# Post-Filtering of Noisy Images Compressed by HEIF
## 4 Post-Filtering Opportunities and Results
### Post-Filtering Performance Analysis
\begin{table}
\begin{tabular}{c c c c c c c c} \hline \(\alpha^{2}\) & 20–30 & 31–44 & 45–64 & 65–90 & 91–130 & 131–180 & 185 \\
**QF for HEIF** & 43 & 41 & 39 & 37 & 36 & 34 & 32 \\
**PSNR\({}_{\text{p}\text{f}}\)** & 32–42 & 31–41 & 30–40 & 29–39 & 28–38 & 27–37 & 26–36 \\
**PSNR-HVS-M\({}_{\text{p}\text{f}}\)** & 36–39 & 34.5–37 & 33–36 & 31–34 & 29.5–32 & 28–30.5 & 27–29.5 \\
**CR\({}_{\text{p}\text{f}}\)** & 3–6 & 3.5–7 & 4–8 & 4.5–9 & 4.5–10 & 4.5–9.5 & 4.5–9 \\
**QF\({}_{\text{OOP}}\)** & \(\approx\)33 & \(\approx\)31 & \(\approx\)29 & \(\approx\)27 & \(\approx\)26 & \(\approx\)25 & \(\approx\)24 \\
**PSNR\({}_{\text{OOP}}\)** & 29–38.5 & 28–37.5 & 27–36.5 & 26.5–36 & 26–35 & 25.5–34 & 25–33 \\
**PSNR-HVS-M\({}_{\text{OOP}}\)** & 33.5–37 & 32–35.5 & 30.5–34 & 29–33 & 27.5–32 & 26.5–30.5 & 26–30 \\
**CR\({}_{\text{OOP}}\)** & 5.5–24 & 6–27 & 7–30 & 8–31 & 8–27 & 7–23 & 6–17 \\ \hline \end{tabular}
\end{table} 
Table 6: The image compression parameters depending on the AWGN variance. The first conclusion that can be drawn from the comparison of data in Tables 6 and 7 is that the PSNR\({}_{\text{COP}}\) and PSNR\({}_{\text{pf}}\) given in Table 7 were within the corresponding limits presented in Table 6. It can be clearly seen that the PSNR\({}_{\text{COP}}\) was always by about 3 dB smaller than the PSNR\({}_{\text{f}}\) (i.e., the noise filtering due to lossy compression was considerably less efficient than the traditional filtering applied to the uncompressed image). A comparison of the PSNR\({}_{\text{COP}}\) and PSNR\({}_{\text{pf}}\) for the same image and the same noise variance in Table 7 demonstrated that PSNR\({}_{\text{pf}}\) was 0.9-2.1 dB larger than the PSNR\({}_{\text{COP}}\), where the largest benefit for the proposed approach was observed for simple structure images contaminated by intensive noise. The second conclusion is that the optimal \(\beta\) was in the range of 2.1 to 3.0, \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline \multirow{2}{*}{**Image**} & **Noise** & **Original** & **PSNR\({}_{\text{f}}\) (BM3D with \(\beta=2.6\)** & **PSNR\({}_{\text{OOP}}\), dB** & **PSNR\({}_{\text{pf}}\), dB** & **Optimal \(\beta\) for** \\  & **Variance** & **PSNR, dB** & **Without Compression, dB** & **P**SNR\({}_{\text{OOP}}\) & **P**SNR\({}_{\text{pf}}\), dB** & **Post-Filtering** \\ \hline \multirow{3}{*}{Pepper} & 25 & 34.1 & 37.67 & 34.39 & 36.24 & 2.2 \\ \cline{2-7}  & 100 & 28.1 & 35.0 & 32.12 & 34.07 & 2.3 \\ \cline{2-7}  & 196 & 25.2 & 33.93 & 30.99 & 33.07 & 2.5 \\ \hline \multirow{3}{*}{Baboon} & 25 & 34.1 & 35.25 & 32.82 & 33.95 & 2.2 \\ \cline{2-7}  & 100 & 28.1 & 30.57 & 28.08 & 28.99 & 2.3 \\ \cline{2-7}  & 196 & 25.2 & 28.58 & 25.3 & 27.29 & 2.1 \\ \hline \multirow{3}{*}{Goldhill} & 25 & 34.1 & 37.18 & 34.04 & 35.51 & 2.3 \\ \cline{2-7}  & 100 & 28.1 & 33.67 & 30.48 & 32.67 & 2.2 \\ \cline{2-7}  & 196 & 25.2 & 32.26 & 29.2 & 31.4 & 2.2 \\ \hline \multirow{3}{*}{Med5} & 25 & 34.1 & 37.86 & 35.14 & 37.03 & 2.1 \\ \cline{2-7}  & 100 & 28.1 & 35.54 & 33.34 & 35.0 & 2.4 \\ \cline{2-7}  & 196 & 25.2 & 34.61 & 32.34 & 34.03 & 2.9 \\ \hline \multirow{3}{*}{Med7} & 25 & 34.1 & 41.59 & 38.71 & 40.33 & 2.1 \\ \cline{2-7}  & 100 & 28.1 & 37.0 & 34.43 & 36.03 & 2.5 \\ \cline{2-7}  & 196 & 25.2 & 34.67 & 32.37 & 33.82 & 2.5 \\ \hline \multirow{3}{*}{Med8} & 25 & 34.1 & 42.78 & 40.27 & 42.1 & 2.4 \\ \cline{2-7}  & 100 & 28.1 & 40.16 & 37.67 & 39.56 & 2.8 \\ \cline{1-1} \cline{2-7}  & 196 & 25.2 & 38.77 & 36.28 & 38.18 & 3.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 7: Results for the additional test images. Figure 8: Test images Med5 (**a**), Med7 (**b**), and Med8 (**c**) used in the additional experiments. where the recommendation to set \(\beta\approx 2.3\) can be given for practice. Finally, the obtained data showed that: (1) the proposed approach is quite general and applicable to images of different origin; and (2) the results depend more on the image complexity than on the image origin domain. Let us present some compression results.