## 2 Image/Noise Model and Compression Efficiency Criteria
Then, with further \(Q\) increasing, three options are possible: (1) the image quality starts to improve and, after attaining the OOP (associated with the dependence maximum), the quality starts to quickly decrease; (2) the quality continues to be almost the same and even local maxima are possible, and then a fast reduction takes the place; (3) the quality starts to decrease. The first situation takes place for the metric \(MS-SSIM^{tc}\) for all five images except the test image, Diego (Figure 3b), and for the metric \(PSNR-HVS-M^{tc}\) for the test image, Frisco (Figure 3a). The second situation is observed with the metric \(PSNR-HVS-M^{tc}\) for some middle complexity test images (Figure 3a). The third situation takes place for the complex structure image, Diego, according to both visual quality metrics (Figure 3a,b). Obviously, the compression in the OOP can be recommended for the first situation, lossy compression in the neighborhood of the local maxima is reasonable in the second situation, and it is unclear what to do in the third situation. Thus, we come to several questions. The first and the most complicated deals with the fact that in practice one does not have the true image and, thus, the dependence \(Metr^{tc}(Q)\) cannot be attained; therefore, the \(Q\) in the OOP cannot be determined and it is impossible to understand if the OOP exists or not. Another, less important, problem is what to do if an OOP does not exist. The arisen question will become clearer after an additional analysis performed below. The plots in Figure 4 visualizes what CR can be provided by a BPG encoder applied to noisy images. Note that the plots in Figure 4 are given for six single-channel RS images of different complexity. The conclusions are the following. First, the CR (for the same \(Q\)) depends on the image complexity. For example, for \(Q=28\) and a noise variance equal to 64, the attained CR for the simplest and most complex test images, Frisco and Diego, differ sufficiently (they are about 4.3 and 3.7, respectively, i.e., one deals with a near-lossless compression). For \(Q=35\), the situation is the following: the CR for Frisco is about 35.3 whilst the CR for Diego is about 6.8, i.e., the CR differs from a near-lossless case and it is considerably larger for the simple structure image. Second, from comparing the data in Figure 4a,b it is possible to state that the CR for noisier images is smaller. For example, for \(\sigma^{2}=196\) and \(Q=35\), the CR values for the images Frisco and Diego are equal to 5.5 and 6.4, respectively. A sharp increase in the CR starts when \(Q\) occurs larger than \(Q_{\text{OOP}}\), especially for simple structure images. Figure 2: Dependences of the considered metrics (calculated between the original and compressed images) on \(Q\): \(PSNR^{nc}(Q)\) for noise variance equal to 64 (**a**), \(PSNR^{nc}(Q)\) for noise variance equal to 100 (**b**), \(PSNR-HVS-M^{nc}(Q)\) (**c**) and \(MS-SSIM^{nc}(Q)\) (**d**) for noise variance equal to 64. Third, there is no sufficient difference in the general tendencies for images of different origin. To partly prove this, we have obtained the dependences of the considered visual quality metrics on \(Q\) for two well-known test images, Lenna and Baboon, as well as an artificial image, RSA (the images are presented in Figure 5). These dependences are given in Figure 3c,d. As one can see, the results for the highly textural images, Diego and Baboon, are very close. Similarly, the results for the simple structure images, RSA, Frisco and Lena are similar as well. Figure 4: Dependences of CR on \(Q\) for noise variance equal to 64 (**a**) and 196 (**b**). Figure 3: Dependences of the considered metrics (calculated between the true and compressed images on \(Q\): \(PSNR-HVS-M^{tc}(Q)\) (**a,c**) and \(MS-SSIM^{tc}(Q)\) (**b,d**) for noise variance equal to 196.