## 3 Results
### Simulation Results for Filtering Additive Gaussian Noise and Speckle Noise  
Below, we provide the results of filtering noisy grayscale images corrupted by additive Gaussian noise (Tables 3-5) and multiplicative speckle noise (Tables 6-8) at various levels using MLMVN and popular filtering methods for comparison. For the BM3D filter, we considered multiple values of the noise standard deviation parameter because, in real-world tasks, this information is usually unknown. For both Lee and Frost filters, we used a filter window size of 3 \(\times\) 3. The damping coefficient for the Frost filter was set to 2. We utilized MLMVN-G-1, MLMVN-G-2, and MLMVN-G-3, and MLMVN-S-1, MLMVN-S-2, and MLMVN-S-3 for filtering additive Gaussian noise and speckle noise, respectively. The evaluation of the MLMVN filtering capabilities was performed using images from the test dataset (Figure 5). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & Train station & Music hall & Big Ben & Cambridge bldg & Gaudi bldg \\ \hline Noisy image PSNR & 25.5579 & 26.385 & 23.9992 & 28.1943 & 25.9918 \\ BM3D 0.5\(\sigma\) & 28.1544 & 28.133 & 26.35 & 29.491 & 27.6905 \\ BM3D 1\(\sigma\) & **33.7363** & **31.2135** & **30.1307** & **32.4521** & **30.6029** \\ BM3D 1.5\(\sigma\) & 32.5886 & 29.5837 & 28.4187 & 30.8032 & 28.7687 \\ MLMVN-G-1 & 31.701 & 30.1547 & 28.5963 & 30.6758 & 29.6646 \\ MLMVN-G-2 & 31.6896 & 30.1126 & 28.5971 & 30.6637 & 29.6592 \\ MLMVN-G-3 & 31.6822 & 30.1428 & 28.5871 & 30.74 & 29.678 \\ \hline \multicolumn{5}{c}{Barcelona plaza} & Sailboat & Manhattan & Airplane & Fighter-jet \\ \hline Noisy image PSNR & 25.5342 & 28.9904 & 27.4271 & 29.0807 & 29.1782 \\ BM3D 0.5\(\sigma\) & 27.0836 & 31.2659 & 29.6573 & 31.6465 & 31.345 \\ BM3D 1\(\sigma\) & **29.8181** & **35.8772** & **34.288** & **38.2263** & **36.4391** \\ BM3D 1.5\(\sigma\) & 28.0811 & 33.8929 & 32.6458 & 37.3526 & 35.4323 \\ MLMVN-G-1 & 29.3247 & 34.2611 & 32.4535 & 35.729 & 35.685 \\ MLMVN-G-2 & 29.3218 & 34.2212 & 32.4293 & 35.6606 & 35.669 \\ MLMVN-G-3 & 29.3326 & 34.2682 & 32.4706 & 35.7497 & 35.6836 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Results of filtering grayscale images affected by additive Gaussian noise with a standard deviation of 0.2\(\sigma\), where \(\sigma\) is the standard deviation of the ideal image. The best results are shown in bold, and the best results using our approach are highlighted in color. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & Train station & Music hall & Big Ben & Cambridge bldg & Gaudi bldg \\ \hline Noisy image PSNR & 31.4534 & 32.4111 & 29.6285 & 34.1881 & 31.8786 \\ BM3D 0.5\(\sigma\) & 33.3987 & 33.6144 & 31.2485 & 34.9508 & 32.9469 \\ BM3D 1\(\sigma\) & **37.4319** & **35.6073** & **34.2258** & **36.6911** & **34.6627** \\ BM3D 1.5\(\sigma\) & 36.119 & 34.2096 & 33.1037 & 35.5024 & 33.7332 \\ MLMVN-G-1 & 35.4248 & 32.3049 & 31.6007 & 32.0308 & 31.7697 \\ MLMVN-G-2 & 35.4047 & 32.2347 & 31.5742 & 32.0051 & 31.7367 \\ MLMVN-G-3 & 35.4594 & 32.3276 & 31.6167 & 32.1362 & 31.8193 \\ \hline \multicolumn{5}{c}{Barcelona plaza} & Sailboat & Manhattan & Airplane & Fighter-jet \\ \hline Noisy image PSNR & 31.4643 & 34.9899 & 33.4381 & 35.0817 & 35.1862 \\ BM3D 0.5\(\sigma\) & 32.3518 & 36.7648 & 35.1181 & 37.1523 & 36.7276 \\ BM3D 1\(\sigma\) & **33.8656** & **40.1366** & **38.2556** & **41.7187** & **39.8033** \\ BM3D 1.5\(\sigma\) & 32.403 & 38.3289 & 36.94 & 41.1617 & 38.9479 \\ MLMVN-G-1 & 31.555 & 35.5246 & 34.306 & 37.7523 & 37.2053 \\ MLMVN-G-2 & 31.5289 & 35.5263 & 34.2972 & 37.7286 & 37.2716 \\ MLMVN-G-3 & 31.6001 & 35.5819 & 34.3893 & 37.8504 & 37.2804 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Results of filtering grayscale images affected by additive Gaussian noise with a standard deviation of 0.1\(\sigma\), where \(\sigma\) is the standard deviation of the ideal image. The best results are shown in bold, and the best results using our approach are highlighted in color.