## 5 Experimental Results and Analyses
### Testing and Comparing the Denoising Performance of Different Algorithms  
Six images are selected from the original images of Figure 3 for testing, on which impulse noise at levels from 20% to 50%, to compare the denoising effects of ATV, ITV, TGV, OGS-L1, OGS-Lp, and Ours algorithms (six in total). To ensure the objectiveness and fairness of the evaluation, the above algorithms all adopt the following iterative condition: \begin{table}
\begin{tabular}{c c c c c c c} \hline
**Image** & **House** & **Lena** & **Woman** & **Milk Drop** & **Girl** & **Shoulder** \\ \hline
**Parameter** & & & & & & \\
**Level** & & & & & & \\ \hline
20\% & 0.14/0.45 & 0.15/0.45 & 0.15/0.65 & 0.14/0.65 & 0.14/0.65 & 0.14/0.65 \\
30\% & 0.15/0.45 & 0.15/0.65 & 0.15/0.7 & 0.16/0.65 & 0.15/0.65 & 0.15/0.65 \\
40\% & 0.15/0.45 & 0.16/0.65 & 0.15/0.7 & 0.19/0.65 & 0.15/0.65 & 0.15/0.65 \\
50\% & 0.18/0.55 & 0.17/0.65 & 0.2/0.7 & 0.2/0.65 & 0.18/0.55 & 0.18/0.55 \\ \hline \end{tabular}
\end{table} 
Table 1: The optimal parameters for different images with the noise level from 20% to 50%. Figure 4: Comparison of peak signal-to-noise ratio (PSNR) under different K values (Noise Level = 30%). The test images (“Girl”, “House”, “Lena”) are corrupted by 30% impulse noise. Figure 5: Comparison of structural similarity (SSIM) under different K values (Noise Level = 30%). The test images (“Girl”, “House”, “Lena”) are corrupted by 30% impulse noise. \[\|\mathbf{F}^{(k+1)}-\mathbf{F}^{(k)}\|_{2}\cdot\|\mathbf{F}^{(k)}\|_{2}^{-1}<10^{-4}. \tag{42}\]  
Regularization parameters of these algorithms are adjusted to ensure the best denoising effect of each, which ensured the fairness of the test. For methods based on the OGS model, the group size is set to K = 5. The test results on different images are given in Tables 2-5. The best indicator values are labeled as black and bold. By observing the data in each table, the following conclusions could be made:  
1. With the introduction of different levels of noise to the images, our model generates higher PSNR and SSIM values for the reconstructed images than other methods, indicating its superior denoising effect. The recovered images also resemble the original ones more. 2. The proposed model works better at lower noise levels. For example, at a 20% noise level, as shown in Table 2, the PSNR value of the \"House\" image (37.72 dB) given by our model is 5.91 dB higher than that given by the ITV model (31.81 dB) and 5.4 dB higher than that of the TGV model (32.32 dB). Even at high noise levels, our model still performs better than the others, which shows the clear advantages that total variation with overlapping group sparsity has over the classic anisotropic TV model. 3. Compared to OGS-L1, our proposed method incorporates the Lp-pseudo-norm shrinkage, which adds another degree of freedom to the algorithm and improves the depiction of the gradient-domain sparsity of the images, achieving a better denoising effect. For example, at a 20% noise level, as shown in Table 2, the PSNR value of the \"Girl\" image (32.34 dB) given by our model is 1.67 dB higher than that given by the OGS-L1 model (30.67 dB). Even at a noise level of 50%, as shown in Table 5, the PSNR value of the \"Girl\" image (27.35 dB) given by our model is still 0.90 dB higher than that given by the OGS-L1 model (26.45 dB). This proves that the Lp-pseudo-norm is more suitable as a regularizer for describing the sparsity of images than the L1-norm. 4. In terms of the runtime of the six models, the OGS-based method is more time consuming than ATV, ITV, and TGV. This is mainly because the OGS model considers the gradient information of the neighborhood in an image undergoing reconstruction, thus making the computation more complex. 5. Comparing the values of PSNR and SSIM in Tables 2-5, OGS-Lp-FAST and OGS-Lp have the same denoising effect. However, by observing the value of runtime of all testing images, we find that convergence is sped up in the OGS-LP-FAST method with the use of accelerated ADMM with a restart.