# Binary Ghost Imaging Based on the Fuzzy Integral Method
## 2 Theory and Methods  
The schematic of BGI system is shown in Figure 1. As Figure 1 shows, the beam emitted from the laser source is illuminated on a rotating ground glass (RGG), which can generate the dynamic speckles to simulate the pseudo-thermal light source. Following the RGG, a beam splitter (BS) is placed in the optical path to split the light into two different optical paths, which are the reference optical path and object optical path, respectively. In the reference optical path, the beam propagates freely, and the spatial distributions of the beam are recorded by a charge coupled device (CCD). The CCD transmits the recorded speckle spatial distribution information to the computer for subsequent processing. In the object optical path, the split beam illuminates on the target. In order to guarantee the speckle spatial distribution recorded by the CCD in the reference optical path is the same as that illuminating the target in the object optical path, the distance from the laser source to the CCD equals that between the laser source and target. A lens is located behind the target to focus the light passing through the target on the active area of the single-pixel detector (SPD). The SPD is connected to a data acquisition system to measure the total transmitted laser intensity. The obtained intensity results are also sent to the computer to reconstruct the image of the target. Here, the kth speckle intensity distribution measured by CCD is denoted as \(I^{(k)}(x,y)\) and the total transmitted light intensity recorded by SPD is denoted as b(k). Therefore, the target transmission distribution can be calculated through the correlation operation between the two detectors. In order to enhance SNR, the reconstruction process utilizes the differential ghost imaging (DGI) technique. The reconstruction can be expressed as:  
\[G(x,y)=\frac{1}{N}\underset{k=\cdot}{\sum}\left(I^{(k)}(x,y)-\left\langle I^{ (k)}(x,y)\right\rangle\right)\left(b^{(k)}-\frac{\left\langle b^{(k)}\right\rangle }{\left\langle O^{(k)}\right\rangle}O^{(k)}\right), \tag{1}\]  
where \(O\) is the total intensity of speckle distribution, and can be expressed as:  
\[O^{(k)}=\int I^{(k)}(x,y)\mathrm{d}x\mathrm{d}y. \tag{2}\]  
In the BGI system, binary signals with two quantization levels are applied at reference and object optical paths in order to compress the data size. The information recorded by the CCD camera in the reference path is more sensitive to the reconstructed result than that recorded by the SPD in the signal path in the BGI system [35]. Therefore, the binarization algorithm is applied to transform the grayscale speckle images to binary speckle images. The binarization algorithm is the key for the BGI system. The binary speckle obtained by the appropriate binarization algorithm can effectively improve the reconstruction quality of the BGI system. The conventional binarization algorithms adopted in the BGI system include the global binarization algorithm and local adaptive binarization algorithm. In reference [38], the reconstruction results of BGI with different binarization methods are compared. It can be concluded that the reconstruction quality of BGI with local binarization method is relatively higher than that of BGI with global binarization method. In the proposed method, the Bradley algorithm [46] is modified to select the binarization threshold value. The flow chart of the proposed algorithm is shown as Figure 2.  
Figure 1: The schematic of BGI system. Contrary to the conventional local adaptive binarization methods, such as PPBGI, which calculate the binary threshold pixel by pixel on every speckle image, the proposed method selects the binary threshold according to the fuzzy integral image, which can help avoid uncertainty in binarization and beyond. In addition, the fuzzy integral image is calculated combined with the integral image, which is calculated through the summed-area table algorithm (SAT) [47].