# Recognizing and Recovering Ball Motion Based on Low-Frame-Rate Monocular Camera
## 4 Experimental Results
### Motion Prediction Results  
This experiment was done to evaluate the accuracy of our method for predicting the velocity and direction of sphere motion. We compared the errors in direction and velocity between our approach and the methods in [11; 12].  
The predicted direction \(\overrightarrow{PD}\) of a trajectory was the line through its end points in the camera coordinate system. The direction prediction error \(\varphi_{direction}\) was the vectorial angle between \(\overrightarrow{PD}\) and the true direction\(\overrightarrow{TD}\) that was marked manually on vertical planes. Thus, \(\varphi_{direction}\) could be calculated as:  
\[\varphi_{direction}=\arccos\left(\frac{\overrightarrow{PD}\cdot\overrightarrow {TD}}{\left|\overrightarrow{PD}\right|\times\left|\overrightarrow{TD} \right|}\right) \tag{25}\]  
We defined the error in velocity as the ratio of the speed difference to the true velocity (TV). The true velocity equaled the distance between the two ends of the trajectory divided by the duration of the exposure. Different methods have contributed various approaches to calculating speed. The velocity prediction of the method in [11] was calculated based on the  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{**Distance (R)**} & \multicolumn{3}{c}{**Average Error (R)**} \\ \cline{2-4}  & **Method in [11]** & **Method in [12]** & **The Proposed Method** \\ \hline
20–110 & 33.2 & 21.5 & **20.7** \\
110–130 & 37.4 & **22.5** & 23.6 \\
130–150 & 79.6 & 40.7 & **25.2** \\
150–200 & 42.7 & 30.6 & **27.0** \\ \hline \hline \end{tabular}
\end{table}
Table 4: The average errors (multiples of the ball radius R) of the trajectory reconstructions.  
Figure 15: The result of our sphere position mapping method: (**a**) original coupled points; (**b**) coupled points projected into their spline curves; (**c**) a reconstructed trajectory using our sphere position mapping method.  
distance traveled and the duration of exposure. In contrast, our method and the method in [12] output predicted velocity (PV) that was the median of the velocity prediction sequence. Then, the velocity prediction error could be calculated as:  
\[\varphi_{velocity}=\frac{|TV-PV|}{TV} \tag{26}\]  
We used 45 images that contained streaks to test these methods. We classified trajectory orientations according to their angle size to the camera's imaging plane. Table 5 shows three methods' direction prediction errors. It can be observed that under different trajectory orientations, the prediction error of our method was the smallest compared with the other two methods. As the trajectory deviated from the camera's imaging plane, the error of our method and method in [12] in predicting the trajectory direction increased. This was because sphere imaging was not sensitive to changes in sphere depth information relative to lateral or longitudinal motion. Therefore, the sphere should move parallel to the imaging plane as much as possible to ensure accurate direction prediction.  
Table 6 shows the velocity prediction errors for the three methods. It can be observed that our method was better than the other methods in different speed intervals. When the speed was below 12 m/s, all three methods had significant errors in their predictions. The streak length increased with the ball speed, and longer streaks would provide more coupled pair points for analyzing the direction and velocity. Thus, the performances of our method and the method in [24] became progressively better as the speed increased.