# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method
Figure 3: Images used in the simulation: (**a**) Fingerprint \(256\times 256\); (**b**) Butterfly \(512\times 512\); (**c**) Baboon \(512\times 512\); (**d**) Boat \(512\times 512\); (**e**) Bird \(512\times 512\); (**f**) Hill \(512\times 512\); (**g**) Truck \(512\times 512\); (**h**) Plane \(512\times 512\); (**i**) Satellite \(512\times 512\). the best \(p\) values were obtained and are listed in Table 1, with the results showing that the \(p\) value was relatively stable for the same image. In addition, to better show the best selection of the \(p\) and \(\mu\) values, we used a three-dimensional (3D) figure. The test results of the \"Hill\" image treated with the Gaussian blur kernel at 30% noise level are presented in Figure 5.

### Numerical Performance Comparison

To test the effectiveness of the proposed algorithm, the following four algorithms were compared: an FTVd model, a TGV model, an OGSTV model, and the proposed frequency-domain TGV-Lp model. The algorithms were adapted from the authors' web pages or shared by the authors. It is noteworthy that all these algorithms are state-of-the-art algorithms in the field of image restoration. In the experiments, regularization parameters were adjusted until each algorithm achieved its best performance for image restoration to ensure an objective and fair comparison. #### 4.3.1 Numerical Performance Comparison for Image Recovery in the Presence of Gaussian Blur

The results of the image restoration tests performed on nine test images degraded with Gaussian blur kernel \((G,7,5)\) and 30-60% impulse noise are listed in Table 2. In the table, the optimal indicators are highlighted in bold. \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline
**Noise Level** & **Fingerprint** & **Butterfly** & **Baboon** & **Boat** & **Bird** & **Hill** & **Truck** & **Plane** & **Satellite** \\ \hline
30\% & 0.30 & 0.45 & 0.50 & 0.35 & 0.50 & 0.55 & 0.50 & 0.55 & 0.75 \\
40\% & 0.30 & 0.45 & 0.50 & 0.35 & 0.50 & 0.55 & 0.50 & 0.55 & 0.75 \\
50\% & 0.30 & 0.45 & 0.50 & 0.35 & 0.50 & 0.55 & 0.50 & 0.55 & 0.75 \\
60\% & 0.25 & 0.50 & 0.50 & 0.35 & 0.50 & 0.55 & 0.55 & 0.55 & 0.75 \\ \hline \end{tabular}
\end{table} 
Table 1: Values of \(p\) that give optimal peak signal-to-noise ratio (PSNR) values for different test images. Figure 5: Effect of the \(p\) and \(\mu\) values on PSNR: PSNR (dB) for the recovery of image “Hill” degraded with a Gaussian blur kernel \((G,7,5)\) and 30% impulse noise. The results of image restoration tests performed on nine test images degraded with a Gaussian blur kernel \((G,15,5)\) and 30% noise are shown in Table 3.

\begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Images**} & **Noise** & \multicolumn{3}{c}{**FTVd**} & \multicolumn{3}{c}{**TGV**} & \multicolumn{3}{c}{**OGSTV**} & \multicolumn{3}{c}{**Proposed**} \\  & **Level** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** \\ \hline \multirow{4}{*}{**Finger**} & 30 & 30.49 & 0.950 & 21.90 & 36



An in-depth analysis of the data in Tables 2 and 3 leads to the following conclusions:

1. All four methods can effectively recover images that were degraded by Gaussian blur and different degrees of impulse noise. The PSNR, SSIM, and SNR values of the proposed method are generally higher than those of other competitive methods, which indicates that the proposed method has better deblurring and denoising effects. In particular, for images with many lines and complex textures, such as \"Fingerprint\" in Figure 3a, \"Butterfly\" in Figure 3b, and \"Baboon\" in Figure 3c, the proposed method performed well, especially at high noise levels. For example, for the \"Fingerprint\" image of Figure 3a treated with a Gaussian blur kernel \((G,7,5)\) and 50% noise, the PSNR value (35.59 dB) of the proposed method was higher by 7.22 dB than that of the FTVd method (28.37 dB), and was higher by 2.55 dB and 1.6 dB than that of the TGV method (33.04 dB) and the OGSTV method (33.99 dB), respectively. 2.