## 3 Properties of Optimal Operation Point  
We have already mentioned that we focused on applying the BPG encoder [35]. It has several obvious advantages stimulating to consider its application for RS and other types of images. First, BPG provides a higher compression ratio compared to JPEG and many other encoders for the same quality. Second, BPG is supported by most web browsers. Third, it supports the same formats as JPEG (grayscale, YCbCr4 4:2:0, 4:2:2, and 4:4:4), and the most popular RGB, YCgCo and CMYK color spaces are supported. The available versions are able to work with data from 8 to 14 bits per channel. In this paper, we present the results obtained using the grayscale and color (4:2:2) BPG version 0.9.8 given at [[https://bellard.org/bpg/](https://bellard.org/bpg/)]([https://bellard.org/bpg/](https://bellard.org/bpg/)), accessed on 25 July 2022.  
Our main intention in this section is to understand what the Qoop is and how it can be determined for a given image, metric, and noise variance. The preliminary observations that follow from the plots in Figure 3 are the following. For all images for which OOPs are observed, they take place for approximately the same \(Q\). The OOPs for the metrics, PSNR-HVS-M and MS-SSIM, are observed for practically the same \(Q\). This is a good property that allows to assume that an OOP does not depend on the image at hand and metric used, but, probably, depends on the noise variance.  
This hypothesis is based on the results obtained earlier for other coders [24; 43; 44; 45]. In [43], it has been demonstrated that for an OOP (according to \(PSNR^{tc}\)) the following condition is satisfied:  
\[PSNR^{nc}(Q)\approx PSNR^{n}. \tag{5}\]  
This means that for \(\sigma^{2}=64\)\(PSNR^{nc}(Q)\approx 30\) dB, for \(\sigma^{2}=100\)\(PSNR^{nc}(Q)\approx 28\) dB and for \(\sigma^{2}=196\)\(PSNR^{nc}(Q)\approx 25\) dB. Taking into account the expression (4), this should occur for \(Q=33\), 35, and 38, respectively. The plots presented in Figure 6 for noise variances equal to 64 and 100 show that this really is the case. Additional studies carried out for other variance values (25, 144, 196, and 289) and other test images have shown that expression (5) is really valid for \(Q=Q_{OOP}\). Substituting (4) into (5) and carrying out simple transformations, it is also possible to obtain (for 8-bit images):  
\[Q_{OOP}\approx 14.9+20 log_{10}(\sigma). \tag{6}\]  
Then, knowing the noise variance a priori or estimating it with appropriate accuracy, it is possible to predict for what value of \(Q\) the OOP is possible. Meanwhile, the values of \(PSNR^{tc}\) for \(Q\) calculated according to (6) can differ significantly. For example, for data in Figure 6a, \(PSNR^{tc}\) is from 28.5 dB to 36.5 dB where the first case (image Diego) corresponds to an OOP absence whilst the second case (image Frisco) relates to an \"obvious\" OOP.  
The following has been demonstrated in [45] for DCT-based coders. First, OOPs according to visual quality metrics are observed less often than OOPs according to PSNR. Second, an OOP according to visual quality metrics is observed for slightly (by about 5...10%) smaller quantization steps than for an optimal PSNR.  
Figure 5: Test images Lenna (**a**), Baboon (**b**), and RSA (**c**).  
Thus, we can suppose that OOPs according to all considered metrics take place for the same \(Q\) determined according to (6); however, it might be so that, for a given image and noise variance, an OOP exists according to \(PSNR^{tc}\) but does not exist according to \(PSNR-HVS-M^{tc}\). Even if the image is compressed in an OOP and the noise is partly removed, residual distortions can be clearly visible (\(PSNR-HVS-M^{tc}\) is smaller than 41 dB and \(MS-SSIM^{tc}\) is smaller than 0.99).  
The carried-out analysis also shows that it is worth analyzing many test images and many values of noise variance to obtain statistical data that allow for making certain conclusions.