# Recognizing and Recovering Ball Motion Based on Low-Frame-Rate Monocular Camera
## 4 Experimental Results
### Trajectory Reconstruction Results  
Comparisons were made using our trajectory reconstruction method with the original method in [11,12]. Since the streak axis is able to reflect the matching effect of the point pairs, we first compared these streak axes output by CPM and ICPPM (both are introduced in Section 3.3) before comparing the reconstruction error. The precise streak axes should be smooth, so they could be well fitted with four-order B splines. Therefore, to compare the streak axes, we compared their goodness of fit (\(R^{2}\)). For each streak axis point \((x,y)\), \(x\) was the input value, \(y\) was the true value, and \(\widehat{y}\) was the predicted value in the spline fitting process. Then, the total sum of squares (TSS), the residual sum of squares (RSS), and \(R^{2}\) were calculated as:  
\[\text{TSS}=\sum_{i=1}^{n}(y_{i}-\overline{y})^{2} \tag{22}\]  
\[\text{RSS}=\sum_{i=1}^{n}(y_{i}-\widehat{y})^{2} \tag{23}\]  
\[R^{2}=1-\frac{\text{RSS}}{\text{TSS}} \tag{24}\]  
The real position value of a trajectory was difficult to determine. We found that given the trajectory's vertical plane, the trajectory's position could be determined based on camera imaging principles. For this reason, we used ArUco [23] to mark vertical planes beforehand and then threw the sphere along the marked vertical planes. By this means, we obtained the real position of each trajectory. An easy way to represent the position of a trajectory was to use the positions of the two ends of the trajectory. Therefore, we defined the trajectory reconstruction error as the average distance from the ends of the trajectory to the vertical plane. In the experiment, we divided the streak axes of the detected streaks into four length intervals and calculated average \(R^{2}\) for each interval as reported in Table 3. It can be observed that our ICPPM outperforms CPM in different streak length intervals, especially for short streaks. This means that the streak axis optimization method in Section 3.3 is effective. Figure 14 lists three comparisons. The streaks are marked in three cropped images. CPM and ICPPM use these streaks as data sources and output their axes. The first set of comparisons used a short streak in Figure 14a. By comparing Figure 14a and Figure 14b, it can be confirmed that our ICPPM method performs much better when the streak is short.The second set of comparisons have used a medium-length streak in Figure 14d. By comparing Figure 14e and Figure 14f, we can observe that our method has better results at the end points of the streak. This is due to the improvements we made to the axis fitting at the ends of the streak in Section 3.3. The third set of comparisons have used a long streak in Figure 14g. By comparing Figure 14h and Figure 14g, We can observe that both methods have good results. However, our ICPPM has output more axis points, which help to improve the accuracy of the trajectory reconstruction. Then, we focused on the reconstruction error. We divide the 78 trajectories into 4 sets based on their distance (measured as multiples of the ball radius) from the camera. For each set, we compared the average error of our trajectory reconstruction method and the methods used in [11,12]. Figure 14: Comparison of the streak axes of CPM and ICPPM: (**a**) a cropped image that contains a short streak; (**b**) the streak axis (disorder green points in image) of CPM; (**c**) the streak axis of our ICPPM; (**d**) a cropped image that contains a medium-length streak; (**e**) the streak axis of CPM; (**f**) the streak axis of our ICPPM; (**g**) a cropped image that contains a long streak; (**h**) the streak axis of CPM; (**i**) the streak axis of our ICPPM. \begin{table}
\begin{tabular}{c c c} \hline \hline \multirow{2}{*}{**Streak Length Interval (pixel)**} & \multicolumn{2}{c}{\(R^{2}\)} \\ \cline{2-3}  & **CPM** & **ICPPM** \\ \hline
60–120 & 0.417 & **0.979** \\
120–160 & 0.632 & **0.917** \\
160–200 & 0.731 & **0.940** \\
200+ & 0.829 & **0.990** \\ \hline \hline \end{tabular}
\end{table}