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# A non-convex PDE-constrained denoising model for impulse and Gaussian noise mixture reduction

• *Corresponding author: Amine Laghrib
• This paper introduces a novel optimization procedure to reduce a mixture of Gaussian and impulse noise from images. This framework is based on a non-convex PDE-constrained with two diffusion operators: a local weickert and a fractional-order ones. The non-convex norm is applied to remove the impulse component, while the local and fractional operators are introduced to preserve image texture and edges. In the first part, we study the theoretical properties of the proposed PDE-constrained, and we show some well-posedness results. In a second part, after having demonstrated how to numerically find a minimizer, a proximal linearized algorithm combined with a Primal-Dual approach is introduced with local convergence results. Finally, we show extensive denoising experiments on various images and noise intensities which confirms the validity of the non-convex PDE-constrained, its analysis and also the proposed optimization procedure.

Mathematics Subject Classification: Primary: 65K10, 90C26; Secondary: 68U10.

 Citation: • • Figure 1.  Illustration of the $L^p$ norms regularization with diiferents values of $p$

Figure 2.  The behaviour of the function $\Psi$ using low values of $\gamma$ for the two choices: $\frac{t}{t+\gamma}$ and $\frac{t}{1+\frac{t}{\gamma}}$

Figure 3.  The behaviour of the function $\Psi$ using high values of $\gamma$ for the two choices: $\Psi(t) = \frac{t}{t+\gamma}$ and $\Psi(t) = \frac{t}{1+\frac{t}{\gamma}}$

Figure 4.  The influence of the parameter $\gamma$ on the obtained solution when the noise is a mixture of Gaussian and impulse noise with parameter $\sigma^2 = 0.05$ and $r = 10^{-5}$, respectively

Figure 5.  The influence of the parameter $\gamma$ on the obtained solution when the noise is a mixture of Gaussian and impulse noise with parameter $\sigma^2 = 0.02$ and $r = 0.1$, respectively

Figure 6.  The evolution of the restored image with respect to the parameter $\gamma$. The first line presents the restored image compared with the noisy and clean ones, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $\sigma^2 = 0.02$ and $r = 0.5$

Figure 7.  The evolution of the restored image with respect to the parameter $\gamma$. The first line presents the restored image compared with the noisy and clean ones, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $\sigma^2 = 0.03$ and $r = 0.6$

Figure 8.  The computed impulse noise component and the restored image compared to the non-convex norm, when the impulse component is $r = 10^{-3}$

Figure 9.  The computed impulse noise component and the restored "Star" image compared to the non-convex norm, when the impulse component is $r = 10^{-1}$

Figure 10.  The first line presents the restored function compared with the L$^1$ norm and clean image of (Phantom, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $\sigma^2 = 0.002$ and $r = 0.03$

Figure 11.  The first line presents the restored function compared with the L$^1$ norm and clean image of (Phantom, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $\sigma^2 = 0.02$ and $r = 0.2$

Figure 12.  The first line presents the restored function compared with the L$^1$ norm and clean image of (Phantom, while the second line presents the associated 3D surfaces in the same order. Note that the noisy image is constructed using a mixture of Gaussian and impulse noise with parameters $\sigma^2 = 0.03$ and $r = 0.5$

Figure 13.  Comparisons with the CODE  method using four images and using different levels of noise. Note that for the first test, we consider a mixture of Gaussian noise with $\sigma^2 = 0.03$ and impulse noise with parameter $r = 0.4$, while for the second test $\sigma^2 = 0.03$ and $r = 0.5$. For the third test $\sigma^2 = 0.04$ and $r = 0.5$ where for the last test $\sigma^2 = 0.04$ and $r = 0.6$

Figure 14.  Comparison between some denoising PDEs and the proposed fractional one for the (Dune image). Note that the mixed noise is considered with $r = 0.3$ for the impulse parameter and $\sigma^2 = 0.01$ for the Gaussian variance noise

Figure 15.  Comparison between some denoising PDEs and the proposed fractional one for the (Barbara image). Note that the mixed noise is considered with $r = 0.6$ for the impulse parameter and $\sigma^2 = 0.06$ for the Gaussian variance noise

Figure 16.  Comparison between some denoising PDEs and the proposed fractional one for the (Barbara image). Note that the mixed noise is considered with $r = 0.2$ for the impulse parameter and $\sigma^2 = 0.03$ for the Gaussian variance noise

Figure 17.  Comparison between some denoising PDEs and the proposed fractional one for the (Zebra image). Note that the mixed noise is considered with $r = 0.4$ for the impulse parameter and $\sigma^2 = 0.035$ for the Gaussian variance noise

Figure 18.  Comparison between some denoising PDEs and the proposed fractional one for the (Knee MRI image). Note that the mixed noise is considered with $r = 0.2$ for the impulse parameter and $\sigma^2 = 0.03$ for the Gaussian variance noise

Figure 19.  Comparison between some denoising PDEs and the proposed fractional one for the (Brain 2 MRI image). Note that the mixed noise is considered with $r = 0.5$ for the impulse parameter and $\sigma^2 = 0.04$ for the Gaussian variance noise

Figure 20.  The comparison with competitive denoising model for the (Pirate image). Note that we consider a mixture of Gaussian and impulse noise with parameters $\sigma^2 = 0.02$ and $r = 0.35$

Figure 21.  The comparison with competitive denoising model for the (Penguin image). Note that we consider a mixture of Gaussian and impulse noise with parameters $\sigma^2 = 0.05$ and $r = 0.6$

Figure 22.  The efficiency of the fractional order operator in fixing the diffusion with respect to the parameter $\alpha$ for the (Cameraman image)

Figure 23.  The efficiency of the fractional order operator in restoring sharp edges with respect to the parameter $\beta$ for the (Star image)

Figure 24.  The efficiency of the fractional order operator in restoring sharp edges with respect to the parameter $\beta$ for the (Woman image)

Figure 25.  The denoising process using the proposed Primal-Dual compared to Euler-Lagrange iterations as optimization approach for the PDE-constrained with final peak-signal-to-noise ratios for each restoration method. We can see the robustness of the proposed Primal-Dual on both the quality of the restored image and the speed of convergence, compared to the Euler-Lagrange approach

Figure 26.  The PSNR value with respect to the first $1000$ iterations for the four denoising tests

Figure 27.  The logarithm of the relative error $E_k = \log_{10}(RelErr)$ with respect to the first $1000$ iterations for the four previous denoising tests using the primal-dual and Euler-Lagrange algorithms

Table 1.  The PSNR and SSIM table for the four examples in Fig. 13

 Image Method PSNR SSIM Tiger CODE 28.33 0.7922 Our 28.84 0.8027 Penguin CODE 25.61 0.6661 Our 26.03 0.6880 Butterfly CODE 25.22 0.6228 Our 25.43 0.6433 Squirrel CODE 25.02 0.6118 Our 25.69 0.6344

Table 2.  The set of parameters being used in denoising results presented in the three tests

 Parameters Method TV+TV2 TV$^{\alpha}$ TGV$^2$ Nonconvex-TV Our Method Iteration number $N$ 2000 2000 2000 2000 2000 The parameter $\alpha_0$ 0.02 — 0.04 — — The parameter $\alpha_1$ 0.06 — 0.065 — — Spatially decaying effect $(k_1, k_2)$ — — — — $(35, 35)$ The fractional order derivative $\beta$ — 1.35 — — 1.77 Regularization parameter $\lambda$ — 0.01 — 0.02 — The concavity parameter $\gamma$ — — — 42 86

Table 3.  PSNR and SSIM results obtained by applying different denoising methods with different levels of Gaussian and impulse noise to nine selected images. In bold the best (highest) score of each line is shown

 Image Method $\sigma$ noise Metric TV$^{\alpha}$ non-convex-TV TV+TV$^2$ TGV TGV$^2$ BM3D proposed Bridge 0.1 PSNR 31.08 32.33 31.03 32.54 32.72 ${\bf{33.14}}$ 32.94 SSIM 0.883 0.901 0.878 0.899 0.893 0.922 ${\bf{0.922}}$ 0.2 PSNR 30.57 31.43 30.45 31.17 31.66 ${\bf{32.09}}$ 32.05 SSIM 0.641 0.685 0.598 0.612 0.631 0.688 ${\bf{0.722}}$ Castle 0.3 PSNR 29.44 29.57 29.12 30.05 ${\bf{30.18}}$ 30.14 30.08 SSIM 0.784 0.791 0.778 0.800 0.804 0.819 ${\bf{0.821}}$ 0.4 PSNR 28.81 28.94 28.75 29.22 29.85 ${\bf{30.04}}$ 30.00 SSIM 0.711 0.725 0.699 0.737 0.738 0.768 ${\bf{0.772}}$ Baboon 0.2 PSNR 26.88 27.19 24.93 25.40 25.44 27.30 ${\bf{28.84}}$ SSIM 0.781 0.788 0.714 0.737 0.748 0.789 ${\bf{0.811}}$ 0.5 PSNR 23.88 24.22 22.55 23.11 23.96 25.19 ${\bf{26.07}}$ SSIM 0.641 0.685 0.598 0.612 0.631 0.688 ${\bf{0.722}}$ Fly 0.2 PSNR 27.60 28.08 26.72 27.58 27.73 29.03 ${\bf{30.48}}$ SSIM 0.808 0.802 0.767 0.748 0.741 0.794 ${\bf{0.834}}$ 0.4 PSNR 26.18 26.43 24.89 25.98 26.04 27.02 ${\bf{27.88}}$ SSIM 0.720 0.733 0.668 0.696 0.695 0.742 ${\bf{0.747}}$ Bird 0.1 PSNR 31.12 32.26 30.45 31.53 31.32 33.60 ${\bf{34.44}}$ SSIM 0.886 0.880 0.788 0.805 0.822 0.868 ${\bf{0.909}}$ 0.3 PSNR 29.45 30.22 28.89 30.08 30.15 32.52 ${\bf{33.37}}$ SSIM 0.826 0.838 0.738 0.794 0.810 0.850 ${\bf{0.876}}$ Woman 0.1 PSNR 33.66 32.93 34.04 33.08 33.88 ${\bf{34.39}}$ 34.27 SSIM 0.911 0.922 0.906 0.902 0.920 ${\bf{0.942}}$ 0.937 0.4 PSNR 24.71 25.12 24.66 24.99 25.83 26.04 ${\bf{26.43}}$ SSIM 0.598 0.604 0.620 0.672 0.677 ${\bf{0.684}}$ 0.682 Eyes 0.2 PSNR 29.87 29.97 28.01 29.68 29.62 30.30 ${\bf{31.77}}$ SSIM 0.812 0.842 0.786 0.796 0.780 0.840 ${\bf{0.857}}$ 0.4 PSNR 27.44 28.15 26.93 27.94 27.55 28.55 ${\bf{29.03}}$ SSIM 0.758 0.759 0.650 0.692 0.700 0.774 ${\bf{0.783}}$ Gazelle 0.5 PSNR 25.19 25.36 23.55 24.02 24.12 25.04 ${\bf{26.49}}$ SSIM 0.597 0.612 0.509 0.547 0.520 0.616 ${\bf{0.678}}$ 0.6 PSNR 23.45 23.50 22.11 22.44 22.22 23.29 ${\bf{23.66}}$ SSIM 0.478 0.481 0.403 0.409 0.413 0.510 ${\bf{0.608}}$ Goose 0.1 PSNR 33.49 33.70 30.88 31.01 31.44 34.17 ${\bf{34.65}}$ SSIM 0.908 0.920 0.865 0.890 0.896 0.919 ${\bf{0.921}}$ 0.3 PSNR 31.08 31.13 28.91 29.17 29.34 31.01 ${\bf{32.36}}$ SSIM 0.818 0.851 0.708 0.713 0.709 0.825 ${\bf{0.866}}$
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