# Post-Filtering of Noisy Images Compressed by HEIF
## 2 Image/Noise Model and Basic Rate/Distortion Dependences for HEIF
Second, there was an interval of QF values (19-39) where the CR increased rapidly when the QF decreased, and this interval contained the QF\({}_{\text{COP}}\) for the considered values of \(\sigma^{2}\). Recall that the noise-filtering effect due to lossy compression is associated with the appearance of considerably more zero-valued DCT coefficients after quantization, and then the CR becomes significantly larger. Figure 2 presents the dependences of the MSE calculated between the compressed and true images on the CR for two test images and two values of \(\sigma^{2}\), 25 and 196, for five coders. As can be observed, OOPs were observed for all five coders (including the advanced DCT-based coder ADCT [58]) for both noise variances for the test image Frisco as well as for three coders (including HEIF) for the test image Diego, contaminated by AWGN with \(\sigma^{2}=196.7\). The plots presented in Figure 2 also reflect the main tendencies typical for the lossy compression of noisy images. First, the presence of the OOP is more probable for simpler structure images--compare the results in Figure 2a,b, where OOPs were absent for all coders for the test image Diego and present for all five coders for the test image Frisco. Figure 2: Dependences of MSE\({}_{\text{tc}}\) (calculated between the true and compressed images) for images Diego (**a**,**c**) and Frisco (**b**,**d**) for \(\sigma^{2}=25\) (**a**,**b**) and \(\sigma^{2}=196\) (**c**,**d**) for five coders. Second, the OOP presence is more likely for larger noise intensity--compare the results for the test image Diego for two values of the noise variance. Third, OOPs could be observed for approximately the same PCC values for different images contaminated by AWGN with the same variance--this follows from data presented in Table 1 (Notes column). Fourth, OOPs are \"more obvious\" for better coders-see data in Figure 2d where the MSE\({}_{\text{tc}}\) in the OOPs (minimal values of MSE\({}_{\text{tc}}\)) was smaller for modern coders than for JPEG (tc relates to metrics calculated between the compressed and true images). Fifth, if an OOP exists, then the CR in it will be quite large--from about 10 for complex structure images (Figure 2c) to 20-60 for simple structure images depending on the noise variance. One more positive feature of HEIF is that it performed better than AVIF (see data in Figure 2c,d) and approximately at the same level as the BPG coder. Also recall that HEIF was slightly faster than BPG. The values of MSE\({}_{\text{tc}}\) were non-zero for all of the QF, test images, and noise variance values for HEIF (as well as other coders) for the plots in Figure 2. This means that there was some residual noise in the compressed image as well as some distortions of information content. A question is whether these residual noise and distortions can be additionally diminished by decompressed image post-filtering. Available results for the BPG coder have shown that this is possible [47; 48], so why then should it not be possible for HEIF? Note that the choice of a proper filter and its parameters depends on the properties of the noise to be removed. Therefore, similarly to [59], it was worth carrying out a statistical analysis for the mixture of residual noise and distortions for HEIF.