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On the lifting of deterministic convergence rates for inverse problems with stochastic noise
Non-convex TV denoising corrupted by impulse noise
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 |
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.
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. |






Algorithm 1 IR |
Update |
1. |
2. |
Algorithm 1 IR |
Update |
1. |
2. |
image size |
Noise Type |
Noisy PSNR |
Results Method Iter(Total inners) time PSNR | |||
lori 256×256 |
salt-and-pepper (20%) |
12.00 | TV |
57 | 0.21 | 29.55 |
AMM | 51(255) | 0.60 | 29.63 | |||
NCTV | 5(100) | 0.88 | 31.24 | |||
RVIN (20%) |
13.35 | TV |
56 | 0.21 | 29.31 | |
AMM | 50(250) | 0.59 | 29.59 | |||
NCTV | 5(100) | 0.86 | 30.72 | |||
boat 512×512 |
salt-and-pepper (20%) |
12.31 | TV |
49 | 1.27 | 29.88 |
AMM | 44(220) | 3.75 | 30.31 | |||
NCTV | 4(74) | 4.80 | 33.74 | |||
RVIN (20%) |
13.81 | TV | 48 | 1.17 | 30.39 | |
AMM | 43(215) | 3.54 | 30.42 | |||
NCTV | 4(76) | 4.89 | 33.65 | |||
cameraman 512×512 |
salt-and-pepper (20%) |
12.06 | TV | 46 | 1.14 | 32.09 |
AMM | 47(235) | 3.90 | 33.05 | |||
NCTV | 4(70) | 4.61 | 39.02 | |||
RVIN (20%) |
13.43 | TV | 44 | 1.08 | 32.05 | |
AMM | 45(225) | 3.76 | 32.65 | |||
NCTV | 4(73) | 4.83 | 37.99 | |||
lighthouse 512×512 |
salt-and-pepper (20%) |
9.48 | TV | 55 | 1.44 | 27.43 |
AMM | 52(260) | 4.52 | 27.00 | |||
NCTV | 5(100) | 6.63 | 27.85 | |||
RVIN (20%) |
11.07 | TV | 54 | 1.34 | 27.52 | |
AMM | 50(250) | 4.17 | 26.96 | |||
NCTV | 5(100) | 6.56 | 27.80 |
image size |
Noise Type |
Noisy PSNR |
Results Method Iter(Total inners) time PSNR | |||
lori 256×256 |
salt-and-pepper (20%) |
12.00 | TV |
57 | 0.21 | 29.55 |
AMM | 51(255) | 0.60 | 29.63 | |||
NCTV | 5(100) | 0.88 | 31.24 | |||
RVIN (20%) |
13.35 | TV |
56 | 0.21 | 29.31 | |
AMM | 50(250) | 0.59 | 29.59 | |||
NCTV | 5(100) | 0.86 | 30.72 | |||
boat 512×512 |
salt-and-pepper (20%) |
12.31 | TV |
49 | 1.27 | 29.88 |
AMM | 44(220) | 3.75 | 30.31 | |||
NCTV | 4(74) | 4.80 | 33.74 | |||
RVIN (20%) |
13.81 | TV | 48 | 1.17 | 30.39 | |
AMM | 43(215) | 3.54 | 30.42 | |||
NCTV | 4(76) | 4.89 | 33.65 | |||
cameraman 512×512 |
salt-and-pepper (20%) |
12.06 | TV | 46 | 1.14 | 32.09 |
AMM | 47(235) | 3.90 | 33.05 | |||
NCTV | 4(70) | 4.61 | 39.02 | |||
RVIN (20%) |
13.43 | TV | 44 | 1.08 | 32.05 | |
AMM | 45(225) | 3.76 | 32.65 | |||
NCTV | 4(73) | 4.83 | 37.99 | |||
lighthouse 512×512 |
salt-and-pepper (20%) |
9.48 | TV | 55 | 1.44 | 27.43 |
AMM | 52(260) | 4.52 | 27.00 | |||
NCTV | 5(100) | 6.63 | 27.85 | |||
RVIN (20%) |
11.07 | TV | 54 | 1.34 | 27.52 | |
AMM | 50(250) | 4.17 | 26.96 | |||
NCTV | 5(100) | 6.56 | 27.80 |
image size |
Noise Type |
Noisy PSNR |
Results Method Iter(Total inners) time PSNR | |||
lori 256×256 |
salt-and-pepper (30%) |
10.23 | TV | 65 | 0.25 | 27.51 |
AMM | 58(290) | 0.69 | 28.04 | |||
NCTV | 6(120) | 1.05 | 29.57 | |||
RVIN (30%) |
11.57 | TV | 65 | 0.24 | 28.10 | |
AMM | 57(285) | 0.69 | 28.25 | |||
NCTV | 6(120) | 1.07 | 29.49 | |||
boat 512×512 |
salt-and-pepper (30%) |
10.55 | TV | 58 | 1.48 | 28.59 |
AMM | 54(270) | 4.87 | 28.84 | |||
NCTV | 5(93) | 6.21 | 31.44 | |||
RVIN (30%) |
12.04 | TV | 58 | 1.42 | 28.71 | |
AMM | 50(250) | 4.21 | 28.87 | |||
NCTV | 5(93) | 5.93 | 31.15 | |||
cameraman 512×512 |
salt-and-pepper (30%) |
10.29 | TV | 55 | 1.34 | 30.41 |
AMM | 57(285) | 4.83 | 31.00 | |||
NCTV | 5(91) | 5.92 | 35.57 | |||
RVIN (30%) |
11.7 | TV | 54 | 1.33 | 30.65 | |
AMM | 52(260) | 4.39 | 30.72 | |||
NCTV | 5(92) | 5.94 | 35.22 | |||
lighthouse 512×512 |
salt-and-pepper (30%) |
10.75 | TV | 64 | 1.65 | 25.98 |
AMM | 61(305) | 5.33 | 25.98 | |||
NCTV | 6(120) | 7.98 | 26.64 | |||
RVIN (30%) |
12.31 | TV | 64 | 1.6 | 25.96 | |
AMM | 57(285) | 4.75 | 25.92 | |||
NCTV | 6(120) | 7.87 | 26.53 |
image size |
Noise Type |
Noisy PSNR |
Results Method Iter(Total inners) time PSNR | |||
lori 256×256 |
salt-and-pepper (30%) |
10.23 | TV | 65 | 0.25 | 27.51 |
AMM | 58(290) | 0.69 | 28.04 | |||
NCTV | 6(120) | 1.05 | 29.57 | |||
RVIN (30%) |
11.57 | TV | 65 | 0.24 | 28.10 | |
AMM | 57(285) | 0.69 | 28.25 | |||
NCTV | 6(120) | 1.07 | 29.49 | |||
boat 512×512 |
salt-and-pepper (30%) |
10.55 | TV | 58 | 1.48 | 28.59 |
AMM | 54(270) | 4.87 | 28.84 | |||
NCTV | 5(93) | 6.21 | 31.44 | |||
RVIN (30%) |
12.04 | TV | 58 | 1.42 | 28.71 | |
AMM | 50(250) | 4.21 | 28.87 | |||
NCTV | 5(93) | 5.93 | 31.15 | |||
cameraman 512×512 |
salt-and-pepper (30%) |
10.29 | TV | 55 | 1.34 | 30.41 |
AMM | 57(285) | 4.83 | 31.00 | |||
NCTV | 5(91) | 5.92 | 35.57 | |||
RVIN (30%) |
11.7 | TV | 54 | 1.33 | 30.65 | |
AMM | 52(260) | 4.39 | 30.72 | |||
NCTV | 5(92) | 5.94 | 35.22 | |||
lighthouse 512×512 |
salt-and-pepper (30%) |
10.75 | TV | 64 | 1.65 | 25.98 |
AMM | 61(305) | 5.33 | 25.98 | |||
NCTV | 6(120) | 7.98 | 26.64 | |||
RVIN (30%) |
12.31 | TV | 64 | 1.6 | 25.96 | |
AMM | 57(285) | 4.75 | 25.92 | |||
NCTV | 6(120) | 7.87 | 26.53 |
image size |
Noise Type |
Noisy PSNR |
Results Method Iter(Total inners) time PSNR | |||
lori 256×256 |
salt-and-pepper (40%) |
8.99 | TV | 80 | 0.30 | 26.8 |
AMM | 76(380) | 0.92 | 27.23 | |||
NCTV | 7(140) | 1.21 | 28.67 | |||
RVIN (40%) |
10.46 | TV | 77 | 0.29 | 27.16 | |
AMM | 71(355) | 0.85 | 27.24 | |||
NCTV | 7(140) | 1.21 | 28.40 | |||
boat 512×512 |
salt-and-pepper (40%) |
9.33 | TV | 70 | 1.75 | 27.22 |
AMM | 68(340) | 6.01 | 27.54 | |||
NCTV | 6(115) | 7.34 | 29.63 | |||
RVIN (40%) |
10.76 | TV | 70 | 1.69 | 27.04 | |
AMM | 64(320) | 5.28 | 27.42 | |||
NCTV | 5(100) | 6.45 | 29.31 | |||
cameraman 512×512 |
salt-and-pepper (40%) |
9.06 | TV | 67 | 1.61 | 28.44 |
AMM | 69(345) | 5.72 | 28.96 | |||
NCTV | 5(100) | 6.48 | 32.65 | |||
RVIN (40%) |
10.43 | TV | 67 | 1.63 | 28.68 | |
AMM | 63(315) | 5.23 | 28.62 | |||
NCTV | 6(115) | 7.51 | 32.09 | |||
lighthouse 512×512 |
salt-and-pepper (40%) |
9.48 | TV | 78 | 2.02 | 24.62 |
AMM | 79(395) | 6.79 | 24.91 | |||
NCTV | 7(140) | 9.27 | 25.54 | |||
RVIN (40%) |
11.07 | TV | 77 | 1.92 | 24.78 | |
AMM | 71(355) | 6.01 | 24.95 | |||
NCTV | 7(140) | 9.16 | 25.47 |
image size |
Noise Type |
Noisy PSNR |
Results Method Iter(Total inners) time PSNR | |||
lori 256×256 |
salt-and-pepper (40%) |
8.99 | TV | 80 | 0.30 | 26.8 |
AMM | 76(380) | 0.92 | 27.23 | |||
NCTV | 7(140) | 1.21 | 28.67 | |||
RVIN (40%) |
10.46 | TV | 77 | 0.29 | 27.16 | |
AMM | 71(355) | 0.85 | 27.24 | |||
NCTV | 7(140) | 1.21 | 28.40 | |||
boat 512×512 |
salt-and-pepper (40%) |
9.33 | TV | 70 | 1.75 | 27.22 |
AMM | 68(340) | 6.01 | 27.54 | |||
NCTV | 6(115) | 7.34 | 29.63 | |||
RVIN (40%) |
10.76 | TV | 70 | 1.69 | 27.04 | |
AMM | 64(320) | 5.28 | 27.42 | |||
NCTV | 5(100) | 6.45 | 29.31 | |||
cameraman 512×512 |
salt-and-pepper (40%) |
9.06 | TV | 67 | 1.61 | 28.44 |
AMM | 69(345) | 5.72 | 28.96 | |||
NCTV | 5(100) | 6.48 | 32.65 | |||
RVIN (40%) |
10.43 | TV | 67 | 1.63 | 28.68 | |
AMM | 63(315) | 5.23 | 28.62 | |||
NCTV | 6(115) | 7.51 | 32.09 | |||
lighthouse 512×512 |
salt-and-pepper (40%) |
9.48 | TV | 78 | 2.02 | 24.62 |
AMM | 79(395) | 6.79 | 24.91 | |||
NCTV | 7(140) | 9.27 | 25.54 | |||
RVIN (40%) |
11.07 | TV | 77 | 1.92 | 24.78 | |
AMM | 71(355) | 6.01 | 24.95 | |||
NCTV | 7(140) | 9.16 | 25.47 |
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