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

Hanwool Na; Myeongmin Kang; Miyoun Jung; Myungjoo Kang

doi:10.3934/ipi.2019007

Article Contents

2019, Volume 13, Issue 1: 117-147. Doi: 10.3934/ipi.2019007

This issue Previous Article The regularized monotonicity method: Detecting irregular indefinite inclusions Next Article Note on Calderón's inverse problem for measurable conductivities

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

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
* Corresponding author: Myungjoo Kang

Received: December 31, 2017

Revised: April 30, 2018

Published: December 2018

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Abstract

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.

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Mathematics Subject Classification: 68U10, 68T05.

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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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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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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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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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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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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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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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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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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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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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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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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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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
Boat	25.30 / 0.6851	25.54 / 0.6985	24.93 / 0.6762	25.22 / 0.6853	25.65 / 0.7085
Elaine	27.05 / 0.7646	27.06 / 0.7688	27.54 / 0.7977	27.21 / 0.7792	27.91 / 0.8104
Face	26.70 / 0.8158	27.08 / 0.8508	27.99 / 0.8779	26.84 / 0.8455	28.28 / 0.8851
Girl	28.99 / 0.8199	28.79 / 0.8218	30.39 / 0.8724	29.06 / 0.8655	30.57 / 0.8788
Mountain	24.73 / 0.6491	24.70 / 0.6539	24.42 / 0.6486	24.16 / 0.6348	24.82 / 0.6666
Peppers	27.10 / 0.8064	27.22 / 0.8161	27.10 / 0.8143	27.12 / 0.8124	27.51 / 0.8303
Remote1	24.97 / 0.7091	25.07 / 0.7072	24.96 / 0.7095	24.23 / 0.6885	25.20 / 0.7098
Remote2	24.70 / 0.6912	24.74 / 0.7010	24.66 / 0.6775	23.89 / 0.6650	24.83 / 0.7026
Remote3	30.33 / 0.7690	30.36 / 0.7701	30.60 / 0.7816	30.01 / 0.7583	30.82 / 0.7846
Remote4	24.62 / 0.6287	24.64 / 0.6611	24.72 / 0.6442	24.33 / 0.6311	24.89 / 0.6756
average	26.45 / 0.7338	26.52 / 0.7449	26.73 / 0.75	26.21 / 0.7365	27.05 / 0.7652

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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
Boat	23.84 / 0.6207	23.89 / 0.6309	23.62 / 0.6165	23.41 / 0.6081	24.17 / 0.6515
Elaine	25.54 / 0.7122	25.41 / 0.7140	26.01 / 0.7552	25.17 / 0.7173	26.25 / 0.7638
Face	25.02 / 0.7707	25.53 / 0.8065	26.12 / 0.8405	24.51 / 0.7974	26.50 / 0.8493
Girl	27.63 / 0.7806	27.29 / 0.7728	28.87 / 0.8394	26.90 / 0.8131	29.00 / 0.8446
Mountain	23.52 / 0.5869	23.40 / 0.5843	23.16 / 0.5791	22.66 / 0.5544	23.52 / 0.6013
Peppers	25.68 / 0.7601	25.82 / 0.7763	25.60 / 0.7732	25.17 / 0.7621	26.16 / 0.7959
Remote1	23.53 / 0.6329	23.60 / 0.6220	23.54 / 0.6422	22.55 / 0.6005	23.70 / 0.6461
Remote2	23.38 / 0.6119	23.39 / 0.6171	23.34 / 0.5835	22.22 / 0.5755	23.58 / 0.6249
Remote3	29.32 / 0.7288	29.36 / 0.7286	29.45 / 0.7351	28.20 / 0.6977	29.76 / 0.7434
Remote4	23.53 / 0.5588	23.42 / 0.5787	23.63 / 0.5750	22.91 / 0.5500	23.75 / 0.6075
average	25.10 / 0.6763	25.11 / 0.6831	25.33 / 0.6939	24.37 / 0.6676	25.64 / 0.7128

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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
Boat	22.93 / 0.5786	23.01 / 0.5837	22.71 / 0.5704	21.92 / 0.5355	23.11 / 0.6005
Elaine	24.27 / 0.6706	23.95 / 0.6689	24.50 / 0.7117	23.06 / 0.6436	24.80 / 0.7240
Face	23.50 / 0.7347	24.12 / 0.7624	24.66 / 0.8058	22.45 / 0.7389	25.00 / 0.8181
Girl	26.32 / 0.7465	26.25 / 0.7332	27.32 / 0.8038	25.30 / 0.7447	27.50 / 0.8125
Mountain	22.58 / 0.5420	22.57 / 0.5384	22.30 / 0.5330	21.12 / 0.4885	22.63 / 0.5600
Peppers	24.16 / 0.7327	24.42 / 0.7371	24.27 / 0.7368	23.21 / 0.7074	24.78 / 0.7534
Remote1	22.61 / 0.5760	22.65 / 0.5723	22.64 / 0.5731	20.77 / 0.5142	22.75 / 0.5754
Remote2	22.43 / 0.5525	22.45 / 0.5526	22.46 / 0.5327	20.32 / 0.4972	22.57 / 0.5555
Remote3	28.37 / 0.6908	28.41 / 0.6931	28.56 / 0.7043	27.08 / 0.6514	28.75 / 0.7138
Remote4	22.81 / 0.5116	22.71 / 0.521	22.92 / 0.5309	21.95 / 0.4849	23.02 / 0.5513
average	24.00 / 0.6336	24.05 / 0.6362	24.23 / 0.6502	22.72 / 0.6006	24.49 / 0.6665

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