# Binary Ghost Imaging Based on the Fuzzy Integral Method
## 4 Discussion  
There are two hyper-parameters, \(\alpha\) and \(w\), which influence the reconstruction quality of the FIBGI. In this section, the effects of these two hyper-parameters on the quality of reconstruction will be discussed in detail, and the way to set these two hyper-parameters, when the prior information of the target is unknown based on the discussion results, will be summarized. Firstly, the influence of the hyper-parameter \(\alpha\), the threshold adjustment parameter, on the imaging quality is analyzed. When the value of \(\alpha\) is close to 1, most of the values in the binarization speckle image are 1. When the value of this parameter is close to 0, most pixel values of the binarization speckle image are 0. The value range of the hyper-parameter is [0, 1]. The exact value needs to be set empirically, or traverse the interval [0, 1], for the best imaging quality. According to the experiment comparison in Section 3, the SSIM is more consistent with the visual effects. Hence, the SSIMs of reconstructed results of the image of a Chinese character and the image of a panda are calculated by increasing the hyper-parameter \(\alpha\) from 0.005 to 1 with the step of 0.005 when the number of measurements is 8000, and the hyper-parameter \(w\) is set as 10. The calculated SSIMs are shown in Figure 8. The blue line in Figure 8a represents the SSIM of the image of the Chinese character, and the red represents the SSIM of the image of the panda. It can be  
Figure 6: The reconstructed results of grayscale target with different methods. Figure 7: The quantitative comparisons of grayscale target with different methods. seen from Figure 8a that the overall trend of SSIMs of the two targets both decrease with the increase in hyper-parameter \(\alpha\). In addition, there is an optimal value in the interval of [0.005, 0.055]. In order to determine the optimal value more precisely, the hyper-parameter increases from 0.005 to 0.055 with the step of 0.0005. The SSIMs of the image of the Chinese character and the image of the panda changing with \(\alpha\) in the interval of [0.005, 0.055] are shown in Figure 8b,c, respectively. It is obvious that there is an optimal value existing in the SSIM curve. The optimal values of the images of the Chinese character and panda are 0.0155 and 0.019, respectively. When the hyper-parameter is set to the optimal value, the SSIM of reconstructed result is the maximum. Moreover, it can be concluded that the optimal value of hyper-parameter is not the same for different targets. Therefore, it is not appropriate to choose the same value for the parameter for different targets. Fortunately, there is no significant difference in SSIM when the value of hyper-parameter \(\alpha\) is in the interval [0, 0.03]. In the case of low requirements for SSIM, it is feasible to select random value in the interval [0, 0.03]. The influence of the hyper-parameter \(w\) on the imaging quality is also analyzed. This parameter determines the size of the local adaptive binarization window. The larger the window is, the more pixels affect the binarization threshold. Especially when the hyper-parameter equals the size of the target, the local adaptive binarization is equivalent to the global binarization. Here, the number of measurements is set as 8000, the hyper-parameter \(\alpha\) is 0.019, and the hyper-parameter \(w\) is selected as 2, 12, and 32, respectively. The reconstruction results of the image of the panda with the proposed method are shown in Figure 9. The first row of Figure 9 is the binarization speckle when the hyper-parameter selects different values. The images in the second row are the corresponding reconstruction results. It can be seen that the binarization speckle has a finer structure when the value of hyper parameter \(w\) is relatively small. Therefore, the resolution of the corresponding reconstruction results was higher. However, the signal-to-noise ratio is low, and there is a huge noise in the reconstruction result.