## 2 Overview of Similarity-Based Full-Reference Image Quality Assessment Methods
One of the most successful similarity-based metrics, known as Feature Similarity (FSIM), has been proposed in two versions (for grayscale and color images--denoted as FSIMc) [34] and verified as one of the best metrics for most IQA databases. It utilizes a similar formula as the SSIM but is applied for low-level features, namely phase congruency (PC--as a significance measure of a local structure) and gradient magnitude (GM), a complementary feature extracted using the Scharr filter. For simplicity, the equal importance of these two factors (\(\alpha=\beta=1\)) has been assumed in the similarity formula between the images \(A\) and \(B\):  
\[S_{L}(\mathbf{x})=\left(\frac{2\cdot PC_{A}(\mathbf{x})\cdot PC_{B}(\mathbf{x })+T_{1}}{PC_{A}^{2}(\mathbf{x})+PC_{B}^{2}(\mathbf{x})+T_{1}}\right)^{\alpha }\cdot\left(\frac{2\cdot GM_{A}(\mathbf{x})\cdot GM_{B}(\mathbf{x})+T_{2}}{ GM_{A}^{2}(\mathbf{x})+GM_{B}^{2}(\mathbf{x})+T_{2}}\right)^{\beta}\, \tag{3}\]  
where \(T_{1}\) and \(T_{2}\) are the stability constants preventing the division by zero and \(\mathbf{x}\) is the sliding window position. The final formula obtained as the averaged \(S_{L}\) for all locations is additionally weighted by the phase congruency, assuming that the Human Visual System (HVS) is highly sensitive to phase congruent structures and may be expressed as  
\[FSIM=\frac{\sum_{\mathbf{x}\in A}S_{L}(\mathbf{x})\cdot PC_{m}(\mathbf{x})}{ \sum_{\mathbf{x}\in A}PC_{m}(\mathbf{x})}\, \tag{4}\]  
where \(PC_{m}(\mathbf{x})=max(PC_{A}(\mathbf{x}),PC_{B}(\mathbf{x}))\) and \(\mathbf{x}\) denotes each position of the local window on the image plane \(A\) (or \(B\)). Its extension for color IQA utilizes the YIQ color space with two chrominance components (I and Q), used as the two other features, weighted using an additional exponential parameter \(\gamma\) used for tuning the importance of the chromatic data. A similar approach based on the comparison of gradient images, known as Gradient Similarity (GSM) has been proposed in the paper [35], where the gradient similarity term, calculated in the same way as the luminance or contrast factor in the SSIM, is combined with the luminance comparison (calculated as the \(l(x,y)\) term the Formula (1)). As the authors have assumed that the typical Prewitt, Sobel, or Scharr kernels are too small the gradient extraction has been conducted using the following \(5\times 5\) pixels mask:  
\[Mask=\left[ \begin{matrix}0&0&0&0&0\\ 1&3&8&3&1\\ 0&0&0&0&0\\ -1&-3&-8&-3&-1\\ 0&0&0&0&0\end{matrix} \right]\, \tag{5}\]  
rotated by \(90^{\circ}\) to obtain four directional masks for gradient calculations. Further metrics inspired by the idea of SSIM include a fast and effective spectral residual similarity (SR-SIM), based on specific visual saliency model [36] correlated with perceived image quality, as well as the Edge Strength Similarity (ESSIM) [37], assuming the edge detection based on Scharr kernel and further calculation of edge strength maps determined from the directional derivatives calculated for each pixel. Obtained edge strength maps for the reference and distorted images are compared using the SSIM-like formula, similarly as for the other metrics. The similarity analysis in the frequency domain has led to the idea of the DCT subband similarity (DSS) proposed by [PERSON] et al. [38]. This approach considers various importance of structural distortions in different DCT subbands from a perceptual point of view. Hence, the similarity values, obtained for individual subbands according to the formula similar to the other mentioned metrics based on the local variances (i.e., their product divided by the sum of squares with stability constants), are weighted providing the final DSS result. Alternatively, good performance may also be obtained using the SSIM-like formula for measurement of contrast deviation as proposed in the paper [39], where the multiscale contrast similarity deviation (MCSD) has been defined as another relatively fast IQA metric with high accordance with MOS values for the most relevant IQA databases.