## 5 Results and Discussion
### Results with the Peak Signal-to-Noise Ratio (PSNR)  
We evaluated the quality of the filtering using the peak signal-to-noise ratio (PSNR) in a first work. This parameter is used to calculate the presence of noise in an image. Its formulation is described in (31):  
\[PSNR=10\times log_{10}(SNR) \tag{31}\]  
\[ \begin{split} SNR&=\frac{signal}{noise}\\ &=\frac{I_{original}}{I_{original}-I_{noisy}}\\ &=\frac{\sum\limits_{x\,y}l_{original}^{2}(x,y)}{\sum\limits_{ x\,y}\Bigl{(}I_{original}(x,y)-I_{noisy}(x,y)\Bigr{)}^{2}}.\end{split}  \tag{32}\]  
In our study, there are no noiseless images, and so we go back to the formula in (33) to evaluate the quality of the images from the dynamic processing. \[SNR=\frac{I_{filtered}}{I_{filtered}-I_{original}} \tag{33}\]  
The values obtained can be compared with the reference values in Table 1. The values obtained on [PERSON]'s image after varying \(p\) and \(\lambda\) are shown in Table 2. These values can be compared with the reference values given in Table 1.  
The [PERSON] image with the adaptive model provides a \(\text{PSNR}=27.66\), while the tiger image gives a \(\text{PSNR}=24.31\). In order to validate the proposed filter, we chose to use the adaptive model on 100 images extracted from the Berkeley database [19] (see Figure 6), 14 images extracted from the USC-SIPI Image Database [20] (see Figure 7), and 24 images extracted from Kodak Lossless True Color Image Suite [21] (see Figure 8). Images from these databases are widely known and show different scenes with contrasts. \begin{table}
\begin{tabular}{c c c} \hline \hline
**PSNR** & **Image Quality** \\ \hline
30 & Excellent \\
24.99 & Good \\
20 & Reasonable \\
15.05 & Very noisy \\
10 & Unreadable \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Relationship between PSNR and image quality. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**PSNR** & \(p\) **= 0.01** & \(p\) **= 1.00** & \(p\) **= 2.00** \\ \hline \(\lambda\)=0.01 & 28.98 & 29.61 & 26.57 \\ \(\lambda\)=0.10 & 33.80 & 36.46 & 27.49 \\ \(\lambda\)=1.00 & 41.47 & 45.81 & 32.83 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Result of using PSNR on the Lena image (Figure 2) by varying \(p\) and \(\lambda\). From the Berkeley database, Table 3 and Table 4 show the DSSIM and PSNR results obtained, and Figure 9 shows two images that obtained the best PSNR following the application of the adaptive model as well as the two images with the worst PSNR. Figure 8: Illustration of the images contained in the KODAK database. Figure 6: Illustration of the images contained in the Berkeley database. Figure 7: Illustration of the images contained in the USC-SIPI databases. The values presented in Table 4 are quite homogeneous, as can be seen from the standard deviation, which remains at 2.19, and the average PSNR, which remains at 31.13. This last value represents a good image quality according to this parameter. With regard to the sample used, 26.36 corresponds to a minimum PSNR. It places the worst case towards a \begin{table}
\begin{tabular}{c c c c c} \hline \hline  & **Average** & **Min** & **Max** & **Standard** \\  & & & & **Deviation** \\ \hline PSNR & 31.13 & 26.36 & 36.24 & 2.19 \\ \hline \end{tabular}
\end{table} 
Table 4: Statistical results on the PSNR obtained with the adaptive model on the Berkeley database. Figure 9: Illustrations of the results obtained by testing 100 images extracted from the Berkeley database with an alteration (a degraded aspect due to, for instance, bad capture or compression) with a ratio of almost 50%. From top to bottom (**a–d**), we note the two best results and the two worst results obtained using the PSNR and DSSIM parameters. From left to right, the original degraded image followed by the result of the application of the Sobel filter, then the result of the application of the adaptive algorithm on the degraded image, and finally, the result of the Sobel filter. In all cases, there is a decreasing number of pixels with a high gradient. good result.