# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method
In the experiments, the iteration stopping criterion for comparing various algorithms was Equation (24), where \(\mathbf{F}^{(k)}\) and \(\mathbf{F}^{(k+1)}\) denote the objective functions after the \(k\)-th iteration and \((k+1)\)-th iteration, respectively. \[\|\mathbf{F}^{(k+1)}-\mathbf{F}^{(k)}\|_{2}\cdot\|\mathbf{F}^{(k)}\|_{2}^{-1}<10^{-4}. \tag{24}\]

The blur kernels used in the test were Gaussian blur, average blur, and motion blur, all of which were generated by built-in functions of MATLAB. The function fspecial ('gaussian', \(S\),\(\sigma\)) generates an \(S\times S\) Gaussian blur kernel standard deviation of \(\sigma\), and the kernel is hereinafter referred to as \((G,S,\sigma)\) for convenience; fspecial ('average', \(S\)) generates an \(S\times S\) average blur kernel, which is referred to as \((A,S)\); fspecial ('motion', \(L\),\(\theta\)) generates a motion blur kernel with a length of \(L\) and angle of \(\theta\), which is referred to as \((M,L,\theta)\). Noise effects were generated by built-in function immoise (Ib,'salt & pepper', level), where \"Ib\" is the blurred image and \"level\" is the noise density, i.e., the percentage of noise in the total number of pixels. ### Parameter Selection and Sensitivity Analysis

Parameters involved in the algorithm of this study are \(\mu_{0},\ \mu_{1},\ \beta_{i}(i=0,1,2),p,\ \text{and}\ \gamma\). In particular, \(\mu_{0}=\mu\alpha_{0}\) and \(\mu_{1}=\mu\alpha_{1}\) are regularization parameters with values greater than zero, which serve to balance the fidelity term and regularization term. Based on our previous experience with testing, we assumed here that \(\alpha_{0}=2\alpha_{1}\) while gradually increasing the value of \(\mu\) between 0.1 and 10 to find the optimal effect of image restoration. Given the characteristics of impulse noise, the value of the Lp-pseudo-norm was gradually increased from 0.1 to 1 by a step of 0.05, and a loop statement was adopted to traverse all paths to find the \(p\) value for the optimal PSNR. In all tests, the penalty parameter \(\beta_{i}(i=0,1,2)\) of the ALM method was set to have ratios of \(\beta_{0}:\beta_{1}:\beta_{2}=50:1:5\), and the slack variable was set to 1, i.e., \(\gamma=1\). #### 4.2.1 Regularization Parameter \(\mu\)

To test the effect of the regularization parameter \(\mu\) on the image restoration results, different levels of blur and impulse noise were introduced into the test images (a)-(i), and PSNR values were recorded by gradually increasing the value of \(\mu\) in the range of 0.1-10 while fixing other parameters. The experimental results show that the proposed algorithm is not sensitive to \(\mu\); that is, during the recovery process of different images, the value of PSNR does not undergo significant changes with changes in \(\mu\), indicating that the proposed algorithm is robust. The test results of the \"Boat\" image of Figure 3d and \"Hill\" image of Figure 3f treated with the Gaussian blur kernel at 30%, 40%, 50%, and 60% noise levels are presented in Figure 4. The PSNR value in Figure 4 was not optimal for the proposed algorithm, as other fixed parameters remained to be adjusted optimally. In subsequent experiments, the regularization parameter was set to 1, i.e.,\(\mu=1\). #### 4.2.2 Value of \(p\) in Lp-Pseudo-Norm

When it came to selecting the \(p\) value of the Lp-pseudo-norm, given the characteristics of the impulse noise, the \(p\) value was gradually increased from 0.1 to 1 in steps of 0.05, and a loop statement was adopted to traverse all paths so as to find the \(p\) value that could give the optimal PSNR value of image restoration. For restoration of the nine images treated with \((G,7,5)\) at 30-60% noise levels,

Figure 4: Effect of the regularization parameter \(\mu\) on the experimental results. (**a**) Effect of \(\mu\) on peak signal-to-noise ratio (PSNR) at noise levels 30%, 40%, 50%, and 60% with image “Boat”. (**b**) Effect of \(\mu\) on PSNR at noise levels 30%, 40%, 50%, and 60% with image “Hill”.