# Compressive Ghost Imaging of the Moving Object Using the Low-Order Moments
## 2 Methods & Setup
Since the acquisition time of GI is always limited in practice, in order to track and image moving targets in a much time-saving manner, there are two main bottle-necks to be solved at present: one is to reduce the essential measurement number for locating the object in each image period, and the other is to speed up the extraction of the motion information. For reducing the required measurement number for a faithful reconstruction, CS has been  
Figure 1: Experimental setup. P: Polarizer; R. G. G.: Rotating ground glass; A: aperture; BS: beam splitter; L1, L2, L3: Lens. demonstrated as a effective method in GI to obtain a better image quality with fewer number of measurements. From the perspective of CS, the measurement process of GI is written as  
\[\mathbf{y}=\mathbf{AT}+n \tag{3}\]  
where \(\mathbf{T}\) is the vectorized representation of \(T(x,y)\). When \(m\ll N\) in the measurement matrix \(\mathbf{A}\), a given \(\mathbf{B}\) cannot specify a unique \(\mathbf{T}\). However, when the image of object is assumed to be sparse, for \(m\) repeated measurements, the image reconstruction of CS is solving a convex optimization problem [19],  
\[\min_{T}\frac{1}{2}\|\mathbf{y}-\mathbf{AT}\|_{2}^{2}+\tau\|\mathbf{T}\|_{1} \tag{4}\]  
where the \(\|\cdot\|_{p}\) represents the \(l_{p}\) norm, and \(\tau\) is a constant scalar controlling the sparsity constraint. Although many efficient \(l_{1}\)-minimization approaches have been proposed to obtain the sparse solutions of Equation (4), including the convex relaxation or the greedy pursuit approaches, most of these algorithms require nested iterations for a faithful reconstruction, which would be less competitive on the motion tracking and imaging tasks. Besides, the object motion estimation also requires the image reconstruction right after each \(m\) measurements, and the time for the CS reconstruction is also limited. To reduce the time and computation cost, we introduce the gradient projection for sparse reconstruction algorithm (GPSR) to reconstruct the image from small number of measurements. GPSR requires only one level of iteration, and the computation of each iteration costs order \(O(kn)\) flops [21], which is more suitable for this task with limited time. When solving the \(l_{1}\)-minimization problem, the GPSR algorithm starts with an initial guess of the solution \(T\), and then computes the gradient of the Equation (4) by using the backtracking line search and updates in the negative gradient direction to find the solution. The updates in each iteration of GPSR are taken along the path of steepest descent, and the algorithm runs until the convergence criteria is met. To accurately estimate the object's location in each image period, previous methods usually required some pre-processing operations, such as image binarization [7], matching the unknown speed of the object [8; 9], or calculating the coefficient matrix of the images [13], etc., which is still time-consuming for the motion estimation tasks. Here, we consider the low-order moments of the image to directly extract the motion information of the moving object, including both the translation and the rotation. The zero-order moment represents the overall intensity of the image \(T(x,y)\), written as  
\[\mu_{00}=\iint dxdy\;T(x,y), \tag{5}\]  
The first-order moments  
\[ \begin{array}{l}\mu_{10}=\iint dxdy\;xT(x,y),\\ \mu_{01}=\iint dxdy\;yT(x,y),\end{array}  \tag{6}\]  
locate the centroid position \((\tilde{x},\tilde{y})\) of \(T(x,y)\), i.e., \(\tilde{x}=\mu_{10}/\mu_{00}\), \(\tilde{y}=\mu_{01}/\mu_{00}\). And the second-order moments  
\[\begin{array}{l}\mu_{20}=\iint dxdy\;T(x,y)x^{2},\\ \mu_{11}=\iint dxdy\;T(x,y)xy,\\ \mu_{02}=\iint dxdy  
Here, we take a brief analysis of the computation cost of locating the object within an image period, as an example to evaluate the time consumption of different methods. For comparison, we analyze the computation cost of a single object-locating procedure using the well-behaved scheme of CBGI.