American Institute of Mathematical Sciences

Advanced
February  2019, 13(1): 117-147. doi: 10.3934/ipi.2019007

Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters

Hanwool Na 1, , Myeongmin Kang 2, , Miyoun Jung 3, and  Myungjoo Kang 4,,

1. 

SK hynix Inc., Icheon, Korea
2. 

Department of Mathematics, Chungnam National University, Daejeon, Korea
3. 

Department of Mathematics, Hankuk University of Foreign Studies, Yongin, Korea
4. 

Department of Mathematical Sciences, Seoul National University, Seoul, Korea

* Corresponding author: Myungjoo Kang

Received  January 2018 Revised  May 2018 Published  December 2018

In this article, we introduce a novel variational model for the restoration of images corrupted by multiplicative Gamma noise. The model incorporates a convex data-fidelity term with a nonconvex version of the total generalized variation (TGV). In addition, we adopt a spatially adaptive regularization parameter (SARP) approach. The nonconvex TGV regularization enables the efficient denoising of smooth regions, without staircasing artifacts that appear on total variation regularization-based models, and edges and details to be conserved. Moreover, the SARP approach further helps preserve fine structures and textures. To deal with the nonconvex regularization, we utilize an iteratively reweighted $\ell_1$ algorithm, and the alternating direction method of multipliers is employed to solve a convex subproblem. This leads to a fast and efficient iterative algorithm for solving the proposed model. Numerical experiments show that the proposed model produces better denoising results than the state-of-the-art models.

Keywords: Image denoising, multiplicative Gamma noise, nonconvex total generalized variation, spatially adaptive regularization parameter, iteratively reweighted $\ell_1$ algorithm, alternating direction method of multiplier.
Mathematics Subject Classification: 68U10, 68T05.
Citation: Hanwool Na, Myeongmin Kang, Miyoun Jung, Myungjoo Kang. Nonconvex TGV regularization model for multiplicative noise removal with spatially varying parameters. Inverse Problems & Imaging, 2019, 13 (1) : 117-147. doi: 10.3934/ipi.2019007
References:
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show all references

References:
[1]

A. Almansa, C. Ballester, V. Caselles and G. Haro, A TV based restoration model with local constraints, Journal of Scientific Computing, 34 (2008), 209-236.  doi: 10.1007/s10915-007-9160-x.  Google Scholar

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[14]

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[15]

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[16]

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[17]

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

[20]

W. Feng, H. Lei and Y. Gao, Speckle reduction via higher order total variation approach, IEEE Transactions on Image Processing, 23 (2014), 1831-1843.  doi: 10.1109/TIP.2014.2308432.  Google Scholar

[21]

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[22]

P. Getreuer, M. Tong and L. A. Vese, A variational model for the restoration of MR images corrupted by blur and Rician noise, in International Symposium on Visual Computing, Springer, 2011,686-698 Google Scholar

[23]

G. Gilboa, N. Sochen and Y. Y. Zeevi, Variational denoising of partly textured images by spatially varying constraints, IEEE Transactions on Image Processing, 15 (2006), 2281-2289.   Google Scholar

[24]

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

[25]

M. L. Gonçalves, J. G. Melo and R. D. Monteiro, Convergence rate bounds for a proximal ADMM with over-relaxation stepsize parameter for solving nonconvex linearly constrained problems, arXiv preprint, arXiv: 1702.01850. Google Scholar

[26]

K. Guo, D. Han and T.-T. Wu, Convergence of alternating direction method for minimizing sum of two nonconvex functions with linear constraints, International Journal of Computer Mathematics, 94 (2017), 1653-1669.  doi: 10.1080/00207160.2016.1227432.  Google Scholar

[27]

W. Guo, J. Qin and W. Yin, A new detail-preserving regularization scheme, SIAM Journal on Imaging Sciences, 7 (2014), 1309-1334.  doi: 10.1137/120904263.  Google Scholar

[28]

M. Kang, M. Kang and M. Jung, Nonconvex higher-order regularization based Rician noise removal with spatially adaptive parameters, Journal of Visual Communication and Image Representation, 32 (2015), 180-193.   Google Scholar

[29]

M. Kang, S. Yun and H. Woo, Two-level convex relaxed variational model for multiplicative denoising, SIAM Journal on Imaging Sciences, 6 (2013), 875-903.  doi: 10.1137/11086077X.  Google Scholar

[30]

F. Knoll, K. Bredies, T. Pock and R. Stollberger, Second order total generalized variation (tgv) for MRI, Magnetic Resonance in Medicine, 65 (2011), 480-491.   Google Scholar

[31]

F. Knoll, G. Schultz, K. Bredies, D. Gallichan, M. Zaitsev, J. Hennig and R. Stollberger, Reconstruction of undersampled radial PatLoc imaging using total generalized variation, Magnetic Resonance in Medicine, 70 (2013), 40-52.   Google Scholar

[32]

D. Krishnan and R. Fergus, Fast image deconvolution using hyper-Laplacian priors, in Advances in Neural Information Processing Systems, 2009, 1033-1041. Google Scholar

[33]

A. Lanza, S. Morigi and F. Sgallari, Convex image denoising via non-convex regularization, in International Conference on Scale Space and Variational Methods in Computer Vision, Springer, 9087 (2015), 666-677. doi: 10.1007/978-3-319-18461-6_53.  Google Scholar

[34]

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

[35]

J.-S. Lee, Digital image enhancement and noise filtering by use of local statistics, IEEE transactions on pattern analysis and machine intelligence, 2 (1980), 165-168.   Google Scholar

[36]

F. Li, M. K. Ng and C. Shen, Multiplicative noise removal with spatially varying regularization parameters, SIAM Journal on Imaging Sciences, 3 (2010), 1-20.  doi: 10.1137/090748421.  Google Scholar

[37]

F. Li, C. Shen, J. Fan and C. Shen, Image restoration combining a total variational filter and a fourth-order filter, Journal of Visual Communication and Image Representation, 18 (2007), 322-330.   Google Scholar

[38]

G. Liu, T.-Z. Huang and J. Liu, High-order TVL1-based images restoration and spatially adapted regularization parameter selection, Computers & Mathematics with Applications, 67 (2014), 2015-2026.  doi: 10.1016/j.camwa.2014.04.008.  Google Scholar

[39]

R. W. Liu, L. Shi, W. Huang, J. Xu, S. C. H. Yu and D. Wang, Generalized total variation-based MRI Rician denoising model with spatially adaptive regularization parameters, Magnetic resonance imaging, 32 (2014), 702-720.   Google Scholar

[40]

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[41]

J. Lu, L. Shen, C. Xu and Y. Xu, Multiplicative noise removal in imaging: An exp-model and its fixed-point proximity algorithm, Applied and Computational Harmonic Analysis, 41 (2016), 518-539.  doi: 10.1016/j.acha.2015.10.003.  Google Scholar

[42]

V. Luminita and T. F. Chan, Reduced non-convex functional approximations for imag restoration & segmentation, UCLA CAM Website 97-56, 1997. Google Scholar

[43]

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.   Google Scholar

[44]

H. Na, M. Kang, M. Jung and M. Kang, An exp model with spatially adaptive regularization parameters for multiplicative noise removal, Journal of Scientific Computing, 75 (2018), 478-509.  doi: 10.1007/s10915-017-0550-4.  Google Scholar

[45]

M. Nikolova, M. K. Ng and C.-P. Tam, Fast nonconvex nonsmooth minimization methods for image restoration and reconstruction, IEEE Transactions on Image Processing, 19 (2010), 3073-3088.  doi: 10.1109/TIP.2010.2052275.  Google Scholar

[46]

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Figure 1.  Original images. First row: Boat $(256 \times 256)$, Elaine $(256 \times 256)$, Face $(255 \times 255)$, Girl $(336 \times 254)$, and Mountain $(400 \times 200)$. Second row: Peppers $(256 \times 256)$, Remote1 $(350 \times 228)$, Remote2 $(350 \times 253)$, Remote3 $(308 \times 236)$, and Remote4 $(275 \times 275)$
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Figure 2.  Evolution of the function $\lambda$ (top) and denoised image $\tilde{u}$ (bottom). Top: (a) initial $\lambda^1$, (b) $\lambda^2$, (c) $\lambda^3$. Bottom: denoised images (a) $\tilde{u}^1$ (PSNR: 22.68), (b) $\tilde{u}^2$ (PSNR: 24.79), (c) $\tilde{u}^3$ (PSNR: 24.83).
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Figure 3.  Denoised images (top) and final $\lambda$ (bottom) with different number of window sizes. (a) $(r_1, r_2) = (7, 21)$ (PSNR : 25.31 dB), (b) $(r_1, r_2) = (7,256)$ (PSNR: 25.63 dB), (c) $(r_1, r_2, r_3) = (7, 21,256)$ (PSNR: 25.65 dB)
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Figure 4.  Denoised images of (a) exp-SARP, (b) our model with a fixed $\lambda$ (without SARP), (c) our NTGV-SARP model. 1st row: $M = 10$, 2nd row: $M = 5$, 3th row: $M = 3$. PSNR: (1st row, left to right) 24.62, 24.77, 24.88; (2nd row) 25.53, 26.42, 26.50; (3rd row) 22.47, 22.49, 22.57.
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29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model">Figure 5.  Denoising results of our model when $M = 10$, and comparisons with other models. (a) data $f$ with $M = 10$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
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29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model">Figure 6.  Denoising results of our model when $M = 10$, and comparisons with other models. (a) data $f$ with $M = 10$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
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29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], and our model">Figure 7.  Denoising results of our model when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$, and comparisons with other models. Top to bottom rows: TwL-4V [29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], and our model
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29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], and our model">Figure 8.  Denoising results of our model when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$, and comparisons with other models. Top to bottom rows: TwL-4V [29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], and our model
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29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model">Figure 9.  Denoising results of our model when $M = 3$, and comparisons with other models. (a) data $f$ with $M = 3$. Denoised images. (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
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29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model">Figure 10.  Denoising results of our model when $M = 3$, and comparisons with other models. (a) data $f$ with $M = 3$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
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29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model">Figure 11.  Denoising results of our model when $M = 10$, and comparisons with other models. (a) data $f$ with $M = 10$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
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29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model">Figure 12.  Denoising results of our model when $M = 5$, and comparisons with other models. (a) data $f$ with $M = 5$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
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29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model">Figure 13.  Denoising results of our model when $M = 3$, and comparisons with other models. (a) data $f$ with $M = 3$. Denoised images: (b) TwL-4V [29]; (c) exp-SARP [44]; (d) SO-TGV [20]; (e) DZ-TGV [54]; (f) our model
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29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], our model">Figure 14.  Denoising results of our model when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$, and comparisons with other models. Top to bottom rows: TwL-4V [29], exp-SARP [44], SO-TGV [20], DZ-TGV [54], our model
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50] when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$. 1st, 3rd rows: our model. 2nd, 4th rows: SAR-BM3D model. PSNR: (1st row, left to right) 27.91, 26.25, 24.80; (2nd row) 28.23, 26.47, 24.99; (3rd row) 30.57, 29.00, 27.50; (4th row) 30.04, 28.35, 26.95">Figure 15.  Comparison of denoised images of our model and SAR-BM3D model [50] when (a) $M = 10$ (b) $M = 5$ (c) $M = 3$. 1st, 3rd rows: our model. 2nd, 4th rows: SAR-BM3D model. PSNR: (1st row, left to right) 27.91, 26.25, 24.80; (2nd row) 28.23, 26.47, 24.99; (3rd row) 30.57, 29.00, 27.50; (4th row) 30.04, 28.35, 26.95
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50] when $M = 3$. (a) our model; (b) SAR-BM3D model. PSNR (top to bottom rows): (a) 22.75, 22.57, 28.75, 23.03; (b) 23.38, 23.03, 28.43, 22.97">Figure 16.  Comparison of denoised images of our model and SAR-BM3D model [50] when $M = 3$. (a) our model; (b) SAR-BM3D model. PSNR (top to bottom rows): (a) 22.75, 22.57, 28.75, 23.03; (b) 23.38, 23.03, 28.43, 22.97
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Table 1.  Comparisons of denoising results when $M = 10$ (PSNR/SSIM)
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat25.30 / 0.685125.54 / 0.698524.93 / 0.676225.22 / 0.685325.65 / 0.7085
Elaine27.05 / 0.764627.06 / 0.768827.54 / 0.797727.21 / 0.779227.91 / 0.8104
Face26.70 / 0.815827.08 / 0.850827.99 / 0.877926.84 / 0.845528.28 / 0.8851
Girl28.99 / 0.819928.79 / 0.821830.39 / 0.872429.06 / 0.865530.57 / 0.8788
Mountain24.73 / 0.649124.70 / 0.653924.42 / 0.648624.16 / 0.634824.82 / 0.6666
Peppers27.10 / 0.806427.22 / 0.816127.10 / 0.814327.12 / 0.812427.51 / 0.8303
Remote124.97 / 0.709125.07 / 0.707224.96 / 0.709524.23 / 0.688525.20 / 0.7098
Remote224.70 / 0.691224.74 / 0.701024.66 / 0.677523.89 / 0.665024.83 / 0.7026
Remote330.33 / 0.769030.36 / 0.770130.60 / 0.781630.01 / 0.758330.82 / 0.7846
Remote424.62 / 0.628724.64 / 0.661124.72 / 0.644224.33 / 0.631124.89 / 0.6756
average26.45 / 0.733826.52 / 0.744926.73 / 0.7526.21 / 0.736527.05 / 0.7652
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TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat25.30 / 0.685125.54 / 0.698524.93 / 0.676225.22 / 0.685325.65 / 0.7085
Elaine27.05 / 0.764627.06 / 0.768827.54 / 0.797727.21 / 0.779227.91 / 0.8104
Face26.70 / 0.815827.08 / 0.850827.99 / 0.877926.84 / 0.845528.28 / 0.8851
Girl28.99 / 0.819928.79 / 0.821830.39 / 0.872429.06 / 0.865530.57 / 0.8788
Mountain24.73 / 0.649124.70 / 0.653924.42 / 0.648624.16 / 0.634824.82 / 0.6666
Peppers27.10 / 0.806427.22 / 0.816127.10 / 0.814327.12 / 0.812427.51 / 0.8303
Remote124.97 / 0.709125.07 / 0.707224.96 / 0.709524.23 / 0.688525.20 / 0.7098
Remote224.70 / 0.691224.74 / 0.701024.66 / 0.677523.89 / 0.665024.83 / 0.7026
Remote330.33 / 0.769030.36 / 0.770130.60 / 0.781630.01 / 0.758330.82 / 0.7846
Remote424.62 / 0.628724.64 / 0.661124.72 / 0.644224.33 / 0.631124.89 / 0.6756
average26.45 / 0.733826.52 / 0.744926.73 / 0.7526.21 / 0.736527.05 / 0.7652
Table 2.  Comparisons of denoising results when $M = 5$ (PSNR/SSIM)
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat23.84 / 0.620723.89 / 0.630923.62 / 0.616523.41 / 0.608124.17 / 0.6515
Elaine25.54 / 0.712225.41 / 0.714026.01 / 0.755225.17 / 0.717326.25 / 0.7638
Face25.02 / 0.770725.53 / 0.806526.12 / 0.840524.51 / 0.797426.50 / 0.8493
Girl27.63 / 0.780627.29 / 0.772828.87 / 0.839426.90 / 0.813129.00 / 0.8446
Mountain23.52 / 0.586923.40 / 0.584323.16 / 0.579122.66 / 0.554423.52 / 0.6013
Peppers25.68 / 0.760125.82 / 0.776325.60 / 0.773225.17 / 0.762126.16 / 0.7959
Remote123.53 / 0.632923.60 / 0.622023.54 / 0.642222.55 / 0.600523.70 / 0.6461
Remote223.38 / 0.611923.39 / 0.617123.34 / 0.583522.22 / 0.575523.58 / 0.6249
Remote329.32 / 0.728829.36 / 0.728629.45 / 0.735128.20 / 0.697729.76 / 0.7434
Remote423.53 / 0.558823.42 / 0.578723.63 / 0.575022.91 / 0.550023.75 / 0.6075
average25.10 / 0.676325.11 / 0.683125.33 / 0.693924.37 / 0.667625.64 / 0.7128
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TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat23.84 / 0.620723.89 / 0.630923.62 / 0.616523.41 / 0.608124.17 / 0.6515
Elaine25.54 / 0.712225.41 / 0.714026.01 / 0.755225.17 / 0.717326.25 / 0.7638
Face25.02 / 0.770725.53 / 0.806526.12 / 0.840524.51 / 0.797426.50 / 0.8493
Girl27.63 / 0.780627.29 / 0.772828.87 / 0.839426.90 / 0.813129.00 / 0.8446
Mountain23.52 / 0.586923.40 / 0.584323.16 / 0.579122.66 / 0.554423.52 / 0.6013
Peppers25.68 / 0.760125.82 / 0.776325.60 / 0.773225.17 / 0.762126.16 / 0.7959
Remote123.53 / 0.632923.60 / 0.622023.54 / 0.642222.55 / 0.600523.70 / 0.6461
Remote223.38 / 0.611923.39 / 0.617123.34 / 0.583522.22 / 0.575523.58 / 0.6249
Remote329.32 / 0.728829.36 / 0.728629.45 / 0.735128.20 / 0.697729.76 / 0.7434
Remote423.53 / 0.558823.42 / 0.578723.63 / 0.575022.91 / 0.550023.75 / 0.6075
average25.10 / 0.676325.11 / 0.683125.33 / 0.693924.37 / 0.667625.64 / 0.7128
Table 3.  Comparisons of denoising results when $M = 3$ (PSNR/SSIM)
TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat22.93 / 0.578623.01 / 0.583722.71 / 0.570421.92 / 0.535523.11 / 0.6005
Elaine24.27 / 0.670623.95 / 0.668924.50 / 0.711723.06 / 0.643624.80 / 0.7240
Face23.50 / 0.734724.12 / 0.762424.66 / 0.805822.45 / 0.738925.00 / 0.8181
Girl26.32 / 0.746526.25 / 0.733227.32 / 0.803825.30 / 0.744727.50 / 0.8125
Mountain22.58 / 0.542022.57 / 0.538422.30 / 0.533021.12 / 0.488522.63 / 0.5600
Peppers24.16 / 0.732724.42 / 0.737124.27 / 0.736823.21 / 0.707424.78 / 0.7534
Remote122.61 / 0.576022.65 / 0.572322.64 / 0.573120.77 / 0.514222.75 / 0.5754
Remote222.43 / 0.552522.45 / 0.552622.46 / 0.532720.32 / 0.497222.57 / 0.5555
Remote328.37 / 0.690828.41 / 0.693128.56 / 0.704327.08 / 0.651428.75 / 0.7138
Remote422.81 / 0.511622.71 / 0.52122.92 / 0.530921.95 / 0.484923.02 / 0.5513
average24.00 / 0.633624.05 / 0.636224.23 / 0.650222.72 / 0.600624.49 / 0.6665
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TwL-4V [29]exp-SARP [44]SO-TGV [20]DZ-TGV [54]Our model
Boat22.93 / 0.578623.01 / 0.583722.71 / 0.570421.92 / 0.535523.11 / 0.6005
Elaine24.27 / 0.670623.95 / 0.668924.50 / 0.711723.06 / 0.643624.80 / 0.7240
Face23.50 / 0.734724.12 / 0.762424.66 / 0.805822.45 / 0.738925.00 / 0.8181
Girl26.32 / 0.746526.25 / 0.733227.32 / 0.803825.30 / 0.744727.50 / 0.8125
Mountain22.58 / 0.542022.57 / 0.538422.30 / 0.533021.12 / 0.488522.63 / 0.5600
Peppers24.16 / 0.732724.42 / 0.737124.27 / 0.736823.21 / 0.707424.78 / 0.7534
Remote122.61 / 0.576022.65 / 0.572322.64 / 0.573120.77 / 0.514222.75 / 0.5754
Remote222.43 / 0.552522.45 / 0.552622.46 / 0.532720.32 / 0.497222.57 / 0.5555
Remote328.37 / 0.690828.41 / 0.693128.56 / 0.704327.08 / 0.651428.75 / 0.7138
Remote422.81 / 0.511622.71 / 0.52122.92 / 0.530921.95 / 0.484923.02 / 0.5513
average24.00 / 0.633624.05 / 0.636224.23 / 0.650222.72 / 0.600624.49 / 0.6665
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