# Binary Ghost Imaging Based on the Fuzzy Integral Method
## 2 Theory and Methods
The vector \(\mathbf{v}\) is a finite set, and the power set of \(\mathbf{v}\) is denoted as \(\mathcal{P}(\mathbf{v})\). Therefore, a set function \(\mu\): \(\mathcal{P}(\mathbf{v})\rightarrow[0,\,1]\) defined on the power set \(\mathcal{P}(\mathbf{v})\) is a fuzzy measure, if the following conditions are satisfied:  
* Boundary condition: \(\mu(\mathcal{O})=0\), \(\mu(\mathbf{v})=1\);
* (Monotonicity: \(A\subseteq B\Rightarrow\mu(A)\leq\mu(B)\) for \(\forall\ A,\,B\in\mathcal{P}(\mathbf{v})\);  
Additionally, due to the assumption of symmetric fuzzy measure, another condition should also be satisfied:  
* \(|A\,|=|\,B\,|\,\Rightarrow\mu(A)=\mu(B)\) for \(\forall\ A,\,B\in\mathcal{P}(\mathbf{v})\). In the condition of CC3, the \(|\,E\,|\) represents the cardinal number of the set \(E\). It is further assumed that \(\mu\) is a uniform fuzzy measure, which satisfies the symmetric fuzzy measure condition, and hence \(\mu\) can be expressed as [48]:  
\[\mu(E)=\frac{|E|}{4}. \tag{9}\]  
According to the expression of Equation (9), the fuzzy measure can be calculated as Table 1.  
Therefore, the fuzzy measure vector \(\mathbf{m}\) is [1, 0.75, 0.5, 0.25]. The discrete Choquet integral can be expressed as:  
\[ \begin{split}(C)\int f\mathrm{d}\mu&=\sum\limits_{i= 1}^{n}(v_{i}-v_{i-1})\mu(E_{i})=[v_{1}-v_{0}]\mathbf{m}[1]+[v_{1}-v_{2}] \mathbf{m}[2]\\ &+[v_{3}-v_{2}]\mathbf{m}[3]+[v_{4}-v_{3}]\mathbf{m}[4].\end{split}  \tag{10}\]  
Here, \(\mathbf{m}[i]\), \(i=1,2,3,4\) is the \(i\)th element in the fuzzy measure vector \(\mathbf{m}\). The computed result of Choquet integral is the value of the fuzzy integral image \(F^{(k)}(x,y)\) at position \((x,y)\). The final step in the proposed method is to utilize the Bradley algorithm based on fuzzy integral images to obtain the binarization threshold of each pixel. In the conventional Bradley algorithm, the size of threshold calculation window \(w\) needs to be set in advance. The average value of the integral image in the selected window is calculated as follows:  
\[A_{w}^{(k)}(x_{1},y_{1};x_{2},y_{2})=\frac{S^{(k)}(x_{2},y_{2})+S^{(k)}(x_{1},y _{1})-S^{(k)}(x_{1},y_{2})-S^{(k)}(x_{2},y_{1})}{w^{2}}, \tag{11}\]  
where positions \((x_{1},\,y_{1})\) and \((x_{2},\,y_{2})\) are the upper left corner and lower right corner of the threshold calculation window, respectively. Hence, the binarization threshold value \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Set** & **Fuzzy Measure \(\mu\)** & **Set** & **Fuzzy Measure \(\mu\)** \\ \hline \([v_{1}]\) & 0.25 & \(\{v_{1},v_{2},v_{3}\}\) & 0.75 \\ \([v_{1},v_{2}]\) & 0.5 & \(\{v_{1},v_{2},v_{3},v_{4}\}\) & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The value of uniform fuzzy measure table. of pixel (\(x_{2}\), \(y_{2}\)) can be calculated based on the average value and threshold sensitivity hyper-parameter \(\alpha\) (in interval [0, 1]), as follows:  
\[T^{(k)}(x_{2},y_{2})=\alpha\cdot A_{w}^{(k)}(x_{1},y_{1};x_{2},y_{2}). \tag{12}\]  
It should be noted that the integral image is corresponding to the fuzzy integral image, which is essentially the representation of the integral image in the fuzzy measure domain. Therefore, contrary to the conventional Bradley algorithm, the average value in the proposed method is calculated based on the fuzzy integral image, rather than the integral image. \(S^{(k)}(x,y)\) in Equation (11) should be changed to \(F^{(k)}(x,y)\). As Step 3 shows in Figure 2, the size of threshold calculation window \(w\) is selected as 3, and the selected calculation window is colored by blue. The values of the four corners of the blue box are chosen to calculate the average value. Based on the average value combining the hyper-parameter \(\alpha\), the threshold of (\(x\), \(y\)) can be finally obtained. The kth binarization speckle is denoted as \(\hat{I}^{(k)}(x,y)\) and it is described as follows:  
\[\hat{I}^{(k)}(x,y)= \begin{cases}1&\text{if}\quad T^{(k)}(x_{2},y_{2})\leq I^{( k)}(x,y)\\ 0&\text{otherwise}\end{cases} .