# Compressive Ghost Imaging of the Moving Object Using the Low-Order Moments
## 1 Introduction  
The emergence of ghost imaging (GI) has provided a novel imaging modality to non-locally extract the information of the object [1]. Being different from the conventional single-snapshot imaging, the image of GI is obtained by measuring the second-order correlation function between the signal from a \"point-like\" bucket detector and the reference light fields [2] or a modulated pattern [3]. The idea of GI has started to play important roles in practical applications, ranging from microscopy [4], tomography [5] to three-dimensional LiDAR [6]. However, GI still requires a large number of repeated measurements to reconstruct a desirable image, which restricts the applications to stationary objects. Besides, when there exists a relative motion or trembling [7] between the GI system and the observed object, the consecutive sampling procedure usually induce the motion blur and cause the visual quality decline in the reconstructed image. Recently, many outperforming methods has empowered GI to image the moving objects. For example, when the object is moving with an unknown speed, tangentially [8], axially [9] or rotationally [10], different kinds of motion blur and the corresponding motion-compensation methods are investigated for the satisfactory GI reconstructions [11; 12]. For a more general situation, with no prior knowledge of the motion, an outstanding idea has been proposed to divide the overall repeated measurements of GI into small image periods, and capture the moving trajectory from the under-sampled images, then gradually obtain the unblurred image of the moving object [13]. It is also demonstrated to be capable of handling the image corruption caused by the random trembling of tracking platform [7]. Overall, the idea relies on the fact that, when the sampling rate of GI is faster than the motion updating rate in the image, i.e., the object's position can be regarded as relativelystatic in a small period of measurements, while changing among the other periods. However, different from the image registration applications in conventional digital image processing [14], the extremely under-sampled reconstructions of GI in a small image period is severely corrupted by the strong statistical fluctuations, which makes it difficult to extract reliable features for registration. To emerge the features for subsequently estimating the motion information of the moving object, it is reported to be hundreds of necessary measurements in each image period [7; 13], which may put forward greater demands either on the sampling rate of GI system or the moving speed of the object in practice. Moreover, the existing methods handling the motion blur in GI usually require some pre-processing operations to extract some \"features\", and then search among those \"features\" to locate the object. Considering the above two problems, the whole procedure is still time-consuming, and also suggest a room for further development of practical real-time tracking and imaging methods of GI. The concept of moment was introduced to the field of image analysis to describe the statistical properties of the image. The low-order moments, also known as, for example, the centroid position or the moment of inertia, have been demonstrated to be the fast and trustworthy solutions to localize the object or estimate the rotation angle from the image sequence [15; 16]. Nevertheless, one of the major obstacles hindering the application of the low-order moments in GI is the background noise in the GI reconstructions, which is inevitable in GI due to the nature of statistical correlation [17] and would reduce the estimation accuracy of object's position and rotation, especially under the small sampling conditions [18]. However, it is also worth noticing that, the compressive sensing (CS) techniques are widely-adopted in GI, which provides an effective way to reconstruct a desirable image with far less measurements than is possible with the conventional GI [19; 20].