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

• * Corresponding author: Myungjoo Kang
• 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.

Mathematics Subject Classification: 68U10, 68T05.

 Citation: • • 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 $(275 \times 275)$

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).

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)

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.

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 ; (c) exp-SARP ; (d) SO-TGV ; (e) DZ-TGV ; (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 ; (c) exp-SARP ; (d) SO-TGV ; (e) DZ-TGV ; (f) 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 , exp-SARP , SO-TGV , DZ-TGV , 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 , exp-SARP , SO-TGV , DZ-TGV , and 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 ; (c) exp-SARP ; (d) SO-TGV ; (e) DZ-TGV ; (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 ; (c) exp-SARP ; (d) SO-TGV ; (e) DZ-TGV ; (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 ; (c) exp-SARP ; (d) SO-TGV ; (e) DZ-TGV ; (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 ; (c) exp-SARP ; (d) SO-TGV ; (e) DZ-TGV ; (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 ; (c) exp-SARP ; (d) SO-TGV ; (e) DZ-TGV ; (f) 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 , exp-SARP , SO-TGV , DZ-TGV , our model

Figure 15.  Comparison of denoised images of our model and SAR-BM3D model  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 16.  Comparison of denoised images of our model and SAR-BM3D model  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

Table 1.  Comparisons of denoising results when $M = 10$ (PSNR/SSIM)

 TwL-4V  exp-SARP  SO-TGV  DZ-TGV  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

Table 2.  Comparisons of denoising results when $M = 5$ (PSNR/SSIM)

 TwL-4V  exp-SARP  SO-TGV  DZ-TGV  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

Table 3.  Comparisons of denoising results when $M = 3$ (PSNR/SSIM)

 TwL-4V  exp-SARP  SO-TGV  DZ-TGV  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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