## 2 Proposed Method
### Construction of Image Co-Occurrence Histogram  
A histogram of an image (IH) is the distribution of the pixel values in the given image. It has several distinct advantages of image without noise, tolerance to image rotation, size variation, and more. A traditional, one-dimensional IH presents occurrence of each image value only, but completely discards information about how image pixels are distributed in a certain location despite its high importance to the perception of an image. Noticeably, a two-dimensional image co-occurrence histogram (ICH) can indicate both occurrence and co-occurrence of image pixels and be used as a means of calculating the salient region. From this fact, let \(I_{h}(x,\ y)=[X_{h}(x,\ y),Y_{h}(x,\ y),Z_{h}(x,\ y)]\) be a pixel in an HDRI. In the proposed method, the ICH is used to modify the salience detection model that is similar to the saliency modeling in [19] in order to process an HDRI. An ICH of one of the CIE XYZ color components, \(X_{h}\), in the intensity domain, \(H_{X}\), is defined as  
\[H_{X}=[h_{X}(m,\ n)],\ 1\leq m,\ n\leq L, \tag{1}\]where \(H_{X}\) is a symmetric square matrix of \(L\times L\), representing occurrence and co-occurrence of a pixel value. The element \(h_{X}(m,\ n)\) indicates the frequency of occurrence of a pixel value at the condition of intensities \(m\) and \(n\) in the local neighborhood window of \(w\times w\) (\(w=3\)). In other words, an ICH, \(H_{X}\), counts how many times a pixel value with intensities \(m\) and \(n\) occurs. Likewise, the ICH can depict a global distribution of intensities as well. Figure 1 shows the product of plotting an ICH. (a) is an original 'tulip' image, and (b) shows an ICH of a gray-scale tulip image. Most pixel pairs are concentrated along the diagonal of the ICH, and the peak is located at the center of the pixel value in the histogram below. The result is similar to what is suggested in [21]. Next, the global and local distributions of intensities are used in detecting image saliency. In general, the principle of contrast states that rare or infrequent visual features in a global image context give rise to high saliency values. From this viewpoint and motivated by the [PERSON]'s entropy theorem [22], a logarithm relationship is applied to Equation (1) as  
\[P_{log}=p_{log}(m,\ n)=-\ln(p(m,\ n)) \tag{2}\]  
where \(p(m,\ n)\) refers to a probability mass function (PMF), which is the product of normalizing the ICH matrix \(H_{X}\). However, saliency is not correlated with occurrence and co-occurrence, so an inverted PMF \(\overline{P}_{v}\) is calculated as  
\[\overline{P}_{v}=[\overline{p}(m,\ n)]= \begin{cases}0,&if\ p_{log}(m,\ n)=0\\ 0,&if\ p_{log}(m,\ n)>\frac{1}{U_{X}}\\ \frac{1}{U_{X}}-\ p_{log}(m,\ n)&if\ p_{log}(m,\ n)\leq\frac{1}{U_{X}}\\ \end{cases}  \tag{3}\]  
In Equation (3), \(\overline{P}_{v}\) is set for \(0\) in the absence of intensity pairs within the given image and if \(P_{log}\) is larger than a threshold, \(\frac{1}{U_{X}}\). Meanwhile, \(U_{X}\) refers to the frequency of occurrence whose value is not zero in \(H_{X}\). Next, saliency is computed by \(\overline{P}_{v}\). For each pixel at a location \((x,\ y)\), image saliency, \(S_{v}(x,\ y)\), is calculated as  
\[S_{v}(x,\ y)=\sum_{x^{\prime}=x-z}^{x+z}\sum_{y^{\prime}=y-z}^{y+z}\overline{ p}(i(x,\ y),\ i(x^{\prime},\ y^{\prime})), \tag{4}\]  
Figure 1: CCH of a tulip image: (**a**) an original image, (**b**) CCH of a grayscale image. where \(z\) represents the size of a neighborhood window. The notations \(i(x,\ y)\) and \(i(x^{\prime},\ y^{\prime})\) define image values at locations \((x,\ y)\) and \((x^{\prime},\ y^{\prime})\), respectively, and \(\overline{p}(i(x,\ y),\ i(x^{\prime},\ y^{\prime}))\) is therefore the element of \(\overline{P}_{v}\) indexed by \(i(x,\ y)\) and \(i(x^{\prime},\ y^{\prime})\). Image brightness and color aside, the ICH can also plot the image gradient orientation which is often closely correlated with the perceptual visual saliency.