## 3 Proposed Method
### SG-ESSIM Method  
As one can observe from the derivation of the ESSIM [19], the impact of visual saliency was not incorporated into these metrics. In this section, we describe how we enhanced the ESSIM by using visual saliency in the measurement of local image quality. Hence, it was highlighted that local image quality degradation was jointly characterized by the objective degradation of edge strength and significance.  
As proposed by the authors of the ESSIM [19], our algorithm first determined the edge strength maps of the reference and distorted images as recommended in the ESSIM [19].  
Subsequently, it calculated the saliency-guided local image quality map. Finally, it pooled the local image quality map to obtain the overall perceptual quality of the distorted image.  
Using Equations (2) and (5), which describe \(E(f,i)\) and \(E(g,i)\) in the ESSIM, our visual saliency-guided local similarity map is defined as  
\[SM(E(f,i),E(g,i))=\frac{2\cdot E(f,i)\cdot E(g,i)+H(V(i))}{(E(f,i))^{2}+(E(g,i) )^{2}+H(V(i))}, \tag{6}\]  
where \(V(i)\) stands for the visual saliency measure at pixel location at pixel location \(i\). Moreover, \(H(\cdot)\) is a decreasing function defined as  
\[H(V(i))=K\cdot e^{-\frac{V(i)}{h}}, \tag{7}\]  
where \(e\) is Euler's number, \(K\) is a scaling factor, and \(h\) is an attenuation factor (Equation (7) is a decreasing function because the similarity metric defined by Equation (9) also needs to be regulated). To be more specific, \(V(i)\) takes the edge strength maps of the reference and distorted images and returns their pixel-wise maximum:  
\[V(i)=max(E(f,i),E(g,i)). \tag{8}\]  
Equation (8) implies that regions with strong edges are more salient to the human visual system than those with weak edges since edge information conveys essential information about the visual scene to humans [47]. A larger value of \(V(i)=max(E(f,i),E(g,i))\) indicated a higher visual saliency at pixel location \(i\). Moreover, the computation of \(V(i)\) did not result in a significant increase in the computational costs since \(E(f,i)\) and \(E(g,i)\) were determined anyway in the ESSIM [19]. Substituting Equations (7) and (8) into Equation (9), we obtain the following equation for visual saliency guided local similarity map:  
\[SM(E(f,i),E(g,i))=\frac{2\cdot E(f,i)\cdot E(g,i)+K\cdot e^{-\frac{max(E(f,i),E(g,i))}{h}}}{(E(f,i))^{2}+(E(g,i))^{2}+K\cdot e^{-\frac{max(E(f,i),E(g,i))}{h}}}. \tag{9}\]  
Finally, the overall predicted perceptual quality score of the image was estimated by averaging the visual saliency-guided (SG) local similarity map  
\[SG-ESSIM=\frac{1}{N}\sum_{i=1}^{N}SM(E(f,i),E(g,i)), \tag{10}\]  
where \(N\) is the total number of pixels.  
The hyperparameters of the SG-ESSIM, which have to be set, were \(K\) and \(h\). These parameters were determined on a subset of the TID2008 [21] database, which contained 5 random reference images and 340 corresponding distorted images. Specifically, those hyperparameters were chosen where the highest Spearman's rank order correlation coefficient between the ground truth and the predicted values was obtained. As a result, we chose \(K\) = 51,000 and \(h\) = 0.5 in our MATLAB implementation. Since the SG-ESSIM was not a machine-learning-based approach, this random subset was also used in the evaluation process.