# Recognizing and Recovering Ball Motion Based on Low-Frame-Rate Monocular Camera
## 3 Method
### Reconstructing Trajectories
#### 3.4.1 Determining Ball Positions  
Given the sphere radius and the imaging circle, the sphere's position can be calculated using camera imaging principles. Unfortunately, the coupled points differ from the points at the ends of the diameter on the imaging circle. Thus, mapping the point pairs using the method in [4] will produce a significant error. Our idea is to use two points on the circumference of the circle to map the spatial position of the sphere, and these two points do not need to fall at the ends of a diameter. Fortunately, we observed that the sphere position could be determined with its image and the projecting plane that tangents to the surface of the sphere. Thus, exploiting this observation, we propose a novel sphere position mapping method which consists of three steps: (1) estimating the projecting line of the sphere's center, (2) estimating the projecting plane that tangents to the surface of the sphere, and (3) estimating the sphere's center with our projection model. We first focus on estimating the projecting line of the sphere's center. According to camera imaging principles, the projecting line could be derived from the projection of the sphere's center. Through the single-view perspective projection model in [19], we find that the sphere center projection is roughly located at the intersection of the vertical line. Meanwhile, it is also close to the angular bisector of the angle formed by the tangents of the two points in a coupled pair. Taking Figure 10a for an example, \(p_{1}\) and \(p_{2}\) are coupled points on the lateral contours. Their tangents intersect at point \(c\); \(l_{1}\) is the angular bisector of the angle formed by \(\overrightarrow{Cp_{1}}\) and \(\overrightarrow{Cp_{2}}\). The intersection \(p_{3}\) of two vertical lines at \(p_{1}\), \(p_{2}\) is the sphere center projection and falls on \(l_{1}\). The ideal intersection is \(p_{3}\) as the projection should be close to \(l_{1}\). However, due to calculation deviation, \(p_{3}\) may deviate significantly from \(l_{1}\), resulting in a large error in the sphere mapping result. To eliminate the deviations, the projection of point \(p_{3}\) onto the line is regarded as the projection of the sphere's center. Taking Figure 10b as an example, \(p_{3}\) deviates significantly from \(l_{1}\), and thus its projection \(c\) is considered as the projection of the sphere's center. After correcting the image for distortion, a mapping can be established between the pixel points and their spatial coordinates. Given a pixel point \((u,v)\), we can map its projecting line \((X,Y,Z)\) in 3D space and vice versa. The conversion relationship is as follows:  
Figure 10: Estimation of the projection position of the center of the ball that the coupled point pair corresponds: (**a**) the intersection falls on the angular bisector in ideal situation; (**b**) the intersection deviates significantly from the angular bisector. \[A\left[ \begin{array}{c}u\\ v\\ 1\end{array} \right]=\left[ \begin{array}{ccc}f/dx&0&u_{0}&0\\ 0&f/dy&v_{0}&0\\ 0&0&1&0\end{array} \right]\left[ \begin{array}{c}X\\ Y\\ Z\\ 1\end{array} \right]=\left[ \begin{array}{c}K\mid 0\end{array} \right]\left[ \begin{array}{c}X \\ Y\\ Z\\ 1\end{array} \right] \tag{16}\]  
where \(K\) is the camera intrinsic matrix. Using Equation (16), the projecting line of the sphere's center is determined. We then estimate the projecting plane that tangents to the sphere's surface. According to the camera imaging principle, the projecting plane could be derived from a line that tangents to the sphere's image. Using Equation (16), a straight line \(y=\theta_{0}+\theta_{1}x\) in the image plane is capable of mapping to a plane in space. The conversion relationship is as follows:  
\[\theta_{1}\frac{f}{dx}X-\frac{f}{dy}Y+Z(\theta_{0}-v_{0}+\theta_{1}u_{0})=0 \tag{17}\]  
Therefore, the two projecting planes that tangent to the surface of the sphere can be determined with \(\overrightarrow{Cp_{1}}\) and \(\overrightarrow{Cp_{2}}\).