## 4 Experiment Study
### Centroid-Based Moving Ball Tracking Subjected to Three Motions
\begin{array}{c}x_{k-1}+\dot{x}_{k-1} \Delta t+1/2\dot{x}_{k-1}\Delta t^{2}\\ y_{k-1}+\dot{y}_{k-1}\Delta t+1/2\dot{y}_{k-1}\Delta t^{2}\\ \dot{x}_{k-1}+\dot{x}_{k-1}\Delta t\\ \dot{y}_{k-1}+\dot{y}_{k-1}\Delta t\end{array} \right] \tag{20}\]  
which can be simplified as:  
\[\mathrm{x}_{k}=\left[ \begin{array}{cccc}1&0&\Delta t&0\\ 0&1&0&\Delta t\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\mathrm{x}_{k-1}+\left[ \begin{array}{cc}1/2\Delta t ^{2}&0\\ 0&1/2\Delta t^{2}\\ \Delta t&0\\ 0&\Delta t\end{array} \right]a_{k-1} \tag{21}\]  
where the transitional matrices \(F\), \(\Gamma\) and \(H\) are defined as  
\[\mathrm{F}=\left[ \begin{array}{cccc}1&0&\Delta t&0\\ 0&1&0&\Delta t\\ 0&0&1&0\\ 0&0&0&1\end{array} \right]\]  
\[\Gamma=\left[ \begin{array}{cccc}1/2\Delta t^{2}&0\\ 0&1/2\Delta t^{2}\\ \Delta t&0\\ 0&\Delta t\end{array} \right] \tag{22}\]  
\[\mathrm{H}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\end{array} \right]\text{. }\]  
The performance of the proposed AKF-ALS-based VOT algorithm is compared to that of the traditional KF-based VOT algorithm, as introduced in [64], across three types of motion: projectile motion, free-fall with a bouncing trajectory, and linear back-and-forth motion. The KF-based VOT algorithm uses the same kinematic equation in (19), but the noise covariance is initially determined as a constant value, i.e., \(Q=1\) and \(R=0.5\). The comparative results are illustrated in Figure 4.  
As indicated from the above figure, the proposed AKF-ALS algorithm shows improved robustness in handling constantly changing speed and direction, rapidly changing velocity, and the abrupt change in velocity when the object changes direction, while continuing tracking accurately after the change ends. As the tracking progresses, the performance of KF-based VOT approaches that of AKF-ALS-based VOT because the state covariance matrix approximates the steady-state value. This highlights the advantage of the proposed AKF-ALS-based VOT method in handling challenging conditions in visual object tracking. The performance of the proposed ALS-KF-based VOT algorithm is evaluated using the root mean square error (RMSE) (in pixels) between the estimated position \(\hat{x}_{1:2,t}\) (the first two elements) and the true position \(x_{1:2,t}\) of the moving ball subjected to projectile motion, free-fall bouncing motion, and back-and-forth linear motion across the video, as shown in Table 1.  
\[RMSE=\sqrt{\frac{1}{N}\sum_{t=1}^{N}(x_{1:2,t}-\hat{x}_{1:2,t})^{2}} \tag{23}\]  
Figure 4: VOT using the proposed AKF-ALS algorithm and traditional KF algorithm. The green, blue, and red circles, respectively, represent the circle of the ball with true centroid, with estimated centroid using AKF-ALS algorithm, and with estimated centroid using KF algorithm. The quantitative analysis of the table above reveals that AKF-ALS consistently achieves lower root mean square error (RMSE) values across various motion patterns. The RMSE for AKF-ALS is 1.05 pixels for projectile motion, significantly better than the 4.67 pixels of the standard KF. For free-fall bouncing motion, AKF-ALS records an RMSE of 2.76 pixels, in contrast to 8.35 pixels for the KF. Additionally, in back-and-forth linear motion scenarios, AKF-ALS demonstrates an RMSE of 0.89 pixels, outperforming the KF's 2.31 pixels. The results show that the proposed AKF-ALS algorithm achieves a lower RMSE compared to a standard KF VOT algorithm with fixed noise covariance, demonstrating the proposed method outperformed the standard Kalman filter in terms of RMSE.