# 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}{l c c c c c c c c c c} \hline \hline \multirow{2}{*}{**Images**} & **Blur** & **Noise** & \multicolumn{3}{c}{**TGV**} & \multicolumn{3}{c}{**OGSTV**} & \multicolumn{3}{c}{**Proposed**} \\  & **Size** & **Level** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** & **PSNR** & **SSIM** & **SNR** \\ \hline \multirow{3}{*}{**Fingerprint**} & 30 & 36.46 & 0.983 & 27.91 & 36.73 & 0.980 & 29.19 & **38.31** & **0.989** & **29.77** \\  & 60 & 28.74 & 0.905 & 20.20 & 29.30 & 0.907 & 21.24 & **30.11** & **0.910** & **21.57** \\  & 30 & 30.40 & 0.918 & 24.15 & 30.99 & 0.933 & 24.74 & **32.21** & **0.947** & **25.97** \\
**Butterfly** & 60 & 25.83 & 0.784 & 19.59 & 26.45 & 0.803 & 20.66 & **28.59** & **0.895** & **22.35** \\  & 30 & 23.01 & 0.761 & 17.54 & 23.29 & 0.778 & 17.83 & **24.11** & **0.814** & **18.65** \\
**Baboon** & 60 & 21.14 & 0.622 & 15.68 & 21.51 & 0.657 & 16.10 & **22.31** & **0.715** & **16.85** \\  & 30 & 29.11 & 0.894 & 23.77 & 29.60 & 0.907 & 24.26 & **31.50** & **0.933** & **26.16** \\
**Boat** & 60 & 25.11 & 0.755 & 19.77 & 26.24 & 0.776 & 20.52 & **27.22** & **0.855** & **21.88** \\
**Bird** & \((A,7)\) & 30 & 32.75 & 0.933 & 23.67 & 33.53 & 0.948 & 24.47 & **34.53** & **0.957** & **25.46** \\  & 60 & 27.05 & 0.785 & 17.99 & 29.06 & 0.853 & 20.35 & **30.19** & **0.899** & **21.13** \\  & 30 & 30.33 & 0.899 & 23.97 & 30.85 & 0.910 & 24.49 & **32.07** & **0.927** & **25.71** \\
**Hill** & 60 & 25.96 & 0.757 & 19.60 & 26.10 & 0.762 & 20.07 & **27.96** & **0.852** & **21.60** \\  & 30 & 32.79 & 0.925 & 25.52 & 33.61 & 0.940 & 26.34 & **34.32** & **0.947** & **27.06** \\
**Truck** & 60 & 27.25 & 0.780 & 19.97 & 30.28 & 0.848 & 22.32 & **31.38** & **0.904** & **24.12** \\  & 30 & 33.87 & 0.945 & 30.68 & **36.45** & **0.975** & **33.27** & 36.37 & 0.969 & 33.08 \\
**Plane** & 60 & 28.15 & 0.793 & 24.97 & 31.87 & 0.922 & 28.69 & **32.31** & **0.926** & **29.13** \\
**Satellite** & 30 & 33.95 & 0.956 & 20.28 & **36.85** & **0.989** & **23.18** & 36.52 & 0.981 & 23.10 \\  & 60 & 28.06 & 0.814 & 14.38 & 28.52 & **0.933** & 14.84 & **28.54** & **0.933** & **14.87** \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Numerical results of PSNR (dB), SSIM, and SNR (dB) for the recovery of nine test images degraded with an average blur kernel \((A,7)\) and 30–60% impulse noise. For different images treated with varying degrees of motion blur and impulse noise, the data in Table 5 reveal that the proposed method outperformed the TGV and OGSTV methods in removing motion blur and impulse noise. When restoring the four images, the proposed method achieved 0.52-5.50 dB and 0.21-2.83 dB higher PSNR values than the TGV method and the OGSTV method, respectively. ### Comparison of Visual Effects

To better observe the test results, some restored images are presented below for visual comparison purposes. #### 4.4.1 Comparison of the Visual Effects of Restored Images in the Case of Gaussian Blur

Figures 6 and 7 present image restorations by the four algorithms of the \"Boat\" image degraded with a Gaussian blur kernel \((G,7,5)\) and 30% impulse noise, and zoomed-in images of the restored images for better comparison, respectively. With respect to the visual effects of the restored images, with the FTvd method (c), a loss of some image details occurred during restoration of the image, and there was an obvious staircase effect. The TGV method (d) and the OGSTV method (e) performed well in suppressing the staircase effect, but slight ringing artefacts were generated when the TGV method was used to restore the image. Furthermore, the OGSTV method performed well in smooth image regions but relatively poorly preserved image edges. The proposed method, however, not only preserved image edges, but also restored smooth image regions well, achieving the best overall visual effects for image restoration.