## 2 Image/Noise Model and Compression Efficiency Criteria  
It is a well-known fact that compression characteristics depend on sufficient image properties. Depending on the image complexity, the CR can vary by several (or even tens) of times for the same PSNR or, equivalently, the quality of compressed images can be rather different for the same CR. Because of this, to ensure universality of the conclusions and recommendations, the corresponding analysis and method synthesis should be performed for a set of images with a very wide variation of properties. A set of test images needs to contain simple, medium, and complex structure images where the complexity can be described or characterized by a percentage of pixels that belong to an image's homogeneous regions or entropy for a noise-free case and so on. For a better understanding, we give six images as the examples of images of different complexity (Figure 1). More than a half of the pixels in Figure 1a belong to quasi-homogeneous regions, a certain part of the pixels in Figure 1c-f relate to homogeneous regions, whilst there are practically no homogeneous regions in the image in Figure 1b. There are different ways to characterize image complexity. We have noticed that the performance of the lossy compression of noisy images correlates with the image entropy for noise-free cases. Entropy E is the following: 5.82 for Frisco, 7.33 for Diego, 7.46 for Fr01, 7.40 for Fr02, 7.38 for Fr03, and 7.29 for Fr04. It will be shown later that the plots for the test images having similar E values for the noise-free versions usually have a similar behaviour. Note that the marginal cases are of the most interest; however, \"average\" cases (of middle complexity) are also worth considering. Because of this, there is a tendency in the image processing community for the creation of image databases and for their use in the design and verification of image processing methods [38; 39]. As a starting point for analyzing the lossy compression of noisy images using BPG, we concentrate on the case of grayscale images. As a noise model, we consider the AWGN, which is known to be the simplest model. Just such a model is commonly used as a starting point in research, for example, in the prediction of the potential efficiency of image denoising [40]; thus, one can present an observed noisy grayscale image as:  
\[I_{ij}^{n}=I_{ij}^{true}+n_{ij}, \tag{1}\]  
where \(I_{ij}^{true}\), \(i=1,...\), \(I_{Im}\), \(j=1,...\), \(I_{Im}\) is the true or noise-free image, \(n_{ij}\) denotes AWGN in the \(ij\)-th pixel, and \(I_{Im}\) and \(J_{Im}\) define the considered image size. It is possible to assume the noise mean equals to zero and the AWGN variance is equal to \(\sigma^{2}\). Moreover, we assume that \(\sigma^{2}\) is either a priori known or pre-estimated with a high accuracy [41]. Quality of the original noisy image can be characterized in different ways. One standard way is to calculate the peak signal-to-noise ratio as:  
\[PSNR^{n}=10 log_{10}\left(\frac{255^{2}}{\sigma^{2}}\right), \tag{2}\]  
under the assumption that an image is represented as an 8-bit two-dimensional (2D) data array. Another way is to employ some visual quality metrics, here, we use two of them. The first one is PSNR-HVS-M [26] (a peak signal-to-noise ratio taking into account the human vision system (HVS) and masking (M)), and the multi-scale structural similarity metric (MS-SSIM) [27]. Both are among the best in characterizing the visual quality of images with distortions typical for remote sensing [42], in particular those due to noise and lossy compression. They can be calculated quickly, are applicable to a single-channel (grayscale images), and are based on different principles. The latter is important since no elementary full-reference metric is perfect and, thus, while carrying out an analysis and making conclusions, it is desired to rely on the results obtained for several visual quality metrics.