# Post-Filtering of Noisy Images Compressed by HEIF
## 4 Post-Filtering Opportunities and Results
### Post-Filtering Performance Analysis
An analysis of the data in Table 4 allowed us to draw conclusions similar to the previous case of \(\sigma^{2}=50\):  
1. If the QF is reduced (which corresponds to an increase in CR), the PSNR and PSNR-HVS for the post-filtered images decrease, and the optimal \(\beta\) also decreases; in other \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{6}{c}{**BM3D**} \\ \hline \multirow{2}{*}{**QF**} & \multicolumn{4}{c}{**Frisco**} & \multicolumn{4}{c}{**Fr03**} \\ \cline{2-7}  & \(\beta_{\text{opt}}\) & **PSNR, dB** & **PSNR-HVS-M, dB** & \(\beta_{\text{opt}}\) & **PSNR, dB** & **PSNR-HVS-M, dB** \\ \hline
23 & 2 & 34.66 & 31.86 & 1.7 & 27.45 & 28.12 \\
25 & 2 & 35.24 & 32.60 & 1.7 & 28.01 & 29.04 \\
27 & 2 & 35.73 & 33.23 & 1.7 & 28.61 & 29.88 \\
29 & 2.4 & 36.18 & 33.80 & 1.7 & 29.16 & 30.64 \\
31 & 2.4 & 36.53 & 34.20 & 2 & 29.64 & 31.30 \\
33 & 2.4 & 36.85 & 34.61 & 2.1 & 30.05 & 31.93 \\
35 & 2.4 & 37.11 & 35.00 & 2.3 & 30.42 & 32.40 \\
37 & 2.4 & 37.26 & 35.27 & 2.3 & 30.70 & 32.81 \\
39 & 2.4 & 37.36 & 35.48 & 2.3 & 30.92 & 33.13 \\
41 & 2.4 & 37.45 & 35.64 & 2.3 & 31.10 & 33.38 \\
43 & 2.4 & 37.63 & 35.78 & 2.3 & 31.25 & 33.59 \\
45 & 2.4 & 37.64 & 35.94 & 2.3 & 31.37 & 33.79 \\
47 & 2.4 & 37.75 & 36.07 & 2.3 & 31.45 & 33.96 \\
49 & 2.4 & 37.78 & 36.12 & 2.3 & 31.54 & 34.12 \\
51 & 2.4 & 37.71 & 36.18 & 2.3 & 31.60 & 34.21 \\ \hline \hline \end{tabular}
\end{table} 
Table 4: Post-filtering performance characteristics, \(\sigma^{2}=100\). Figure 4: Dependences of the output metrics on QF and \(\beta\) for two test images and three metrics, \(\sigma^{2}=50\). Plots of PSNR **(a,b)**, PSNR-HVS **(c,d)** and FSIM **(e,f)** for the test images Frisco **(a,c,e)** and Fr03 **(b,d,f)**. words, the positive effect of post-filtering becomes smaller, and starting from a certain QS, post-filtering becomes useless;
2. At the same time, there exist QF values for which the use of post-filtering is reasonable; for example, for QF = 39, the values of PSNR\({}_{\text{tc}}\) for the compressed images were equal to 28.66 dB for Frisco and 28.11 dB for Fr03, respectively. In turn, the PSNR values after post-filtering were equal to 37.36 dB for Frisco and 30.92 dB for Fr03, respectively;
3. Post-filtering is less efficient for complex structure images; for any QF, the output PSNRs were significantly smaller for test image Fr03 than for test image Frisco;
4. The PSNR-HVS-M metric showed the same dependencies;
5. A reduction in \(\beta_{\text{opt}}\) is observed if the QF decreases; thus, one needs to set a smaller threshold to provide the filter operation in a quasi-optimal manner. The functions of the metrics on QF and \(\beta\) are represented in Figure 5. As seen, the surfaces are again quite smooth; however, the maxima in the neighborhood of optimal \(\beta\) have become more \"obvious\" compared with the case of \(\sigma^{2}\) = 50 (Figure 4); nevertheless, for \(\beta\), while it slightly differed from \(\beta_{\text{opt}}\), its performance was almost the same as for \(\beta_{\text{opt}}\). Thus, the post-filtering efficiency remained high for \(\beta\) within the limits of 2.0 to 2.5. Again, the reduction in metric output values with a decrease in QF was not rapid for QF = 36, while the reduction speed increased for a smaller QF. Figure 5: _Cont._  
Consider now the case of \(\sigma^{2}=200\). The obtained data are shown in Table 5.  
The analysis of the data in Table 5 resulted in the following conclusions:  
1. If the QF is reduced (i.e., the CR increases), both the PSNR and PSNR-HVS for the post-filtered images become worse, and the optimal \(\beta\) also reduces; this shows that the positive effect of post-filtering reduces, and starting from a QS around 23, post-filtering becomes useless;
2. There are QF values for which the post-filtering is beneficial; for example, for QF = 37, the values of \(\mathrm{PSNR_{tc}}\) for the compressed images were equal to 25.65 dB for Frisco and 25.39 dB for Fr03, respectively.