## 3 Methodology
### EW-FCM Segmentation Technique  
In this section, we first present a short analysis of the Otsu technique. Then, we introduce EW and histogram accumulation moment, including FCM. Finally, the new image threshold technique for image segmentation is presented. Otsu is one of the most commonly referenced thresholding methods. Let \(I=g(x,y)\) be an image with the gray value belonging to the interval \([0,1,\,\ldots,\,L-1]\). Assign \(n_{i}\) as the pixel numbers with the gray value \(i\) and the total pixel numbers in \(g(x,y)\) as \(N\). The existence probability of the gray level \(i\) is given by  
\[P_{i}=\frac{n_{i}}{N}. \tag{1}\]  
Assume the threshold \(th\) (\(0\leq th\leq L-1\)) divides \(g(x,y)\) into two classes: background \(C_{O}=\{(x,y)|0\leq g(x,y)\leq th\}\) and object \(C_{B}=\{(x,y)|th+1\leq g(x,y)\leq L-1\}\). The medium gray level of individual class and the probability of category occurrence, respectively, are calculated as follows:  
\[\mu_{o}(th)=\sum_{i=0}^{th}\frac{iP_{i}}{\omega_{O}(th)},\,\,\mu_{B}(th)=\sum_{ i=th+1}^{L-1}\frac{iP_{i}}{\omega_{B}(th)}, \tag{2}\]  
where  
\[\omega_{O}(th)=\sum_{i=0}^{th}P_{i},\,\,\omega_{B}(th)=\sum_{i=th+1}^{L-1}P_{ i}. \tag{3}\]  
In [PERSON]'s technique, the resulting threshold performance is measured by analyzing the difference between the background and foreground. The optimal threshold \(th^{*}\) when applying this criterion must maximize between-class variance as follows:  
\[ \begin{split} th^{*}=argmax\big{\{}&\omega_{O}(th )\mu_{O}^{2}(th)+\omega_{B}(th)\mu_{B}^{2}(th)\big{\}}.\\ & 0\leq th\leq L-1\end{split}  \tag{4}\]  
The basic idea to improve [PERSON]'s technique is the addition of a weight \(W\) to the objective function in Equation (4) to regulate the output threshold, given as follows:  
\[ \begin{split} th^{*}=argmax\big{\{}& W\big{(} \omega_{O}(th)\mu_{O}^{2}(th)+\omega_{B}(th)\mu_{B}^{2}(th)\big{)}\big{\}}.\\ & 0\leq th\leq L-1\end{split}  \tag{5}\]  
Image entropy describes the properties of an image; it is a mathematical measure of randomness. Images with low entropy values possess minimal information and hold many pixels with similar intensity values. An image with zero entropy means that all pixels hold the same gray value. Reference [56] suggested an EW scheme by substituting weight \(W\) with the entropy objective function \(\psi(th)\) in Equation (5) to create a new objective function  
\[ \begin{split} th^{*}=argmax\big{\{}&\psi(th)\big{(} \omega_{O}(th)\mu_{O}^{2}(th)+\omega_{B}(th)\mu_{B}^{2}(th)\big{)}\big{\}}.\\ & 0\leq th\leq L-1\end{split}  \tag{6}\]  
The entropy objective function is defined as follows:  
\[\psi(th)=ln(\omega_{O}(th)\omega_{B}(th))+\frac{H_{th}}{\omega_{O}(th)}+\frac {H_{n}}{\omega_{B}(th)}, \tag{7}\]  
where  
\[H_{th}=-\sum_{i=0}^{th}P_{i}\ln(P_{i}),\,\,\,H_{n}=-\sum_{i=0}^{L-1}P_{i}\ln(P_ {i}). \tag{8}\]Next, we discuss the first order of the cumulative moment. Let \(M_{O}(th)\) denotes the FCM of the gray histogram, which is the mean gray value from 0 to \(th\) given by  
\[M_{O}(th)=\sum_{i=0}^{th}iP_{i}. \tag{9}\]  
The mean gray of the entire image is \(M_{T}\), defined by  
\[M_{T}=\sum_{i=0}^{L-1}iP_{i}. \tag{10}\]  
The FCM \(M_{T}\) helps the optimal threshold avoid dropping into the local optimum [57]. We combine the EW \(\psi\)(th) and FCM to obtain the optimal threshold and create a new objective function for image segmentation as follows:  
\[ \begin{split} th^{*}&=argmax\{\psi(th)(M_{T})\}.\\ & 0\leq th\leq L-1\end{split}  \tag{11}\]  
We adopt the segmentation process in reference [58], including texture filtering, threshold and binarize, and plot the boundaries. [PERSON] [58] uses the Otsu method to calculate the threshold; meanwhile, our framework uses the new objective function in Equation (11) to determine the threshold. Our objective function provides a better segmentation technique due to the reservation of all properties of EW-FCM. Figure 2 shows the segmentation results of the original Otsu technique, EW scheme [56], and the proposed EW-FCM segmentation method.