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December  2017, 11(6): 949-974. doi: 10.3934/ipi.2017044

## Multiplicative noise removal with a sparsity-aware optimization model

 1 College of Mathematics and Statistics, Shenzhen University, Shenzhen 518060, China 2 Department of Mathematics, Syracuse University, Syracuse, NY 13244, USA 3 School of Data and Computer Science, Guangdong Provincial Key Lab of Computational Science, Sun Yat-sen University, Guangzhou 510275, China 4 Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, USA

* Corresponding author: Lixin Shen

Received  April 2016 Revised  August 2017 Published  September 2017

Restoration of images contaminated by multiplicative noise (also known as speckle noise) is a key issue in coherent image processing. Notice that images under consideration are often highly compressible in certain suitably chosen transform domains. By exploring this intrinsic feature embedded in images, this paper introduces a variational restoration model for multiplicative noise reduction that consists of a term reflecting the observed image and multiplicative noise, a quadratic term measuring the closeness of the underlying image in a transform domain to a sparse vector, and a sparse regularizer for removing multiplicative noise. Being different from popular existing models which focus on pursuing convexity, the proposed sparsity-aware model may be nonconvex depending on the conditions of the parameters of the model for achieving the optimal denoising performance. An algorithm for finding a critical point of the objective function of the model is developed based on coupled fixed-point equations expressed in terms of the proximity operator of functions that appear in the objective function. Convergence analysis of the algorithm is provided. Experimental results are shown to demonstrate that the proposed iterative algorithm is sensitive to some initializations for obtaining the best restoration results. We observe that the proposed method with SAR-BM3D filtering images as initial estimates can remarkably outperform several state-of-art methods in terms of the quality of the restored images.

Citation: Jian Lu, Lixin Shen, Chen Xu, Yuesheng Xu. Multiplicative noise removal with a sparsity-aware optimization model. Inverse Problems & Imaging, 2017, 11 (6) : 949-974. doi: 10.3934/ipi.2017044
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##### References:
Five gray-level test images. (a) "Cameraman" ($512\times 512$). (b) "Lena" ($512\times 512$). (c) "Peppers" ($512\times 512$). (d) "Remote1" ($768\times 574$). (e) "Remote2" ($632\times 540$)
(a) PSNR versus number of iterations. (b) Relative error versus number of iterations. Here, various $x^{(0)}$s are used as initial estimates in Algorithm 1. From top to bottom: the test images are the degraded "Cameraman" with multiplicative noise at levels $L=10$, $6$, $4$, and $2$
(a) PSNR versus number of iterations for $L=4$. (b) PSNR versus number of iterations for $L=2$. Here, various $x^{(0)}$s are used as initializations in Algorithm 1 for the degraded "Cameraman" with multiplicative noise
(a) PSNR versus number of iterations. (b) Relative error versus number of iterations. Here, two different $x^{(0)}$s are used as initial estimates in Algorithm 1. The solid lines are plotted firstly by selecting the parameters of Algorithm 1 to obtain (nearly) optimal PSNR values (marked by '$\circ$') under the prescribed tolerance $\text{TOL}=3\times 10^{-4}$; then the dashed lines are plotted by using the same parameters as those of corresponding solid lines. The test images are the degraded "Cameraman" with multiplicative noise at various noise levels ($L=10$, $6$, $4$, and $2$)
Results of various denoising methods on "Cameraman" image corrupted by multiplicative noise with $L=2$ (the first column) and $L=4$ (the second column). From top to bottom: Noisy images (8.63 dB, 11.64 dB), DZ (24.27 dB, 25.85 dB), TwL-4V (25.28 dB, 26.72dB), Ⅰ-DIV (24.98 dB, 26.52 dB), HMNZ (25.30 dB, 27.33 dB), and Ours (25.47 dB, 27.38 dB)
Results of various denoising methods on "Remote1" image corrupted by multiplicative noise with $L=2$ (the first column) and $L=4$ (the second column). From top to bottom: Noisy images (9.28 dB, 12.28 dB), DZ (21.80 dB, 23.01 dB), TwL-4V (22.57 dB, 23.68 dB), Ⅰ-DIV (22.39 dB, 23.48 dB), HMNZ (22.63 dB, 24.03 dB), and Ours (22.76 dB, 23.98 dB)
 Algorithm 1 Fixed-point algorithm based on the proximity operators for model (3). Input: noisy image $f>0$ in $\mathbb{R}^{n}$; parameters $\lambda>0$, $\mu$, $\beta>1$; $\alpha>0$, Initialization: $x^{(0)}$ and $y^{(0)}=0$; positive numbers $\sigma$ and $\rho$ such $\mu<\sigma$ and $\frac{\rho}{\mu}>8\sin^2\frac{(\sqrt{n}-1)\pi}{2\sqrt{n}}$. repeat (a) $x^{(k+1)}\leftarrow\mathrm{prox}_{\frac{1}{\rho}\Phi}(x^{(k)}-\frac{\mu}{\rho}H^\top(Hx^{(k)}-y^{(k)}))$, (b) $y^{(k+1)}\leftarrow\mathrm{prox}_{\frac{\lambda}{\sigma}\psi}(y^{(k)}+\frac{\mu}{\sigma}(Hx^{(k+1)}-y^{(k)}))$, until converges or satisfies a stopping criteria. Write the output of $x^{(k+1)}$ from the above iteration as $\overline{x}$. The restored image is $u^\star=e^{\overline{x}}$.
 Algorithm 1 Fixed-point algorithm based on the proximity operators for model (3). Input: noisy image $f>0$ in $\mathbb{R}^{n}$; parameters $\lambda>0$, $\mu$, $\beta>1$; $\alpha>0$, Initialization: $x^{(0)}$ and $y^{(0)}=0$; positive numbers $\sigma$ and $\rho$ such $\mu<\sigma$ and $\frac{\rho}{\mu}>8\sin^2\frac{(\sqrt{n}-1)\pi}{2\sqrt{n}}$. repeat (a) $x^{(k+1)}\leftarrow\mathrm{prox}_{\frac{1}{\rho}\Phi}(x^{(k)}-\frac{\mu}{\rho}H^\top(Hx^{(k)}-y^{(k)}))$, (b) $y^{(k+1)}\leftarrow\mathrm{prox}_{\frac{\lambda}{\sigma}\psi}(y^{(k)}+\frac{\mu}{\sigma}(Hx^{(k+1)}-y^{(k)}))$, until converges or satisfies a stopping criteria. Write the output of $x^{(k+1)}$ from the above iteration as $\overline{x}$. The restored image is $u^\star=e^{\overline{x}}$.
Parameter values in our algorithm (Algorithm 1) at various noise levels
 $\lambda$ $\alpha$ $\beta$ $\mu$ $\rho$ $\sigma$ $L=10$ 0.306 0.0015 6.06 30 250 150 $L=6$ 0.406 0.00085 15.02 30 250 150 $L=4$ 0.506 0.000108 250 30 255.5 90.26 $L=2$ 0.8 0.00001 1655.05 30 290 156.26
 $\lambda$ $\alpha$ $\beta$ $\mu$ $\rho$ $\sigma$ $L=10$ 0.306 0.0015 6.06 30 250 150 $L=6$ 0.406 0.00085 15.02 30 250 150 $L=4$ 0.506 0.000108 250 30 255.5 90.26 $L=2$ 0.8 0.00001 1655.05 30 290 156.26
Parameter values for all testing algorithms
 $L$ Method $\lambda$ $\alpha$ $\beta$ $\mu$ $\rho$ $\sigma$ 10 Ours 0.569 1.0002 1.00053 20.1 425.5 20.140 TwL-4V 3.6/$L$ 1.0 $-$ $-$ 0.3 $-$ Ⅰ-DIV 0.31 378.0 $-$ $-$ $-$ $-$ DZ 0.07 19 3.0 3.0 $-$ $-$ HMNZ 0.1 $-$ 10 17.5 $-$ $-$ 6 Ours 0.555 0.3885 1.005 19.6 285.5 19.667 TwL-4V 2.9/$L$ 1.0 $-$ $-$ 0.29 $-$ Ⅰ-DIV 0.45 918.0 $-$ $-$ $-$ $-$ DZ 0.06 $3.8$ $3.0$ $3.0$ $-$ $-$ HMNZ 0.1 $-$ 10 9 $-$ $-$ 4 Ours 0.659 0.18515 1.0105 29.658 255.5 29.918 TwL-4V 2.4/$L$ 1.0 $-$ $-$ 0.3 Ⅰ-DIV 0.55 658.0 $-$ $-$ $-$ $-$ DZ 0.05 $1.59$ $3.0$ $3.0$ $-$ $-$ HMNZ 0.1 $-$ 10 6 $-$ $-$ 2 Ours 0.8 0.000001 115.0 21.0 168 26.26 TwL-4V 1.8/$L$ 1.0 $-$ $-$ 0.3 $-$ Ⅰ-DIV 0.84 1059.0 $-$ $-$ $-$ $-$ DZ 0.065 $0.45$ $3.0$ $3.0$ $-$ $-$ HMNZ 0.1 $-$ 10 1.5 $-$ $-$
 $L$ Method $\lambda$ $\alpha$ $\beta$ $\mu$ $\rho$ $\sigma$ 10 Ours 0.569 1.0002 1.00053 20.1 425.5 20.140 TwL-4V 3.6/$L$ 1.0 $-$ $-$ 0.3 $-$ Ⅰ-DIV 0.31 378.0 $-$ $-$ $-$ $-$ DZ 0.07 19 3.0 3.0 $-$ $-$ HMNZ 0.1 $-$ 10 17.5 $-$ $-$ 6 Ours 0.555 0.3885 1.005 19.6 285.5 19.667 TwL-4V 2.9/$L$ 1.0 $-$ $-$ 0.29 $-$ Ⅰ-DIV 0.45 918.0 $-$ $-$ $-$ $-$ DZ 0.06 $3.8$ $3.0$ $3.0$ $-$ $-$ HMNZ 0.1 $-$ 10 9 $-$ $-$ 4 Ours 0.659 0.18515 1.0105 29.658 255.5 29.918 TwL-4V 2.4/$L$ 1.0 $-$ $-$ 0.3 Ⅰ-DIV 0.55 658.0 $-$ $-$ $-$ $-$ DZ 0.05 $1.59$ $3.0$ $3.0$ $-$ $-$ HMNZ 0.1 $-$ 10 6 $-$ $-$ 2 Ours 0.8 0.000001 115.0 21.0 168 26.26 TwL-4V 1.8/$L$ 1.0 $-$ $-$ 0.3 $-$ Ⅰ-DIV 0.84 1059.0 $-$ $-$ $-$ $-$ DZ 0.065 $0.45$ $3.0$ $3.0$ $-$ $-$ HMNZ 0.1 $-$ 10 1.5 $-$ $-$
PSNR (dB) and CPU time (s) for Ⅰ-DIV[29], DZ[11], TwL-4V[17], HMNZ[14], our algorithm (Algorithm 1 with $x^{(0)}=\log(\text{SAR-BM3D}(f))$), and SAR-BM3D[25] for test images of Fig. 1 corrupted by multiplicative noise with $L=10, 6, 4, 2$, respectively
 Image $L$ Noisy Ⅰ-DIV DZ TwL-4V HMNZ Ours SAR-BM3D Camer. 10 PSNR 15.61 28.69 28.30 28.83 29.96 30.15 28.89 Time $-$ 6.28 86.30 6.23 66.62 117.35+6.03 117.35 6 PSNR 13.39 27.35 26.93 27.59 28.46 28.53 26.35 Time $-$ 11.40 110.36 7.33 62.86 116.59+10.11 116.59 4 PSNR 11.64 26.52 25.85 26.72 27.33 27.38 23.67 Time $-$ 13.57 149.54 7.94 63.69 118.31+13.31 118.31 2 PSNR 8.63 24.98 24.27 25.28 25.30 25.47 16.45 Time $-$ 15.57 190.75 9.72 62.58 124.30+15.20 124.30 Lena 10 PSNR 15.64 28.47 27.51 28.60 29.41 29.65 28.48 Time $-$ 6.57 85.90 6.85 65.15 118.64+6.33 118.64 6 PSNR 13.42 27.34 26.24 27.48 28.07 28.14 25.92 Time $-$ 12.06 106.87 7.29 63.69 116.62+10.42 116.62 4 PSNR 11.68 26.64 25.45 26.72 27.01 27.37 23.63 Time $-$ 13.78 144.34 8.48 63.03 118.32+13.14 118.32 2 PSNR 8.71 25.07 24.01 25.17 25.21 25.50 16.46 Time $-$ 17.16 181.92 10.73 62.88 124.56+15.71 124.56 Pepp. 10 PSNR 15.93 28.83 27.20 28.86 29.13 29.53 28.09 Time $-$ 6.88 84.15 7.47 65.33 116.86+5.62 116.86 6 PSNR 13.70 27.95 26.15 27.92 28.12 28.51 25.78 Time $-$ 12.46 108.44 8.16 60.54 118.31+10.24 118.31 4 PSNR 11.98 27.10 25.10 27.05 27.14 27.76 23.63 Time $-$ 13.30 145.13 8.54 63.75 119.16+12.90 119.16 2 PSNR 8.93 25.57 23.72 25.54 25.52 25.97 16.45 Time $-$ 17.44 188.50 10.59 61.59 115.71+15.03 115.71 Rem.1 10 PSNR 16.27 25.22 25.15 25.33 26.16 26.21 25.43 Time $-$ 12.94 190.94 12.78 108.13 210.80+11.14 210.80 6 PSNR 14.00 24.17 24.01 24.41 25.02 25.07 23.64 Time $-$ 23.09 222.15 13.63 102.92 211.02+23.01 211.02 4 PSNR 12.28 23.48 23.01 23.68 24.03 23.98 22.20 Time $-$ 27.56 286.88 14.95 105.97 211.00+26.75 211.00 2 PSNR 9.28 22.39 21.80 22.57 22.63 22.76 16.45 Time $-$ 32.09 347.21 17.75 99.72 211.89+28.51 211.89 Rem.2 10 PSNR 16.23 25.56 25.59 25.63 26.76 26.69 25.92 Time $-$ 9.16 144.40 9.58 84.96 163.00+8.90 163.00 6 PSNR 14.02 24.55 24.30 24.45 25.25 25.43 24.07 Time $-$ 17.46 164.92 9.94 85.24 162.70+17.03 162.70 4 PSNR 12.27 23.45 23.38 23.63 23.93 24.26 22.26 Time $-$ 22.75 219.99 11.79 78.15 163.07+20.35 163.07 2 PSNR 9.22 22.01 21.86 22.18 22.12 22.72 16.08 Time $-$ 25.86 271.57 14.01 75.17 162.84+24.02 162.84
 Image $L$ Noisy Ⅰ-DIV DZ TwL-4V HMNZ Ours SAR-BM3D Camer. 10 PSNR 15.61 28.69 28.30 28.83 29.96 30.15 28.89 Time $-$ 6.28 86.30 6.23 66.62 117.35+6.03 117.35 6 PSNR 13.39 27.35 26.93 27.59 28.46 28.53 26.35 Time $-$ 11.40 110.36 7.33 62.86 116.59+10.11 116.59 4 PSNR 11.64 26.52 25.85 26.72 27.33 27.38 23.67 Time $-$ 13.57 149.54 7.94 63.69 118.31+13.31 118.31 2 PSNR 8.63 24.98 24.27 25.28 25.30 25.47 16.45 Time $-$ 15.57 190.75 9.72 62.58 124.30+15.20 124.30 Lena 10 PSNR 15.64 28.47 27.51 28.60 29.41 29.65 28.48 Time $-$ 6.57 85.90 6.85 65.15 118.64+6.33 118.64 6 PSNR 13.42 27.34 26.24 27.48 28.07 28.14 25.92 Time $-$ 12.06 106.87 7.29 63.69 116.62+10.42 116.62 4 PSNR 11.68 26.64 25.45 26.72 27.01 27.37 23.63 Time $-$ 13.78 144.34 8.48 63.03 118.32+13.14 118.32 2 PSNR 8.71 25.07 24.01 25.17 25.21 25.50 16.46 Time $-$ 17.16 181.92 10.73 62.88 124.56+15.71 124.56 Pepp. 10 PSNR 15.93 28.83 27.20 28.86 29.13 29.53 28.09 Time $-$ 6.88 84.15 7.47 65.33 116.86+5.62 116.86 6 PSNR 13.70 27.95 26.15 27.92 28.12 28.51 25.78 Time $-$ 12.46 108.44 8.16 60.54 118.31+10.24 118.31 4 PSNR 11.98 27.10 25.10 27.05 27.14 27.76 23.63 Time $-$ 13.30 145.13 8.54 63.75 119.16+12.90 119.16 2 PSNR 8.93 25.57 23.72 25.54 25.52 25.97 16.45 Time $-$ 17.44 188.50 10.59 61.59 115.71+15.03 115.71 Rem.1 10 PSNR 16.27 25.22 25.15 25.33 26.16 26.21 25.43 Time $-$ 12.94 190.94 12.78 108.13 210.80+11.14 210.80 6 PSNR 14.00 24.17 24.01 24.41 25.02 25.07 23.64 Time $-$ 23.09 222.15 13.63 102.92 211.02+23.01 211.02 4 PSNR 12.28 23.48 23.01 23.68 24.03 23.98 22.20 Time $-$ 27.56 286.88 14.95 105.97 211.00+26.75 211.00 2 PSNR 9.28 22.39 21.80 22.57 22.63 22.76 16.45 Time $-$ 32.09 347.21 17.75 99.72 211.89+28.51 211.89 Rem.2 10 PSNR 16.23 25.56 25.59 25.63 26.76 26.69 25.92 Time $-$ 9.16 144.40 9.58 84.96 163.00+8.90 163.00 6 PSNR 14.02 24.55 24.30 24.45 25.25 25.43 24.07 Time $-$ 17.46 164.92 9.94 85.24 162.70+17.03 162.70 4 PSNR 12.27 23.45 23.38 23.63 23.93 24.26 22.26 Time $-$ 22.75 219.99 11.79 78.15 163.07+20.35 163.07 2 PSNR 9.22 22.01 21.86 22.18 22.12 22.72 16.08 Time $-$ 25.86 271.57 14.01 75.17 162.84+24.02 162.84
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