# Reducing Motion Blur in Ghost Imaging Via the Hessian Matrix
## 3 Results  
A brief, related work was published earlier [22]. The results proved that the method worked well when dealing with motion blur. Here, we performed further research. We verified the effect of the method under different rotating speeds of the RGG, compared the method with another deblur algorithm and studied the influence of parameters in our method. Thus, we may provide some reference for others to use this method. Five groups of data were collected in the experiments. The rotation speed of the RGG was set to \(\frac{5}{36}\) rad/s, \(\frac{10}{36}\) rad/s, \(\frac{15}{36}\) rad/s, \(\frac{20}{36}\) rad/s and \(\frac{25}{36}\) rad/s and each group was repeated several times in different rotation speeds of the RGG. We used second-order correlation (SC) to reconstruct the images and compared the results with the images reconstructed by SC with the Hessian matrix and the image filtered by the Lucy-Richardson filter [23]. The PSF Lucy-Richardson filter could be calculated during reconstructing the images (as shown in Figure 2a) and the number of iterations in the Lucy-Richardson was set to be 1000. The comparison was implemented on the platform of a MATLAB 2019b with an Intel Core i7-5500U CPU and 12 GB RAM. The computation time of the SC fluctuated between 96 s and 112 s, while the time of the SC with the Hessian matrix increased to about 153 s to 177 s and the time of the SC filtered by the Lucy-Richardson fluctuated between 122 s to 164 s according to the iteration time. Figure 4 shows the reconstructed image of the Hessian enhanced GI experiment at different rotation speeds of the RGG under 10,000 samplings. As we can see from the figure, compared with the Lucy-Richardson filter, the boundary of the object in the image reconstructed by the SC with the Hessian matrix (II) was much more continuous and clearer, while the noise of the reconstructed images filtered by the Lucy-Richardson filter was much lower. In the condition proposed in the experiment, motion blur was not caused by the moving blur but the high rotational speed of the RGG, making the SC with the Hessian matrix better than the Lucy-Richardson algorithm if we needed a clearer object. When the rotational speed was higher than \(\frac{25}{36}\) rad/s, it was almost impossible to reconstruct the image. When the rotational speed was lower than \(\frac{5}{36}\) rad/s, motion blur was no longer an important factor in reducing the image quality. Figure 3: Change of PSF after being processed by the Hessian matrix. Figure 2: (**a**) Point spread function (PSF) with motion blur; (**b**) projection in the x-z plane; (**c**) projection in the y-z plane. Taking the third line of data shown in Figure 4 as an example where the rotation speed of the RGG was \(\frac{15}{36}\) rad/s, we introduced the signal-to-noise ratio (SNR) to evaluate the performance of the Hessian filter. The SNR can be written as  
\[\text{SNR}=10\log_{10}\left(\frac{\sum_{x}\sum_{y}T(x,y)^{2}}{\sum_{x}\sum_{y}(O (x,y)-T(x,y))^{2}}\right) \tag{4}\]  
where \(T(x,y)\) represented the value of the object expected as shown in Figure 5 and \(O(x,y)\) was the reconstructed image. As is shown in Figure 6, the SNR increased as the measurement number increased. When the measurement number was less than 1000, the information in the data collected was not enough to fully reconstruct the image. In contrast, when the measurement number was over 4000, motion blur became an important factor that reduced the image quality. As shown in Figure 4, the Lucy-Richardson filter reduced the noise in the reconstructed  
Figure 4: Reconstructed images of different methods at different rotations of the RGG under 10,000 samplings: I. Second-order correlation (SC), II. SC with the Hessian matrix, III. SC filtered by the Lucy–Richardson. (**a**) \(\frac{5}{36}\) rad/s, (**b**) \(\frac{10}{36}\) rad/s, (**c**) \(\frac{15}{36}\) rad/s, (**d**) \(\frac{20}{36}\) rad/s, (**e**) \(\frac{25}{36}\) rad/s. Figure 5: The expected object: ‘GI’ pattern.