## 5 Results and Discussion
### Results with the Structural Dissimilarity Measure (DSSIM)  
In the first stage, we use structural dissimilarity DSSIM [22] to assess the quality of filtering, and Table 5 shows the results obtained. This measure was developed to evaluate the visual quality of a compressed image with respect to the original image. It uses the SSIM parameter (structural similarity) to measure the similarity of structure between the two images. Its formula is given by (34):  
\[DSSIM(x,y)=\frac{1-SSIM(x,y)}{2} \tag{34}\]  
with  
\[SSIM(x,y)=\frac{(2\mu_{x}\mu_{y}+c_{1})(2\sigma_{x}\sigma_{y}+c_{2})(2 cov_{xy} +c_{3})}{(\mu_{x}^{2}+\mu_{y}^{2}+c_{1})(\sigma_{x}^{2}+\sigma_{y}^{2}+c_{2})( \sigma_{x}\sigma_{y}+c_{3})}\]  
where \(\mu_{x}\) and \(\mu_{y}\) are averages in the \(x\) and \(y\) direction, and \(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\) are the variances of \(x\) and \(y\), respectively. \(c_{1},c_{2},c_{3}\) are parameters needed to stabilise the division. In order to simulate the images from the network, we compressed all images in the dataset. For this study, we chose to use JPEG compression. The images available in the database were compressed in JPEG format with several compression ratios ranging from 10% to 90% quality. We have already shown the impact of dynamic \(\lambda\) with respect to static \(\lambda\). Table 6 summarises this study. The algorithmic process gives the best values in the dynamic \(\lambda\) case and where \(\lambda\) is 1.0. In terms of the DSSIM criterion, these two cases give better results. However, the \(\lambda\) value of 1.0 leads to too much smoothing over the whole image, as shown in Figure 10.  
In a second step, we compare one side of the original image with the compressed image and the other side with the compressed image after applying our filtering. We used other filtering algorithms in the evaluation process. The method of [PERSON] [23] was used, which consists of regularising an image by an anisot \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Compression/\(\lambda\)** & **0.1** & **0.5** & **1.0** & **Proposal Model** \\ \hline
10\% & 0.124749391 & **0.124685624** & 0.126527000 & 0.126313566 \\
20\% & 0.073399066 & **0.067223833** & 0.067583050 & 0.067538158 \\
30\% & 0.0590243333 & 0.0504784583 & **0.050169525** & 0.050176050 \\
40\% & 0.052676424 & 0.0429842166 & **0.042350825** & 0.042386891 \\
50\% & 0.048671533 & 0.0382977916 & **0.0374521999** & 0.0374591749 \\
60\% & 0.045554683 & 0.0346366000 & **0.0336435749** & 0.033697841 \\
70\% & 0.04233888583 & 0.030723008 & **0.02957985** & 0.0296397499 \\
80\% & 0.039299400 & 0.0268465583 & **0.0255539583** & 0.0256166749 \\
90\% & 0.036822691 & 0.023159216 & **0.0215449250** & 0.0216251499 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Illustrations of DSSIM results obtained from 100 images extracted from the Berkeley database. This study was carried out according to the compression ratio in order to simulate degradation, and different values taken for \(\lambda\) were compared to the method used for dynamic \(\lambda\) (bold values represent the best results). \begin{table}
\begin{tabular}{c c c c} \hline \hline
**DSSIM** & \(p\) = **0.01** & \(p\) = **1.00** & \(p\) = 2.00** \\ \hline \(\lambda=0.01\) & 0,07 & 0.06 & 0.11 \\ \(\lambda=0.10\) & 0.02 & 0.01 & 0.09 \\ \(\lambda=1.00\) & 0.004 & 0.001 & 0.02 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: The result of using the DSSIM parameter on the Lena image (Figure 2) by varying \(p\) and \(\lambda\). based on PDEs. This model requires parameterisation for its implementation, so we used standard values given in the literature. Gaussian filtering was used, which consists of applying a Gaussian filter to the rows and then to the columns of the image. Peer group filtering was used, which allows an image to be smoothed by replacing a pixel with the weighted sum of its closest neighbours in terms of colour distance. The purpose was to observe whether our proposal leads to improvements in terms of visual rendering, especially in the case of high losses.