## 2 Image/Noise Model and Compression Efficiency Criteria
PSNR-HVS-M is defined as:  
\[PSNR-HVS-M^{n}=10 log_{10}\left(\frac{255^{2}}{MSE-HVS-M^{n}}\right), \tag{3}\]  
where \(PSNR-HVS-M^{n}\) is determined in \(8\times 8\) blocks in the DCT domain, taking into account the masking effect and lower sensitivity of a human eye to distortions in high spatial frequencies rather than distortions in low spatial frequencies. Note that PSNR and PSNR-HVS-M [26] are both expressed in dB with larger values relating to better quality. Usually, for AWGN and similar distortions, PSNR-HVS-M is slightly larger than PSNR due to masking effect. Figure 1: \(512\times 512\) pixel images having simple (**a**), complex (**b**), and middle complexity (**c**–**f**) structures (namely, Frisco (**a**), Diego (**b**), Fr01 (**c**), Fr02 (**d**), Fr03 (**e**), and FR04 (**f**)). For these two metrics, it is important that the distortion visibility thresholds have been determined [38]. Distortions are usually invisible if PSNR exceeds 36 dB and PSNR-HVS-M is larger than 41 dB. This happens if the noise variance is larger than 15...20. Because of this, in this paper we concentrate on considering \(PSNR^{n}\) smaller than 36 dB which is typical for the aforementioned applications. The metric MS-SSIM [27] extracts and employs structural information from the scene. It is in the limits from zero to unity where larger values correspond to a better visual quality. The distortion invisibility threshold is approximately equal to 0.99. If a lossy compression method is applied, one obtains a compressed image \(I_{ij}^{c}\), \(i=1,...,I_{lm}\), \(j=1,...,J_{lm}\) that differs from \(I_{ij}^{n}\), \(i=1,...,I_{lm}\), \(j=1,...,J_{lm}\) and depends on the compression controlling parameter (CCP). For different coders, different parameters play the role of the CCP. This can be the quality factor, scaling factor, quantization step, and bits per pixel [12; 24; 31; 32]. For the BPG coder, the quality parameter Q allows changing the image quality and compression ratio. Performing a multiple compression of \(I^{n}\) using different Q (which are integers in the limits from 1 to 51 for the BPG encoder), the rate-distortion curve \(Metr^{nc}(Q)\) can be obtained for any image where \(Metr^{nc}\) is a metric under interest calculated between noisy (original) and compressed images. Such metrics behave in a reasonable manner, i.e., their values become worse (smaller for all three considered metrics) if the \(Q\) increases. This is clearly seen in all four plots presented in Figure 2. In Figure 2a,b, the dependences of \(PSNR^{nc}(Q)\) for six test images for two values of the noise variance are presented. For very small \(Q<7\), the original and compressed images practically do not differ and the compression is near-lossless. Then, one has a practically linear part of all curves; thus, it is possible to say that the part starts at \(Q=7\) and ends at such \(Q\) that \(PSNR^{nc}(Q)\approx PSNR^{n}\). For larger \(Q\), the dependences for different images diverge where larger \(PSNR^{nc}\) values are observed for simpler structure images. For the middle part, it is possible to approximate the curves as:  
\[PSNR^{nc}(Q)\approx 63-Q,\ dB. \tag{4}\]  
The dependences \(PSNR-HVS-M^{nc}(Q)\) and \(MS-SSIM^{nc}(Q)\) are monotonous. They are \"going jointly\" for all test images for \(Q<25\). A joint analysis of all four dependences shows that for \(Q<29\) the introduced distortions are invisible. This means that it is possible to carry out a visually lossless compression and then, after image transferring (storage) and decompression, to perform noise post-filtering if necessary. In the case of simulations, i.e., if one has a true image, adds noise to it, and then compresses this image, it is also possible to calculate the metric \(Metr^{tc}\) between the compressed and true images, and to get the dependence \(Metr^{tc}(Q)\). The properties of such dependences are the most interesting. Two examples are given in Figure 3. For \(Q<29\), the image quality is \"stable\" and it is practically the same as for the uncompressed image.