## 2 Overview of Similarity-Based Full-Reference Image Quality Assessment Methods
Due to its potential instability, particularly for black or \"flat\" (with constant luminance) areas of images, the original version of the UIQI has been shortly replaced by the Structural Similarity (SSIM) [29], where the original formula has been extended as  
\[SSIM=l(x,y)\cdot c(x,y)\cdot s(x,y)=\frac{2\bar{x}g+C_{1}}{\bar{x}^{2}+\bar{y} ^{2}+C_{1}}\cdot\frac{2\sigma_{x}\sigma_{y}+C_{2}}{\sigma_{x}^{2}+\sigma_{y}^ {2}+C_{2}}\cdot\frac{\sigma_{xy}+C_{3}}{\sigma_{x}\sigma_{y}+C_{3}}\, \tag{1}\]  
where the default values of the stabilizing constants (added in comparison to UIQI) for 8-bit grayscale images are: \(C_{1}=(0.01\times 255)^{2}\), \(C_{2}=(0.03\times 255)^{2}\) and \(C_{3}=C_{2}/2\). Their role is to prevent the potential division by zero without affecting significantly the overall quality metric's value, calculated as the average similarity value for all positions of the sliding window. The local structural similarity values are calculated as the product of three factors representing luminance distortions \(l(x,y)\), contrast loss \(c(x,y)\), and structural distortions \(s(x,y)\) between the fragments \(x\) and \(y\) of images \(A\) and \(B\), according to the current position of the sliding window. These similarities are based on the local mean values (\(\bar{x}\) and \(\bar{y}\)) of the original and distorted images, the respective variances (\(\sigma_{x}^{2}\) and \(\sigma_{y}^{2}\)), and the covariance \(\sigma_{xy}\). The default window is \(11\times 11\) pixels Gaussian window, considered as appropriate for typical standard definition images. Since the SSIM metric has been widely accepted in the community, its implementation has appeared in OpenCV and MATLAB(r), for example, and many modifications based on similar formulas have been proposed by various researchers. Considering the postulated independence of results of the image size without the necessity of changing the sliding window's size, the multi-scale version of this method has been developed by the same authors known as MS-SSIM [30], where the image details at different resolutions are assessed applying iteratively a low-pass filter and downsampling by a factor of 2. The contract and structure comparisons are conducted for each scale, whereas the luminance comparison is computed only for the highest scale. The default normalized weighting exponents have been provided after cross-scale calibration experiments. During the next several years many other SSIM-based metrics have been proposed, which have also been used in the conducted experiments related to this paper. In 2006 the Quality Index based on Local Variance (QILV) has been proposed [31], where a comparison of the local variance distribution is considered as the extension of the use of mean-variance values in the original SSIM. Finally, the following formula has been proposed, assuming that \(A\) and \(B\) are the input images:  
\[QILV=\frac{2\mu_{V_{A}}\mu_{V_{B}}}{\mu_{V_{A}}^{2}+\mu_{V_{B}}^{2}}\cdot\frac{ 2\sigma_{V_{A}}\sigma_{V_{B}}}{\sigma_{V_{A}}^{2}+\sigma_{V_{B}}^{2}}\cdot \frac{\sigma_{V_{A}}\sigma_{V_{B}}}{\sigma_{V_{A}}\sigma_{V_{B}}}\, \tag{2}\]  
where \(\sigma_{V_{A}V_{B}}\) is the covariance between the variances of two images (\(V_{AAnother modification proposed by [PERSON] et al. [32] is based on the use of the wavelet domain. The metric known as Complex Wavelet Structural Similarity (CW-SSIM) utilizes the assumption that the presence of some distortions causes consistent phase changes in the local wavelet coefficients. The metric is also more robust to small rotations and translations in comparison to the spatial domain SSIM. On the other hand, the application of information content weighting for the multi-scale SSIM (referred to as IW-SSIM), proposed in the paper [33], incorporates the Laplacian pyramid transform to calculate the appropriate weights based on the statistical Gaussian model of natural images. A similar information weighting may also be applied for MSE and PSNR metrics, significantly enhancing their correlation with subjective quality scores.