## 2 Proposed Method
### Construction of Image Co-Occurrence Histogram
If the gradient is unsigned, the histogram channels spread in the range of 0 to 180 degrees, whereas if signed, the histogram channels spread in the range of 0 to 360 degrees. As the edge direction alone is considered in calculating the image gradient orientation, the scope of image gradient orientation is 180 bins, ranging from 1 to 180. A gradient orientation ICH, \(H_{o}\), of 180 \(\times\) 180 bins is plotted like the \(H_{X}\) in Equation (1). The inverted PMF function \(\overline{P}_{o}\) and the saliency \(S_{o}(x,\ y)\) can be similarly constructed as specified in Equations (2)-(4), respectively. The saliency information is extracted by computing a combination of the intensity domain and the gradient orientation domain as  
\[S=S_{\text{var}}+S_{\text{var}} \tag{5}\]  
where  
\[S_{\text{var}}(x,\ y)=\frac{1}{9}\sum_{m=-1}^{1}\sum_{n=-1}^{1}S_{v}(x+m,\ y+n), \tag{6}\]  
\[S_{\text{var}}(x,\ y)=\frac{1}{9}\sum_{m=-1}^{1}\sum_{n=-1}^{1}S_{o}(x+m,\ y+n), \tag{7}\]  
where \(S_{v}\) refers to the saliency of intensity and \(S_{o}\) is the saliency of the gradient orientation. Based on Equation (5), the saliency is further used to calculate a saliency-aware weighting (SAW) \((\Gamma_{b}(x,y))\) for an HDRI. To calculate an edge-aware weighting (EAW) \((\Gamma_{e}(x,\ y))\), the local variance and mean of the given HDRI are used. The final weighting function, \(W(x,\ y)\), is described as  
\[W(x,\ y)=\frac{\Gamma_{e}(x,y)}{\Gamma_{b}(x,y)} \tag{8}\]  
here  
\[\Gamma_{b}(x,\ y)= \begin{cases}1,&if\ S_{F}(x,\ y)\leq 1\\ S_{F}(x,\ y),&otherwise\end{cases}  \tag{9}\]  
and  
\[S_{F}(x,\ y)=\frac{S(x,\ y)}{\frac{1}{N}\sum_{x=1}^{N}\sum_{y=1}^{N}S(x,\ y)} \tag{10}\]  
where \(S(x,\ y)\) refers to the resulting value of Equation (5) and \(N\) is the total number of pixels in the given image. In the process of detecting image saliency, a pixel \((x,\ y)\) with a larger \(\Gamma_{b}(x,\ y)\) takes priority over the others. If a pixel \((x,\ y)\) belongs to an attention-salient region, the value of \(\Gamma_{b}(x,\ y)\) is 1 and above. Therefore, a higher priority is given to a pixel in an attention-salient region. This is similar to the ability of attentional control of our HVS. Our human eyes are more likely to pay attention to information in the attention-salient region than elsewhere. Furthermore, a luminance value, which is defined as one of the CIE XYZ color components, \(Y\), is used in calculating an EAW [23] in Equation (8). The luminance value in the log domain is expressed as \(L_{h}(x,\ y)\). Let \(\sigma_{L_{h},\ \rho_{2}}(x,\ y)\) and \(\mu_{L_{h},\ \rho_{2}}(x,\ y)\) be the standard deviation and the average value of the component \(L_{h}\) in \(\Omega_{\rho_{2}}(x,\ y)\). If \(L_{h}(x,\ y)\) is located at the edge and \(L_{h}(x^{\prime},y^{\prime})\) is in the flat area, \(\sigma_{L_{h},\ \rho_{2}}^{2}(x,\ y)/\mu_{L_{h},\ \rho_{2}}^{2}(x,\ y)\) is larger than \(\sigma_{L_{h},\ \rho_{2}}^{2}(x^{\prime},\ y^{\prime})/\mu_{L_{h},\ \rho_{2}}^{2}(x^{\prime},\ y^{ \prime})\), which implies a higher weighting in Equation (9). From this observation, \(\Gamma_{e}(x,\ y)\) is calculated by using normalized local standard deviations of all the pixels as  
\[\Gamma_{e}(x,\ y)=\frac{1}{N\times N}\sum_{x^{\prime}=1}^{N}\sum_{y^{\prime}=1}^{ N}\frac{ \begin{pmatrix}\sigma_{L_{b},\ \rho_{2}}^{2}\ (x,\ y)+\
u_{1}\\ \mu_{L_{b},\ \rho_{2}}^{2}\ (x,\ y)+\
u_{2}\end{pmatrix} ^{\zeta}}{ \begin{pmatrix}\sigma_{L_{b},\ \rho_{2}}^{2}\ (x^{\prime},\ y^{\prime})+\
u_{1}\\ \mu_{L_{b},\ \rho_{2}}^{2}\ (x^{\prime},\ y^{\prime})+\
u_{2}\end{pmatrix} ^{ \zeta}} \tag{11}\]  
where \(\zeta\) is a constant value, \(\
u_{1}\) is added in order to avoid instability which can occur if \(\sigma_{L_{b},\ \rho_{2}}^{2}\ (x,\ y)\) is zero. The \(\
u_{1}\) is calculated as \(\left(0.001*L\right)^{2}\), with L, the dynamic range of the given image [24]. \(\
u_{2}\) is a small constant that is intended to prevent dividing by 0.