# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Images** & **Blur Size** & **Noise Level** & **FTvd** & **TGV** & **OGSTV** & **Proposed** \\ \hline \multirow{4}{*}{**Baboon**} & \((M,10,10)\) & 30 & 21.7921 & 24.0100 & 22.0647 & **24.8964** \\  & \((M,10,10)\) & 40 & 21.1785 & 22.9020 & 23.2183 & **24.3687** \\  & \((M,20,20)\) & 30 & 21.0863 & 23.0958 & 22.8402 & **23.6167** \\  & \((M,20,20)\) & 40 & 20.7721 & 22.4554 & 22.3522 & **23.2369** \\  & & 30 & 26.0616 & 28.7070 & 28.4671 & **29.3110** \\  & \((M,30,30)\) & 40 & 25.3602 & 27.9250 & 27.8820 & **28.9013** \\  & \((M,30,30)\) & 50 & 24.2406 & 26.8643 & 27.2269 & **28.3973** \\  & & 60 & 22.4797 & 25.2250 & 26.4264 & **27.7279** \\  & & 30 & 30.0618 & 31.7805 & 32.5596 & **33.3166** \\  & \((M,20,20)\) & 40 & 28.7341 & 30.3195 & 31.8788 & **32.6383** \\  & & 50 & 26.2732 & 28.2106 & 31.1538 & **31.7443** \\  & & 60 & 22.9536 & 25.2543 & 30.1680 & **30.3756** \\  & & 30 & 30.3177 & 31.8087 & 32.3495 & **33.3499** \\
**Truck** & \((M,20,20)\) & 40 & 29.1381 & 30.3154 & 31.7592 & **32.7189** \\  & & 50 & 27.5260 & 28.4093 & 31.1104 & **31.9235** \\  & & 60 & 24.7013 & 25.2220 & 30.0642 & **30.7173** \\ \hline \hline \end{tabular}
\end{table} 
Table 5: Numerical results of PSNR (dB) for the image restoration test conducted on four test images degraded with different levels of blur and impulse noise. #### 4.4.2 Visual Effect Comparison for Image Restoration in the Case of Average Blur

Figures 8 and 9 present the \"Fingerprint\" image restored using the four algorithms from the image degraded with average blur kernel \((A,7)\) and 50% impulse noise, as well as the zoomed-in restored images for better comparison. With respect to the visual effects of the restored images, the FTVd method (c) led to a certain degree of staircase effect in the restored image, the TGV method (d) led

Figure 6: Comparison of the visual effects and PSNRs of image restorations of the “Boat” image degraded with a Gaussian blur kernel \((G,7,5)\) and 30% impulse noise: (**a**) blurred image; (**b**) blurred image with 30% impulse noise; (**c**) fast total variation deconvolution (FTVd)-restored image; (**d**) total generalized variation (TGV)-restored image; (**e**) overlapping group sparsity total variation (OGSTV)-restored image; (**f**) proposed algorithm-restored image. Figure 7: Local zoomed-in images from Figure 6 for better comparison: (**a**) selected region of the original image; (**b**) local zoomed-in image of the selected region; (**c**) FTVd-restored image; (**d**) TGV-restored image; (**e**) OGSTV-restored image; (**f**) proposed algorithm-restored image. to blurred image edges in the restored image, and the OGSTV method (e) needed to be improved in preserving image edges. In contrast, the proposed method performed well in restoring the details of the image lines. Figure 8: Comparison of the visual effects and PSNRs of image restorations of the “Fingerprint” image degraded with an average blur kernel \((A,7)\) and 50% impulse noise: (**a**) blurred image; (**b**) blurred image with 50% impulse noise; (**c**) FTvd-restored image; (**d**) TGV-restored image; (**e**) OGSTV-restored image; (**f**) proposed algorithm-restored image. Figure 9: Local zoomed-in images from Figure 7 for better comparison: (**a**) selected region of the original image; (**b**) local zoomed-in image of the selected region; (**c**) FTvd-restored image; (**d**) TGV-restored image; (**e**) OGSTV-restored image; (**f**) proposed algorithm-restored image. #### 4.4.3 Visual Effect Comparison for Image Restoration of Images Degraded with Motion Blur

Figure 10 compares the zoomed-in details of the \"Butterfly\" image restored using the four algorithms with those of the image degraded with a motion blur kernel \((M,20,20)\) and \(30\%\) impulse noise.