# Reducing Motion Blur in Ghost Imaging Via the Hessian Matrix
## 3 Results
images effectively, making the SNR of images reconstructed by SC with the Hessian matrix and those filtered by Lucy-Richardson filter nearly equal. In addition, it is worth noting that there were two important parameters: the standard deviation \(\sigma\) and the convolution window \(w\) when calculating the Gaussian second-order partial derivative, which made a big difference to the reconstructed images. The equation between \(\sigma\) and \(w\) can be written as [24]  
\[w=\left(\frac{\sigma-0.8}{0.3}+1\right)\times 2+1 \tag{5}\]  
where \(w\) determined the range of the pixels where the gradient of intensity was to be calculated; thus, it influenced the performance of the Hessian matrix. In the experiment, the standard deviation \(\sigma\) varied from 1 to 5, making the convolution window \(w\) vary from 4.3 (considering that the number of pixels is integer and \(w\) here was set to be 5) to 31. The reconstructed images of the SC with the Hessian matrix are shown in Figures 7 and 8. As is shown in Figures 7 and 8, the different value of \(\sigma\) led to different reconstructed images. To provide a reference for others to use this method, we studied the relationship between the optimal value of \(\sigma\) and the data collected. As the method changed the shape of the PSF from an oval to a circle, we speculated that the optimal value of \(\sigma\) related to the ratio of the major axis to the minor axis of the projection in the x-y plane. Figure 6: The signal-to-noise ratio (SNR) of an image reconstructed by the SC, SC filtered by Lucy-Richardson and SC with a Hessian filter. Figure 7: Reconstructed images of the SC with the Hessian matrix in different standard deviations. (**a**) \(\sigma=1\); (**b**) \(\sigma=2\); (**c**) \(\sigma=3\); (**d**) \(\sigma=4\); (**e**) \(\sigma=5\). As listed in the Table 1, the ratio of the five sets of data were 1.5, 1.8, 2.1, 2.4 and 2.9 and the optimal value of \(\sigma\), as shown in Figure 9, was 2, 2, 2, 2 and 1, respectively. According to the speculation, the optimal value of \(\sigma\) should have been 2, 2, 2, 2 and 3. Here we calculated the SNR of five sets of data under different rotational speeds of the RGG and the results are shown in Figure 9. With the exception of the last data group, the optimal of \(\sigma\) was about equal to the ratio; thus, our speculation should have been correct. Unfortunately, if the rotational speed of the RGG keeps growing, it is difficult to reconstruct the image no matter what algorithm is used, making it difficult to explore the relationship between the optimal of \(\sigma\) and the rotational speed of the RGG further. \begin{table}
\begin{tabular}{c c c} \hline \hline
**Rotational Speed of the RGG** & **Ratio** & **Optimal Value of \(\sigma\)** \\ \hline
5/36 rad/s & 1.5 & 2 \\
10/36 rad/s & 1.8 & 2 \\
15/36 rad/s & 2.1 & 2 \\
20/36 rad/s & 2.4 & 2 \\
25/36 rad/s & 2.9 & 1 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The ratio and the optimal value of \(\sigma\) under different rotational speeds of the RGG. Figure 8: The SNR of a reconstructed image of SC with the Hessian filter in different values of standard deviation. Figure 9: The SNR of the reconstructed image of the SC with the Hessian matrix in different values of standard deviation at different rotation speeds of the RGG.