# Post-Filtering of Noisy Images Compressed by HEIF
## 1 Introduction and Related Work
Noise also takes place in medical, night light, astronomic, and other images [22; 23]. Some drones use cheap and low quality digital cameras, and as a result, the acquired images can also be noisy [24]. If a noisy image is subject to lossy compression, specific effects, discovered for SAR [25; 26] and other images [27; 28] about three decades ago, take place. The specific effect of noise reduction due to lossy compression was shown for both JPEG and wavelet-based techniques in [29; 30] and [27; 28], respectively. The concept of an optimal operation point (OOP) that can be observed was introduced in [28]. The OOP is the value of a used PCC when a minimal \"distance\" between the compressed and noise-free images is observed according to a considered quality metric. First, a possible OOP existence was demonstrated for conventional metrics such as PSNR; later, the possibility of OOP occurrence was shown [31] for many visual quality metrics including PSNR-HVS-M [32] and MS-SSIM [33]. For most metrics, the OOP relates to the global maximum, but there are a few metrics (MSE, MDSI [34]) for which the OOP corresponds to the global minimum. As above-mentioned, the initial results concerning the OOP were obtained for JPEG and wavelet-based coders including JPEG2000 [35]. In state-of-the-art papers [36; 37], a possible OOP existence was shown for coders based on discrete cosine transform (DCT) including a better portable graphics (BPG) coder [38; 39]. Finally, the existence of an OOP has recently been demonstrated [17] for coders such as HEIF and AVIF [40], which are part of the video coder HEVC. In other words, it is possible to state that the existence of an OOP for some practical situations is typical for the lossy compression of noisy images including not only the case of pure additive noise, but also the case of signal-dependent noise [41]. Note that for many coders and noise types, fully automatic procedures for OOP determination have been proposed [37; 41] based on the assumption that the noise type and characteristics are either known in advance or can be quite accurately estimated by the corresponding blind techniques for signal-dependent [42; 43] and pure additive [44] noise types. The OOP is not only a theoretical phenomenon, but also has considerable practical importance. An attractive advantage of the OOP is that, if an OOP exists for a given noisy image and a used coder, there is a certain sense of the image compressing in the OOP or its close neighborhood. Then, the benefit is twofold: good quality of the compressed image is provided, and the produced CR is quite large [17; 36]. Meanwhile, two questions arise. First, there are images, coders, and noise intensities for which the OOP is absent (e.g., complex structure images corrupted by not very intensive noise). Then, there is how to set the coder PCC for such images to provide reasonably high CR and compressed image quality. Second, although the filtering effect in the OOP can be quite large, denoising due to lossy compression is not perfect. A good filter is able to provide an output PSNR that is a few dB larger than the PSNR in the OOP. Keeping this in mind, an approach presuming a more \"careful\" (compared with the OOP) lossy compression combined with post-filtering applied to a decompressed image could be useful. Such an approach, to the best of our knowledge, was first proposed for DCT-based coders in [45; 46]. Recently, a similar approach was studied for the BPG coder, where the DCT-based post-filtering was first analyzed [47], followed by the consideration of more complex non-local post-processing [48]. The post-processing of decompressed images is widely known [49; 50], but the main goal for the corresponding methods is usually deblocking. The novelty of the approach in [47; 48] consists of the fact that post-filtering after decompression was applied to remove residual noise for noisy images compressed by a modern BPG coder. In particular, the following has been shown:  
1.