# Reducing Motion Blur in Ghost Imaging Via the Hessian Matrix
## 1 Introduction  
As an indirect modality of imaging, ghost imaging (GI) is a new method of imaging that needs multiple measurements for one imaging process. In the experiment system, the pseudo-thermal light produced by a laser passing through a rotating ground glass (RGG) is split into two arms by a beam splitter (BS); the arm with the illuminated object is called the signal arm and the data without spatial resolution from this arm is recorded by the bucket detector, which means that the data from the detector is the sum of all of the light intensities within the captured range of the detector. The other arm is called the reference arm in which a series of data is recorded by the Complementary Metal Oxide Semiconductor (CMOS) camera [1, 2]. After the collection of the data, the image can be reconstructed by some algorithms with data from both arms. When the RGG rotates through the size of the laser spot, the distribution of the laser speckle is gradually changed. Until it rotates out of the entire range of the laser spot, which corresponds to the \"coherence time\" of the pseudo-thermal light [3], the distribution of the laser speckle could be totally different from the original one. In order to observe the second-order intensity correlation, the coherence time should be close to or shorter than the integration time (exposure time) of the detector in the reference arm, which requires a shorter exposure time or a lower rotation speed of the RGG [4].  
Compared with conventional imaging, GI hosts a number of advantages such as high resolution and high sensitivity and could be widely applied in many fields ranging from remote imaging to lidar detection and microscopy [5, 6]. The first visualization of GI was realized using entangled photon pairs in 1995 [3] and then it was proved that the experiment could also be realized with pseudo-thermal light sources and even true-thermal light [7, 8, 9, 10]. Recently, modulated light and other types of light sources were applied in an experiment [11, 12, 13, 14]. Due to its convenience in application, the pseudo-thermal source is still commonly used in GI experiments.  
In recent years, with the development of ghost imaging, the object is no longer limited to static targets [15; 16; 17] and different kinds of algorithms and methods have been investigated to deal with the problems in the imaging process with moving objects [18; 19]. One of these problems is motion blur, which causes the visual quality decline in the reconstructed image. Due to that, ghost imaging requires multiple samplings in one imaging. The position of the object and speckle should remain relatively static in each acquisition process when it is a moving object, which means a short exposure time and coherence time are needed, leading to the requirement of a fast speed of the RGG and a fast speed of the RGG leads to motion blur.  
To overcome this defect, in this paper we introduce the Hessian matrix to compensate for the influence of motion blur caused by the fast speed of the RGG. Based on a table top experimental setup, we measured the image quality of a reconstructed image under different numbers of coincident measurements ranging from under-sampling to over-sampling. The experimental results demonstrated that, with the post-processing method using the Hessian matrix, GI can effectively reconstruct an image with a higher image quality and less influence of motion blur.