# Recognizing and Recovering Ball Motion Based on Low-Frame-Rate Monocular Camera
## 1 Introduction  
Sphere motion reconstruction techniques based on images are widely used in indoor screen-based ball sports such as indoor golf, baseball, soccer, as well as other interdisciplinary application areas including geological engineering [1] and cell biology [2]. Sphere motion reconstruction consists of two basic steps: (1) projection extraction and (2) trajectory reconstruction. The result of projection extraction is a set of pixels, and the accuracy of these pixels significantly impacts space position mapping results in trajectory reconstruction. Therefore, sphere motion reconstruction usually requires expensive high-precision equipment and complex on-site deployment to capture precise sphere projections. The high-cost equipment thus limits the application of sphere motion reconstruction. Researchers have proposed sphere motion reconstruction methods based on low frame rate cameras to reduce hardware costs. When using a still camera to capture a moving sphere, the sphere leaves a visible but blurred streak on the image plane due to its high speed relative to the exposure time. These streaks are referred to as motion blurs. Researchers have proposed a series of methods to extract these blurs and deduce motion trajectories based on them. These methods usually extract streaks based on the shapes of streaks when the projections are circle-like, i.e., the difference between the edge pixels of foreground and background in an image is obvious. In this scenario, a standard method is fitting streaks using circles and ellipses. In [3; 4; 5; 6], by calculating the gradient of images, [PERSON] et al. obtained circle edges directly, thus significantly reducing the difficulty of subsequent analysis of sphere position. In [7], [PERSON] et al. used half circle fitting that considers light direction to extract center points of flying golf balls on images. In [8], ellipses are used to fit a moving sphere and its shadow to triangulate the ball position. In [9], [PERSON] et al. used the YOLOv3-Tiny deep learning algorithm to capture and detect the pitched baseball. These methods prove that using circle and ellipse fitting to find sphere projection is effective. However, when the ball speed is too fast relative to exposure time, its streak edge will blend into the background, forming a highly blurred streak. In this case, the foreground and background edge pixels will be challenging to detect for circle fitting. Therefore, these methods are not effective when using low frame rate cameras. Researchers have devised a series of methods to deal with highly blurred streaks. One approach is to remove blur artifacts. For example, in [10], [PERSON] et al. used the Wiener filter to obtain a sequence of deblurred images to pick the most circle-like objects. However, this method is sensitive to streak length. Another approach is to exploit the blur. In [11; 12], [PERSON] et al. and [PERSON] et al. proposed models based on pixel intensities to obtain the contours of projections. However, the model in [11] also had the limitation of being sensitive to streak length. Meanwhile, the model in [12] lacked the optimization of output points and thus magnified the error in semi-ellipse fitting. In [13], [PERSON] et al. used the Fourier transform to determine the motion state of a fast moving puck. However, due to the inability to detect depth information, this method is only applicable to two-dimensional tabletop ball games. After processing the blurred images and obtaining the streaks, researchers need to infer the ball motion state. Some combined a sequence of frames or used multiple synchronized cameras to analyze ball movements. For example, in [14], the ball motion trajectories were determined based on position variation in frame sequences. However, this method required that different frames had captured the motion processes and that at least a few were valid. In [7], several multiple-exposure pictures from multiple synchronized cameras were used to determine sphere position.