# Image Deblurring under Impulse Noise via Total Generalized Variation and Non-Convex Shrinkage
## 3 Model Proposal and Solution Method
### Solution Method
Figure 11 presents a visual effects comparison of the image restorations using the four algorithms for the following images: the \"Bird\" image degraded with a motion blur kernel \((M,20,20)\) and \(30\%\) noise, the \"Baboon\" image degraded with a motion blur kernel \((M,10,10)\) and \(40\%\) noise, the \"Butterfly\" image degraded with a motion blur kernel \((M,30,30)\) and \(50\%\) noise, and the \"Truck\" image degraded with a motion blur kernel \((M,20,20)\) and \(60\%\) noise. As shown in the figure, the proposed method allowed for better preservation of the smooth image regions and restoration of the details of the image lines with fewer artefacts. Figure 10: Comparison of restored image details and the PSNRs for the “Butterfly” image degraded with a motion blur kernel \((M,20,20)\) and \(30\%\) impulse noise: (**a**) selected region of the original image; (**b**) local zoomed-in image of the selected region; (**c**) blurred image; (**d**) blurred image with \(30\%\) impulse noise; (**e**) FTvd-restored image; (**f**) TGV-restored image; (**g**) OGSTV-restored image; (**h**) proposed algorithm-restored image. Figure 11: Comparison of the visual effects and PSNRs of restored images for images degraded with different levels of motion blur and impulse noise. ### Comparison of Computing Time

Finally, to further test the timeliness of the proposed model, the proposed method was compared with the FTVd, TGV, and OGSTV methods by testing the mean time required to restore the nine images of Figure 3 for different blur kernels at various impulse noise levels. Some test results are shown in Table 6, where the best indicators are highlighted in bold. The calculation times in the table reveal that FTVd, which is a classic algorithm, is still advantageous in terms of computing speed. The proposed method is comparable to the OGSTV method in terms of image restoration time, while they are both faster than the TGV method. The proposed method uses the Lp-pseudo-norm as a regularization constraint, which increases the method's computational burden, thereby leading to a certain degree of increase in the image restoration time. However, the gradient difference is re-expressed in the form of a convolution in this study, where it corresponds to the dot multiplication of two matrix spectra in the frequency domain; moreover, FFT is employed for convolution calculation. These strategies effectively avoid large-scale matrix multiplications, thus speeding up the calculation. ## 5 Conclusions

In this study, a regularized method for image restoration based on the Lp-pseudo-norm and TGV was proposed to de-blur images and remove impulse noise. The method constrains the first-order and second-order gradients of the image and takes advantage of the sparse representation ability of the Lp-pseudo-norm for image gradients. In this way, it effectively removes image blur and noise, and reduces undesirable artefacts (such as staircase and ringing artefacts) in image restoration while better preserving the edges and details of the image. A large number of experimental results revealed that the proposed method is consistently superior to several state-of-the-art methods in terms of numerical indicators and visual effects. The regularization parameters \(\mu\) of the Lp-pseudo-norm used in the experiments were relatively invariant, indicating that the proposed algorithm is robust. In follow-up studies, efforts will be made to further improve the computational efficiency of the proposed method. It should be noted that the proposed algorithm uses a generalized regularization term, which can be easily extended to other algorithms and is suitable for other image restoration problems. Funding acquisition, F.Y. ; Methodology, F.L. and Y.C. ; Software, F.L. and Y.C. ; Writing--original draft, F.L. ; Writing--review and editing, F.L. and Y.C.