pepper
(30%)		 10.29 TV$\ell_1$551.3430.41
AMM57(285)4.8331.00
NCTV$\ell_1$5(91)5.9235.57
RVIN
(30%)		 11.7 TV$\ell_1$541.3330.65
AMM52(260)4.3930.72
NCTV$\ell_1$5(92)5.9435.22
lighthouse
512×512		 salt-and-pepper
(30%)		 10.75 TV$\ell_1$641.6525.98
AMM61(305)5.3325.98
NCTV$\ell_1$6(120)7.9826.64
RVIN
(30%)		 12.31 TV$\ell_1$641.625.96
AMM57(285)4.7525.92
NCTV$\ell_1$6(120)7.8726.53
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image
size		 Noise
Type		 Noisy
PSNR		 Results
Method Iter(Total inners) time PSNR
lori
256×256		 salt-and-pepper
(30%)		 10.23 TV$\ell_1$650.2527.51
AMM58(290)0.6928.04
NCTV$\ell_1$6(120)1.0529.57
RVIN
(30%)		 11.57 TV$\ell_1$650.2428.10
AMM57(285)0.6928.25
NCTV$\ell_1$6(120)1.0729.49
boat
512×512		 salt-and-pepper
(30%)		 10.55 TV$\ell_1$581.4828.59
AMM54(270)4.8728.84
NCTV$\ell_1$5(93)6.2131.44
RVIN
(30%)		 12.04 TV$\ell_1$581.4228.71
AMM50(250)4.2128.87
NCTV$\ell_1$5(93)5.9331.15
cameraman
512×512		 salt-and-pepper
(30%)		 10.29 TV$\ell_1$551.3430.41
AMM57(285)4.8331.00
NCTV$\ell_1$5(91)5.9235.57
RVIN
(30%)		 11.7 TV$\ell_1$541.3330.65
AMM52(260)4.3930.72
NCTV$\ell_1$5(92)5.9435.22
lighthouse
512×512		 salt-and-pepper
(30%)		 10.75 TV$\ell_1$641.6525.98
AMM61(305)5.3325.98
NCTV$\ell_1$6(120)7.9826.64
RVIN
(30%)		 12.31 TV$\ell_1$641.625.96
AMM57(285)4.7525.92
NCTV$\ell_1$6(120)7.8726.53
Table 3.  The performance comparison under the noise level 40%
image
size		 Noise
Type		 Noisy
PSNR		 Results
Method Iter(Total inners) time PSNR
lori
256×256		 salt-and-pepper
(40%)		 8.99 TV$\ell_1$800.3026.8
AMM76(380)0.9227.23
NCTV$\ell_1$7(140)1.2128.67
RVIN
(40%)		 10.46 TV$\ell_1$770.2927.16
AMM71(355)0.8527.24
NCTV$\ell_1$7(140)1.2128.40
boat
512×512		 salt-and-pepper
(40%)		 9.33 TV$\ell_1$701.7527.22
AMM68(340)6.0127.54
NCTV$\ell_1$6(115)7.3429.63
RVIN
(40%)		 10.76 TV$\ell_1$701.6927.04
AMM64(320)5.2827.42
NCTV$\ell_1$5(100)6.4529.31
cameraman
512×512		 salt-and-pepper
(40%)		 9.06 TV$\ell_1$671.6128.44
AMM69(345)5.7228.96
NCTV$\ell_1$5(100)6.4832.65
RVIN
(40%)		 10.43 TV$\ell_1$671.6328.68
AMM63(315)5.2328.62
NCTV$\ell_1$6(115)7.5132.09
lighthouse
512×512		 salt-and-pepper
(40%)		 9.48 TV$\ell_1$782.0224.62
AMM79(395)6.7924.91
NCTV$\ell_1$7(140)9.2725.54
RVIN
(40%)		 11.07 TV$\ell_1$771.9224.78
AMM71(355)6.0124.95
NCTV$\ell_1$7(140)9.1625.47
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image
size		 Noise
Type		 Noisy
PSNR		 Results
Method Iter(Total inners) time PSNR
lori
256×256		 salt-and-pepper
(40%)		 8.99 TV$\ell_1$800.3026.8
AMM76(380)0.9227.23
NCTV$\ell_1$7(140)1.2128.67
RVIN
(40%)		 10.46 TV$\ell_1$770.2927.16
AMM71(355)0.8527.24
NCTV$\ell_1$7(140)1.2128.40
boat
512×512		 salt-and-pepper
(40%)		 9.33 TV$\ell_1$701.7527.22
AMM68(340)6.0127.54
NCTV$\ell_1$6(115)7.3429.63
RVIN
(40%)		 10.76 TV$\ell_1$701.6927.04
AMM64(320)5.2827.42
NCTV$\ell_1$5(100)6.4529.31
cameraman
512×512		 salt-and-pepper
(40%)		 9.06 TV$\ell_1$671.6128.44
AMM69(345)5.7228.96
NCTV$\ell_1$5(100)6.4832.65
RVIN
(40%)		 10.43 TV$\ell_1$671.6328.68
AMM63(315)5.2328.62
NCTV$\ell_1$6(115)7.5132.09
lighthouse
512×512		 salt-and-pepper
(40%)		 9.48 TV$\ell_1$782.0224.62
AMM79(395)6.7924.91
NCTV$\ell_1$7(140)9.2725.54
RVIN
(40%)		 11.07 TV$\ell_1$771.9224.78
AMM71(355)6.0124.95
NCTV$\ell_1$7(140)9.1625.47
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