# Post-Filtering of Noisy Images Compressed by HEIF
## 1 Introduction and Related Work
A positive effect of post-filtering can be provided for images of different complexity under the condition that the CR is smaller than the CR for the OOP (parameter Q, which serves as the PCC for the BPG coder, is smaller than Q\({}_{\text{OOP}}\));
2. The optimal parameters of the DCT-based filter [51] change depending on Q and Q\({}_{\text{OOP}}\) and noise intensity; general dependences can be found in [47];
3. The same has been observed [48] for the BM3D filter [52], where this filter, in general, performs slightly better than the DCT-based filter [47];
4. It is possible to predict the post-filter performance [48] and set its parameters in a way close to optimal; if Q increases, CR increases, but the general efficiency of post filtering applied to decompressed images decreases; in this sense, Q \(\approx\) Q\({}_{\text{OOP}}\)-4 can be recommended for practical use. Note here that the BPG and HEIF encoders have recently been proposed and can both be part of HEVC [7; 53]. Moreover, the characteristics of BPG and HEIF in the case of the lossy compression of noisy images are very similar [17]. Meanwhile, the ways to find the OOP for the BPG and HEIF coders are different [17]. One way to find the OOP for HEIF was proposed in [17]. However, an opportunity to apply post-filtering for noisy images compressed by HEIF, to the best of our knowledge, has not been investigated yet. This was the main goal of this paper, and it deals with the novelty of the obtained results. As a starting point, we considered the case of single-channel images contaminated by additive white Gaussian noise (AWGN) with a zero mean for the following reasons. AWGN is a standard noise model used in image processing applications such as the blind estimation of noise variance [44], filtering [54; 55], and lossy compression [36; 37]. If the noise observed in the original image is signal-dependent, it is possible to convert it to additive by using a proper variance stabilizing transform (VST) [17; 56]. If such a VST is applied, lossy compression is employed after direct VST; the corresponding inverse transform should be used after decompression. Furthermore, although many modern images are multichannel nowadays [41; 57], we focused on single-channel image lossy compression for two reasons. First, component-wise compression is still an easy way for volume reduction. Second, such a preliminary analysis of component-wise compression will, hopefully, bring an understanding of what should be conducted for multichannel data [41; 57]. Summarizing the aforesaid, the novelty of this paper consists of the following:  
1. To the best of our knowledge, the procedure of the post-filtering of noisy images lossy compressed by HEIF has never been considered, and the post-filtering of noisy images compressed by HEIF has several peculiarities given in the next items;
2. The filter choice is based on the presented statistical analysis of residual noise and introduced distortions where, to the best of our knowledge, such analysis has never been carried out;
3. The recommendations on setting the QF for the proposed compression procedure were based on the analysis results presented in Sections 4.1 and 4.2. These recommendations differ from the recommendations for the BPG codec since the HEIF and BPG coders are controlled by different parameters (QF and Q) with different properties (CR increases if the QF becomes smaller for HEIF, and if Q increases for BPG, the QF varies in the limits from 1 to 100 whilst Q varies from 1 to 51, and so on);
4. Although the general tendencies of setting the post-filter parameter \(\beta\) appeared similar for the BPG and HEIF coders (for the larger CR optimal, \(\beta\) should be smaller), the obtained dependencies differed from the dependencies of different metrics on Q and \(\beta\) for the BPG coder presented in our earlier paper [47; 48]. The rest of this paper is structured as follows. The image/noise model and the fundamental rate/distortion dependencies are considered in Section 2.