# Reducing Motion Blur in Ghost Imaging Via the Hessian Matrix
## 2 Materials and Methods  
The experimental setup of GI with the Hessian matrix experiment is shown in Figure 1. A 532 nm laser passing through a RGG (Edmund 100 mm diameter 220 gri) made a pseudo-thermal light source. The pseudo-thermal light was split into two arms by a BS. The signal arm penetrated a transitive object (0.3 mm square \"GI\" pattern) and was summed to be registered by a bucket detector (Thorlabs PDA100/A) while the reference arm, with no object on the path, was recorded by a CMOS camera (xiQ MQ003 MG-CM). In the experiment, we set up the experiment system according to Figure 1 and set the exposure time of the CMOS as 15 ms. Five sets of data were collected including data from the reference arm and the signal arm under different rotation speeds of the RGG, which varied from \(\frac{5}{36}\) rad/s to \(\frac{25}{36}\) rad/s. The data from the reference arm occupied a field of view of 300 \(\times\) 300 pixels and the number of samplings in each set of data was 10,000. The reconstructed image of the object in ghost imaging can be written as [15]:  
\[O(x,y)=\ \frac{1}{N}\sum_{i=1}^{N}(B_{i}-\langle B\rangle)(I_{i}(x,y)) \tag{1}\]  
where \(B_{i}\) was the response of the bucket detector in the \(i\)th sampling, \(\langle B\rangle\) was the ensemble average of \(B_{i}\), \(I_{i}(x,y)\) was the intensity distribution of the light field recorded by the CMOS camera and \(N\) was the number of samplings. Usually, \(B_{i}\) and \(I_{i}(x,y)\) could be written as the integral of the detectors' response during the exposure time \(\tau_{e}\). This assumes that \(\tau_{e}\) is set to be an appropriate value and the reconstructed image has no motion blur. The speed of the RGG was \(v_{0}\) and the speed of the RGG was then set to \(v_{0}+\Delta v\) for moving object imaging. The speckle field was generated by directing the laser through the RGG and the speckle in the current moment was produced by displacement of it in the previous moment; thus, the faster speed led to a longer distance, which is also called motion blur. Figure 1: Experiment system: RGG: rotating ground glass; BS: 50:50 beam splitters; Object: “GI” (ghost imaging) pattern. The coherence time could be calculated according Equation (2):  
\[\tau=\leavevmode\
obreak\ l/(\omega\times r) \tag{2}\]  
where \(\tau\) was the coherence time, \(l\) was the diameter of the laser spot, \(\omega\) was the angular velocity (RGG) and \(r\) was the radius at the laser spot; thus, the coherence time of five sets of data were between 28.64 ms and 5.72 ms (\(l\) = 0.5 mm, \(r\) = 0.04 m, \(\omega\) varied from \(\frac{5}{36}\leavevmode\
obreak\ \mathrm{rad}/\mathrm{s}\) to \(\frac{25}{36}\leavevmode\
obreak\ \mathrm{rad}/\mathrm{s}\)). In order to mimic the case of motion blur caused by fast speed of the RGG, the camera's exposure time was set to be 15 ms and the rotation speed of the RGG was set to different values. This made the coherence time vary from less than the exposure time to over the exposure time, which led to a motion blur in both the reference samples and the reconstructed images. We then introduced the Hessian matrix and applied it in the reference samples as a filtering process to get rid of the GI image degradation. As this is different from the conventional image calculation procedure of GI, before we did the correlation, we first calculated the Hessian matrix of the data in the reference arm. The Hessian matrix can be described as the second-order partial derivative of a matrix [20] and it represents the gradient of the intensity the image. According to the concepts of linear scale-space theory, taking the second-order partial derivative of the image can be written as the convolution of the image and the derivatives of Gaussian functions [21]; thus, Equation (1) can be written as:  
\[O^{\prime}(x,y)=\leavevmode\
obreak\ \frac{1}{N}\sum_{i=1}^{N}(B_{i}-\langle B \rangle)\bigg{(}I_{i}(x,y)\otimes\frac{\partial^{2}G(x,y;\sigma)}{\partial x \partial y}\bigg{)} \tag{3}\]  
where \(\otimes\) is the convolution operator.