# Binary Ghost Imaging Based on the Fuzzy Integral Method
## 5 Conclusions  
Based on BGI system architecture, the FIBGI method is proposed to achieve reconstruction of the target with high quality. Contrary to the conventional binary methods, such as MBGI, OBGI, and PPBGI, the proposed method utilizes the result of fuzzy integrals to determine the binarization threshold. The main structure of the algorithm is the Bradley algorithm, which is a kind of local adaptive binarization algorithm. In the proposed method, the integral image is firstly calculated through the SAT algorithm. According to the integral image, the fuzzy integral image is obtained through the Choquet integral. It is the introduction of fuzzy integrals in the process of binarization that makes the result of speckle binarization more accurate, and retains more speckle information. Finally, similar to the conventional Bradley algorithm, the binarization threshold of each pixel is determined by the calculated fuzzy integral image, threshold sensitivity hyper-parameter \(\alpha\), and the threshold calculation window hyper-parameter \(w\). The effectiveness of the proposed algorithm is verified by theoretical analysis and experiment results. It is compared qualitatively and quantitatively with the DGI, MBGI, OBGI, and PPBGI methods. The proposed method can obtain higher quality reconstruction results than other methods. Although there is noise existing in the reconstruction results of the proposed method, the resolution of the reconstruction result is much higher, and the visual effect is much better. Moreover, the PSNRs and SSIMs of various methods are calculated. Although there is little difference in PSNR among all the methods, the SSIMs of the proposed method are significantly higher than those of other methods with different numbers of measurements. In addition, the influences of the threshold sensitivity hyper-parameter \(\alpha\) and the threshold calculation window hyper-parameter \(w\) are also analyzed in the paper. Through the analysis, it can be found that there is an optimal value for each hyper-parameter. The best reconstruction results can be obtained when the two hyper-parameters are set to their respective optimal values. In addition, the selection of the threshold calculation window hyper-parameter \(w\) needs to be based on the actual imaging requirements. When the image resolution is more important, the parameter can be set to a small value, and when the image signal-to-noise ratio is more important, the parameter can be set to a large value. In the future, the SSIM uncertainty of the reconstruction results with the proposed FIBGI method will be analyzed. A dataset containing grayscale images with different complexities will be built, and the grayscale image in the dataset will be reconstructed with the proposed method. With the reconstructed results, the average and variance of the SSIM will be calculated, and the relationship between SSIM and image information entropy will also be studied.  
Formal analysis, X.Y., L.X. and W.L.; Investigation, C.Y.; Methodology, X.Y., L.X. and W.L.; Supervision, J.H. and L.W.; Writing--original draft, X.Y.; Writing--review and editing, L.W. and W.Z. All authors have read and agreed to the published version of the manuscript.  
This research was funded by National Natural Science Foundation of China (NSFC), grant number 61801429; Key Laboratory Foundation, grant number WDZC2020500208; Natural Science Foundation of Zhejiang Province, grant number LQ20F050010 and LY20F010001; Fundamental Research Funds of Zhejiang Sci-Tech University, grant number 2020Q020.  
Not applicable.  
We thank [PERSON] and [PERSON] for their fruitful discussions. We also thank all the anonymous reviewers for their insightful comments.  
The authors declare no conflict of interest.