# Binary Ghost Imaging Based on the Fuzzy Integral Method
## 3 Experiment Results and Comparative Analysis
In Equation (15), \(MAX_{I}\) is the possible maximum value of the image pixels. The reconstructed results with different methods are normalized to the interval between 0 and 1. Therefore, the maximum value of image pixels is 1, namely \(MAX_{I}=1\). \(MSE\) is the mean square error of the image. For the target image with the pixel number of \(N=n\times m\), the MSE is defined as:  
\[MSE=\frac{1}{nm}\sum_{i=1}^{m}\sum_{j=1}^{n}\left|O(i,j)-R(i,j)\right|^{2}, \tag{17}\]  
where \(O(i,j)\) represents the target, \(R(i,j)\) represents the reconstructed image, and \(i,j\) represent the horizontal and vertical coordinates of pixels, respectively. In F (16), \(\mu_{x}\) and \(\mu_{y}\) are the means of total pixels in the original image and reconstructed image, respectively. \({\sigma_{x}}^{2}\) and \({\sigma_{y}}^{2}\) are the variances of the original image and reconstructed image, respectively. \(\sigma_{xy}\) is the covariance of the original image and recovered image. \(c_{1}\) and \(c_{2}\) are two small positive numbers, which are used to avoid being divided by 0. These two small positive numbers can be calculated as following [49]:  
\[c_{i}=\left(K_{i}L\right)^{2}, \tag{18}\]  
where \(I=1\), \(2\), \(K_{i}<<1\) is a small constant, and \(L\) is the dynamic range of the pixel values, which is 1 for normalized image. According to the analysis in reference [49], \(K_{i}\) can be selected arbitrarily, and the performance of the SSIM index algorithm is fairly insensitive to variations of these values. Therefore, following the reference [49], \(K_{1}\) and \(K_{2}\) are also set as 0.01 and 0.03 in the SSIM index calculation, respectively. The two small positive numbers \(c_{1}\) and \(c_{2}\) are \(10^{-4}\) and \(9\times 10^{-4}\), respectively. The calculated results are shown in Figure 5. Figure 5a-d are the PSNR of reconstructed Chinese character, PSNRs of reconstructed panda, SSIMs of reconstructed Chinese character, and SSIMs of reconstructed panda, respectively, with every method under different sampling rate conditions. The horizontal coordinate of each sub-image in Figure 5 represents the number of measurements, which range from 1000 to 10,000. The vertical coordinate represents the value of quantitative evaluation index. In Figure 5, different colors of curves represent different reconstruction algorithms. As Figure 5 shows, the number of measurements affects the reconstruction quality of all the algorithms. The PSNR and SSIM of all the methods improve with the increase in measurement numbers. It can be concluded from Figure 5a,b that the PSNRs of the reconstructed results by different methods have little difference under the same number of measurements. The PSNR of the proposed method does not exceed those of other methods, and sometimes it is even lower than those of other methods. The reason for this is that the noise in the background is relatively high, although the reconstructed result of the proposed method has a better resolution. The higher the noise is in the background, the bigger the error between the reconstructed results and corresponding pixel points of the original image is. Therefore, the PSNR of the proposed method has no significant difference with other methods, whereas there is a deviation between the quantificationally calculated PSNRs and visual effect. Contrary to the PSNR, the SSIM is more highly consistent with visual effects. From the perspective of image composition, the SSIM defines structural information as an attribute that reflects the structure of objects in the scene. Additionally, the attribute is independent of brightness and contrast. Distortion is modeled as the combination of three different factors: brightness, contrast, and structure. The three factors are estimated by the mean value, standard deviation, and covariance, respectively. The SSIM focuses more on the evaluation of image similarity, and the evaluation accuracy of image similarity is better than PSNR [49].