# Recognizing and Recovering Ball Motion Based on Low-Frame-Rate Monocular Camera
## 3 Method
### Overview  
A canal surface is obtained as the envelope of a family of spheres of constant radius, whose center is swept along a space curve. Such a curve is called an axis [15]. Each streak corresponds to a unique ball trajectory. Therefore, once exposure time, background, and sphere color are known, the 3D sphere motion trajectory and speed can be uniquely determined. This paper reconstructs sphere motion by analyzing streaks based on the above characteristics.  
Our sphere motion reconstruction method consists of three steps: (1) acquiring contours from original images, (2) matching coupled point pairs, and (3) reconstructing trajectories based on the coupled point pairs. Figure 1 shows the overall steps of our proposed method, and Figure 2 is the flow diagram of our sphere motion reconstruction method.  
1. Acquiring contours: contours are the boundaries of an image's streak and background. Each contour is a collection of contour points. When a streak is clear, the Canny algorithm proposed by [PERSON] or the convolutional network-based tube contour detection method [16] proposed by [PERSON] et al., can directly derive an accurate sequence of contour points. However, the image of a high-speed moving sphere can be very blurry,  
Figure 1: The overall steps of our sphere motion reconstruction method.  
Figure 2: Workflow of our sphere motion reconstruction method.  
and the contour line output by Canny becomes cluttered and incomplete. Moreover, these contours contain a lot of noise caused by light and human motion. Figure 1 contains a black streak as the target streak. Figure 1b shows the contour lines output by Canny, and the streak extracted is incomplete. To address the streak fragmentation problem and the effect of noise, we propose exploiting streak shapes and internal pixel features to extract the complete contours.  
2. Matching coupled point pairs: After obtaining contours, a sequence of coupled point pairs can be matched. A canal surface is equivalent to the union of circumferences with a constant radius. Such a circumference is called a cross section, and each cross section is centered on the axis. Two contour points are said to be coupled if they are the image of two points on the same cross section [15]. As an example, in Figure 1b, a coupled point pair is formed by two red points on the lateral contours. This coupled point pair can uniquely determine a ball position in space. A contour point pair obtained using a traditional method (e.g., [15]) has a small fitting error since the profile is not completely perpendicular to the streak direction. The effect of this error will be magnified when reconstructing the corresponding cross section. In practical application scenarios, where high-speed moving spheres are imaged more blurred, the evaluated streak direction will have a more significant error. Therefore, the traditional contour point pairs detection method will fail. To address this problem, this paper proposes an iterative method to update the contour point pairs and thus obtain an accurate sequence of contour point pairs.  
3. Reconstructing trajectories: A trajectory is formed by a sequence of ball positions, directions, and velocities. The midpoint of a coupled point pair is not the image of the sphere's center; thus, the contour point pair cannot be directly mapped to its corresponding sphere in 3D space. Therefore, for each coupled point pair, the image of its corresponding sphere center should be estimated. In this paper, we design a model to estimate the images of sphere centers and map the coupled point pairs to their corresponding sphere locations in space. As shown in Figure 1d, the coupled point pair in red color has been mapped to a ball in space. After mapping all the pairs in the sequences, a sphere motion trajectory composed of the sphere position sequences will be obtained. The sphere motion velocity can be estimated with the trajectory based on pixel intensities.