# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method
\right.. \tag{19}\]  
In the ADMM framework, dual variables can be updated using the ascending gradient method as follows:  
\[\left\{\begin{array}{l}\mathbf{\Lambda_{0}^{(k+1)}}=\mathbf{\Lambda_{1}^{(k)}}+ \gamma\beta_{0}(\mathbf{H}*\mathbf{F}^{(k+1)}-\mathbf{G}-\mathbf{X_{0}^{(k+1)}})\\ \mathbf{\Lambda_{1}^{(k+1)}}=\mathbf{\Lambda_{1}^{(k)}}+\gamma\beta_{1}(\mathbf{K_{h}*\mathbf{F }^{(k+1)}}-\mathbf{V_{h}^{(k+1)}}-\mathbf{X_{1}^{(k+1)}})\\ \mathbf{\Lambda_{```
1:Input:G,H
2:Output:\(F\)
3:Initialize:
4:\(k=1,X_{i}^{(k)}=0,\Lambda_{i}^{(k)}=0,(i=0,1,\ldots,5),F^{(k)}=0,\mu_{0},\mu_{1}, \beta_{i}(i=0,1,2),\gamma,p,tol\).
5:Set \(\|\mathbf{F}^{(k+1)}-\mathbf{F}^{(k)}\|_{2}/\|\mathbf{F}^{(k)}\|_{2}\) as 1;
6:While \(\|\mathbf{F}^{(k+1)}-\mathbf{F}^{(k)}\|_{2}/\|\mathbf{F}^{(k)}\|_{2}>tol\)do
7:Update \(\mathbf{F}^{(k+1)},\mathbf{V}_{h}^{(k+1)},V_{\mathbf{v}}^{(k+1)}\) according to Equation (16);
8:Update \(X_{i}^{(k+1)}(i=0,1,\ldots,5)\) according to Equation (18);
9:Update \(\Lambda_{i}^{(k+1)}(i=0,1,\ldots,5)\) according to Equation (20);
10:\(k=k+1\);
11:End while
12:Return\(F^{(k)}\) as F.
```

**Algorithm 1** Pseudocode TGV_Lp for image restoration

## 4 Numerical Experiments

To verify the appropriateness and validity of the proposed model, standard images of different styles were selected as experimental objects. The images were degraded with Gaussian blur, average blur, motion blur, and different degrees of impulse noise so that a large number of numerical simulation experiments could be performed to make indicator-based comparisons and evaluations in an objective manner. In particular, the image \"Satellite\" was downloaded from the Emory University image database [27], and the other images were downloaded from CVG-UGR (Computer Vision Group, University of Granada) image database [28]. These test images were widely used in other relevant publications, thereby allowing for objective and fair comparison of the test results. All calculations in this study were performed using MATLAB R2018a on a Windows 10 system with an Intel @Core (tm) i7-7700 central processing unit (CPU) (4.2 GHz) with 16.0 GB of memory.

### Evaluation Indicators and Stopping Criteria

For objective evaluation of image quality, PSNR, SSIM, and SNR were used. Their definitions are given below [29].

\[PSNR(\mathbf{X},\mathbf{Y})=10 log_{10}\frac{L^{2}}{\frac{1}{N^{2}}\sum\limits_{i=1}^{ N}\sum\limits_{j=1}^{N}\left(\mathbf{X}_{ij}-\mathbf{Y}_{ij}\right)^{2}}, \tag{21}\]

where \(\mathbf{X}\) represents an original image, and \(\mathbf{Y}\) represents a restored image; \(L\) represents the maximum gray level of the image, which was set as 255 in this study. Generally, a larger PSNR value of a restored image denotes less image distortion and greater similarity between the restored image and the original image. \[SSIM(\mathbf{X},\mathbf{Y})=\frac{(2\mu_{X}\mu_{Y}+L^{2}k_{1}^{2})(2\sigma_{XY}+L^{2} k_{2}^{2})}{\Big{(}\mu_{X}^{2}+\mu_{Y}^{2}+L^{2}k_{1}^{2}\Big{)}\Big{(}\sigma_{X}^{2 }+\sigma_{Y}^{2}+L^{2}k_{2}^{2}\Big{)}}, \tag{22}\]

where \(\mu_{X}\) and \(\mu_{Y}\) represent the means of images \(\mathbf{X}\) and \(\mathbf{Y}\), respectively, \(\sigma_{X}^{2}\) and \(\sigma_{Y}^{2}\) represent the variances of \(\mathbf{X}\) and \(\mathbf{Y}\), respectively, \(\sigma_{XY}\) represents the covariance of \(\mathbf{X}\) and \(\mathbf{Y}\), and the constants \(k_{1}\) and \(k_{2}\) are very small positive numbers introduced to prevent the denominator of Equation (22) from being zero; they were set as 0.01 and 0.03 in this study, respectively. The maximum value of SSIM is 1, which indicates that the original image and the restored image are 100% identical; as the value of SSIM approaches 1, the similarity between the original and restored images increases. \[SNR(\mathbf{X},\mathbf{Y})=10 log_{10}\frac{\|\mathbf{X}\|_{2}^{2}}{\|\mathbf{X}-\mathbf{Y}\|_{2}^{ 2}}. \tag{23}\]Images with low noise have higher SNR values, making SNR an important metric for characterizing the performance of the restoration algorithm.