# Moving Vehicle Tracking with a Moving Drone Based on Track Association
## 3 Multiple Target Tracking
### Track-Track Association  
If multiple measurements are continuously detected on a single object, more than one track can be generated. We develop the track-track association to eliminate redundant tracks. Multiple tracks on the same target have the error dependencies on each other, thus the following track-association hypothesis testing [21] is preceded as  
\[[\hat{\mathbf{x}}_{s}(k|k)-\hat{\mathbf{x}}_{t}(k|k)]^{T}[T_{st}(k)]^{-1}[\hat{\mathbf{x} }_{s}(k|k)-\hat{\mathbf{x}}_{t}(k|k)]^{T}\leq\gamma_{f}, \tag{22}\]  
\[S_{t}=1,\ldots,N(k),s\
eq t \tag{23}\]  
\[P_{st}(k\big{|}k)=[I-b_{s}(k)W_{s}(k)H]\Big{[}FP_{st}(k-1\big{|}k-1)F^{T}+Q \Big{]}[I-b_{t}(k)W_{t}(k)H] \tag{24}\]  
where \(\hat{\mathbf{x}}_{s}(k|k)\) and \(\hat{\mathbf{x}}_{t}(k|k)\) are the state vector of track \(s\) and \(k\), respectively, at frame \(k\); \(P_{s}(k|k)\) and \(P_{t}(k|k)\) are the covariance matrix of track \(s\) and \(k\), respectively, at frame \(k\); \(\gamma_{f}\) is a thresholding value for track association; \(N(k)\) is the number of tracks at frame \(k\); \(b_{s}(k)\) and \(b_{t}(k)\) are binary numbers that become one when track \(s\) or \(t\) is associated with a measurement. If there is no measurement associated, it will be zero. In this case, the state vector and the covariance matrix are replaced by predictions as in Equations (20) and (21). The fused covariance in Equation (24) is a linear recursion and its initial condition is set at \(P_{st}(0\big{|}0)=[0]_{4\times 4}\). When the track association hypothesis is accepted, the most accurate track is selected, the current state is replaced with a fused estimate, and the remaining track is immediately terminated. The selection process is based on the determinant of the covariance matrix because the more accurate track has less error (covariance). A track is selected and fused as  
\[\hat{c}=\underset{s,t}{argmin}[|P_{s}(k|k)|,|P_{t}(k|k)|], \tag{25}\]  
\[\hat{\mathbf{x}}_{\hat{c}}(k\big{|}k)=\hat{\mathbf{x}}_{s}(k\big{|}k)+[P_{s}(k|k)-P_{ st}(k|k)][P_{s}(k|k)+P_{t}(k|k)-P_{st}(k|k)-P_{ts}(k|k)]^{-1}[\hat{\mathbf{x}}_{t}(k|k)- \hat{\mathbf{x}}_{s}(k|k)], \tag{26}\]  
\[P_{\hat{c}}(k\big{|}k)=P_{s}(k\big{|}k)-[P_{s}(k|k)-P_{st}(k|k)][P_{s}(k|k)+P_ {t}(k|k)-P_{st}(k|k)-P_{ts}(k|k)]^{-1}[P_{s}(k|k)-P_{ts}(k|k)]. \tag{27}\]  
Figure 4 illustrates the track-track association process. Assuming that there are two tracks on the same target at frame \(k\) as shown in Figure 4a, the statistical distance between the tracks is tested as shown in Figure 4b. In Figure 4c, the track with the least determinant of the covariance matrix, which is track \(s\), is selected. The state and covariance of track \(s\) are replaced by fused ones and the other track \(t\) is terminated at the same time.  
In the paper, we establish criteria for a valid track and track termination, respectively. One is the minimum track life to become a valid track. Track life is the number of frames between the last frame updated by a measurement and the initial frame [24]. Another criterion is for track termination. The track is terminated if the search for a measurement for a specific number of frames fails. In Figure 5, the track life is six frames, and the track is terminated after six frames without updating measurements.