# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method
When restoring the nine degraded images treated with the Gaussian blur kernel \((G,7,5)\) and 30-60% impulse noise, the proposed method achieved 0.58-2.85 dB higher PSNR values compared with the TGV method. When restoring the nine degraded images treated with the Gaussian blur kernel \((G,15,5)\) and impulse noise, the proposed method achieved 0.45-1.73 dB higher PSNR values compared with the TGV method. For example, for the \"Bird\" image of Figure 3e treated with the Gaussian blur kernel \((G,7,5)\) and two different levels (50% vs. 60%) of noise, the PSNR values of the proposed method (32.31 dB and 29.69 dB) were 2.85 dB and 2.7 dB higher than those of the TGV method (29.46 dB and 26.99 dB), respectively. 3. The OGSTV method is an excellent algorithm for image deblurring and impulse noise removal, and it uses the neighborhood gradient information of the image to form a combined gradient for image restoration, which achieves good image restoration performance for smooth image regions but is less satisfactory in recovering image edge regions containing sharp lines and angles. In the test, it was observed that, for smooth images with non-complex textures, such as the \"Plane\" image of Figure 3h and \"Satellite\" image of Figure 3i, the OGSTV method had particularly excellent performance, which was slightly better than that of the proposed algorithm in the case of low noise levels. In the case of high noise levels, however, the proposed algorithm was still superior to the OGSTV method. For example, when restoring the \"Satellite\" image degraded with \((G,7,5)\) and 30% noise, the OGSTV method achieved a slightly higher PSNR value of 36.97 dB compared to the proposed method (36.54 dB), with a difference of only 0.43 dB; however, in the cases of 50% and 60% noise, the PSNR values of the OGSTV method were lower than those of the proposed method. The above analysis indicates that the use of Lp-pseudo-norm shrinkage in the proposed method increases the degree of freedom by one, making it less difficult to describe image gradient sparsity. The performance of the proposed method in removing Gaussian blur and impulse noise is generally better than that of the three competitive methods, especially for images with complex textures. In \begin{table}
\begin{tabular}{l c c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Images**} & **Degrada** & \multicolumn{3}{c}{**FTVd**} & \multicolumn{3}{c}{**TGV**} & \multicolumn{3}{c}{**OGSTV**} & \multicolumn{3}{c}{**Proposed**} \\  & **-tion** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** \\ \hline \multirow{2}{*}{**Fingerprint**} & \multirow{2}{*}{} & 27.31 & 0.877 & 18.78 & 29.89 & 0.935 & 21.36 & 30.09 & 0.934 & 21.55 & **30.72** & **0.970** & **22.18** \\
**Butterfly** & & 24.91 & 0.705 & 18.67 & 26.96 & 0.861 & 20.72 & 27.05 & 0.864 & 20.80 & **27.47** & **0.873** & **21.22** \\
**Baboon** & Blur: & 19.89 & 0.481 & 14.43 & 20.99 & 0.597 & 15.53 & 21.09 & 0.603 & 15.62 & **21.72** & **0.645** & **16.27** \\
**B boat** & \((G,15,5)\) & 24.73 & 0.679 & 19.20 & 25.63 & 0.815 & 20.29 & 26.37 & 0.835 & 21.02 & **26.86** & **0.851** & **21.82** \\
**Bird** & Noise & 28.15 & 0.741 & 19.08 & 29.60 & 0.885 & 20.54 & 29.81 & 0.892 & 20.75 & **30.05** & **0.906** & **20.91** \\
**Hill** & level: & 25.98 & 0.690 & 19.68 & 27.37 & 0.830 & 21.01 & 27.41 & 0.832 & 21.05 & **28.01** & **0.840** & **21.65** \\
**Truck** & 30\% & 27.56 & 0.707 & 20.29 & 29.42 & 0.862 & 22.16 & 29.57 & 0.866 & 22.31 & **30.30** & **0.872** & **23.01** \\
**Plane** & & 29.76 & 0.815 & 26.58 & 30.67 & 0.904 & 27.65 & 31.34 & 0.938 & 28.15 & **31.82** & **0.948** & **28.64** \\
**Satellite** & & 28.20 & 0.821 & 14.52 & 29.66 & 0.900 & 15.98 & **31.47** & **0.963** & **17.79** & 31.39 & 0.962 & 17.71 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical results of PSNR (dB), SSIM, and SNR (dB) for the recovery of nine test images degraded with a Gaussian blur kernel \((G,15,5)\) and 30% impulse noise.