# Post-Filtering of Noisy Images Compressed by HEIF
## 4 Post-Filtering Opportunities and Results
### Post-Filtering Performance Analysis
In turn, the PSNR values after post-filtering were equal to 35.78 dB for Frisco and 29.41 dB for Fr03, respectively;
3. Again, the post-filtering was less efficient for the complex structure image Fr03 than for the test image Frisco;  
Figure 5: Dependences of the output metrics on QF and \(\beta\) for two test images and three metrics, \(\sigma^{2}=100\). 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**). \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{4}{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 & 2.3 & 34.15 & 30.99 & 1.8 & 27.18 & 27.40 \\
25 & 2.3 & 34.54 & 31.48 & 1.9 & 27.69 & 28.12 \\
27 & 2.3 & 34.85 & 31.86 & 1.9 & 28.15 & 28.72  
* The PSNR-HVS-M metric had similar dependencies;
* A reduction in \(\beta_{\text{opt}}\) is observed if the QF decreases; this means that one has to set a smaller threshold to provide the filter operating in a quasi-optimal manner. * The functions of the metrics on QF and \(\beta\) are represented in Figure 6.  
Figure 6: Dependences of the output metrics on QF and \(\beta\) for two test images and three metrics, \(\sigma^{2}=200\). 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**). Their analysis showed that the surfaces had maxima in the neighborhood of optimal \(\beta\); small deviations of \(\beta\) from \(\beta_{\rm opt}\) did not essentially reduce the performance. The post-filtering efficiency was high for a \(\beta\) of about 2.2. The reduction in the metric output values with QF decreasing was not rapid for QF \(\geq\) 34 but the reduction speed increased for smaller QF. The expedience of decompressed image post-filtering can be also studied in another way. Figure 7 presents the dependencies \(\rm PSNR_{p\,pf}\) = \(\rm PSNR_{pf}-PSNR_{tc}\) (Q = 1) and \(\rm\Delta PSNR_{p\,tc}\) = \(\rm PSNR_{tc}-PSNR_{tc}\) (Q = 1) on the CR obtained by varying the QF and CR, respectively, for two test images and two noise variance values. Here, the \(\rm PSNR_{pf}\) was the PSNR calculated between the decompressed image post-filtered with the optimal \(\beta\) and the true image, and \(\rm PSNR_{tc}\), as we recall, is the PSNR calculated between the compressed and true images. Analysis of \(\rm\Delta PSNR_{p\,tc}\) (Figure 7a-c) showed that in all three cases, OOPs existed (there was a maxima of \(\rm\Delta PSNR_{tc}\) (CR)). Meanwhile, the maximal \(\rm\Delta PSNR_{p\,tc}\) for the test image Fr03 (Figure 7a) was very small (about 0.5 dB). The maximal values of \(\rm\Delta PSNR_{p\,tc}\) were larger for \(\sigma^{2}\) = 200 (\(\sigma^{2}\) is denoted as Var in the Figure title, see Figure 7c), and exceeded 6 dB for the test image Frisco (Figure 7b). As seen, the \(\rm PSNR_{p\,pf}\), in the considered limits of CR, was always larger than \(\rm PSNR_{p\,tc}\). However, the difference between them decreased if the CR increased. In the neighborhood of the OOP (CR \(\approx\) 10 in Figure 7a, CR \(\approx\) 60 in Figure 7b, CR \(\approx\) 14 in Figure 7c) and for a larger CR, the benefit due to post-filtering became negligible (about 0.5 dB and less) (i.e., post-filtering became practically useless). Consider now the dependences \(\rm PSNR\)-\(\rm HVS\)-\(\rm M_{p\,pf}\) = \(\rm PSNR\)-\(\rm HVS\)-\(\rm M_{pf}-PSNR\)-\(\rm HVS\)-\(\rm M_{tc}\) (Q = 1) and \(\rm PSNR\)-\(\rm HVS\)-\(\rm M_{p\,tc}\) = \(\rm PSNR\)-\(\rm HVS\)-\(\rm M_{tc}\) - \(\rm PSNR\)-\(\rm HVS\)-\(\rm M_{tc}\) (Q = 1) on the CR presented in Figure 7d-f. As seen, there were two cases when all \(\rm PSNR\)-\(\rm HVS\)-\(\rm M_{p\,tc}\) were negative (Figure 7d,f) (i.e., there was no OOP for the test image Fr03).