# Compressive Ghost Imaging of the Moving Object Using the Low-Order Moments
## 2 Methods & Setup  
An experimental setup of GI based on pseudo-thermal light source is implemented in Figure 1. The pseudo-thermal light source is formed by a polarized laser of 532 nm (CNI Laser MGL-DS532) passing through a rotating ground glass to simulate the spatial incoherence of the thermal light. (Since the coherence time of the pseudo-thermal light can be easily controlled, this kind of light source is widely-adopted in GI.) Although the spatial light modulators can also be applied to generate the pseudo-thermal light, however, considering the need for GI of moving objects is commonly encountered in remote sensing or LiDAR imaging applications, and the laser intensity required in these applications often exceeds the damage threshold of the modulators. Thus in our implementation, we choose a more flexible and accessible device, i.e., a rotating ground glass diffuser, to generate the pseudo-thermal light. The pseudo-thermal light source is collimated by L1 (f = 50.8 mm) and split into two light paths: on the reference path, the random speckle patterns are imaged by L2 (f = 50.8 mm) with \(z_{3}=z_{4}=100\) mm and then be recorded by a CMOS camera (XiMea MQ003 MG), occupying a field of view (FOV) of \(N=160\times 160\) pixels as \(I_{r}(x,y;t)\); on the test path, the speckle patternsare projected by L3 (f = 50.8 mm) with \(z_{1}=53\) mm, \(z_{2}=1200\) mm onto the object plane, and the intensity reflected by the object is collected by a \"point-like\" bucket detector (Thorlabs PDA100A) at \(z_{5}~{}=~{}500\) mm, registered as \(B(t)\). For the convenience of CS processing, during the procedure of \(M\)th repeated measurements, each speckle pattern \(I_{r}(x,y;t)\) (t\(\in[1,M]\)) is reshaped to a \(N\)-length row vector, forming the measuring matrix \(\mathbf{A}\) of \(M\times N\) elements, and the light intensity obtained by the bucket detector is registered to be a \(M\)-length column vector \(\mathbf{B}\). For a conventional stable GI system, the second-order correlation function between yields the image \(T(x,y)\) of object,  
\[T(x,y)=\frac{\langle I_{r}(x,y;t)\cdot B(t)\rangle_{t}}{\langle I_{r}(x,y;t) \rangle_{t}\langle B(t)\rangle_{t}}. \tag{1}\]  
where \(\langle\cdot\rangle\) denotes the ensemble average over all measurements. For mimicking a translational moving object, a reflective \"paperplane\" (\(4\times 4\) cm) is pinning on a stepping-motorized stage with a 2-D translation \((\Delta x,\Delta y)\). When the object is moving, within a small number of measurements, denoted by \(m\) (\(m\ll M\)), the object location can be regarded as relatively static. Thus the whole imaging process can be divided into \(K\) image periods (\(K=M/m\)), corresponding to object located at different positions. The goal of motion compensation in GI is to accurately estimate the \((\Delta x_{k},\Delta y_{k})\) within the different image period \(k\) (k \(\in\) [1,K]). Likewise, for mimicking a rotational moving object, a reflective \"truck\" object (\(3\times 5\) cm) is pinning on a motorized rotation stage (Thorlabs HDR50/M). To overcome the rotational blur, we need to estimate the rotation angle \(\phi\) in different image periods. Therefore, the image \(T^{\prime}(x,y)\) of the moving object can be viewed as the superposition of the sequential blurry images \(T(x,y)\) compensated with a position displacement \((\Delta x_{k},\Delta y_{k})\) and a rotation operator \(R(x_{k},y_{k};\phi)\) determined by the rotation angle \(\phi\), which can be expressed as  
\[T^{\prime}(x,y)=\frac{1}{M}\sum_{k=1}^{K}T\left(x-\Delta x_{k},y-\Delta y_{k} \right)R(x_{k},y_{k};\theta), \tag{2}\]  
Considering this, we noticed that the procedure of GI to image the moving object can be divided into two steps: one is to reconstruct a reliable image within an under-sampled image period to ensure the accuracy of motion estimation, and the other is to extract the motion information from the under-sampled images.