## 3 Prediction-Driven Measurement Sub-Partitioning
### Performance Evaluation in Tracking of Extended Targets with Typical Tracks

Scenario A is a typical tracking scenario with clutter and missed detection. It contains the tracking of crossed targets, newborn targets, and spawned targets. The true tracks and measurements are illustrated in Figure 3. Both of two targets were born at \(k=1\) s and died at \(k=100\) s. They are crossed at \(k=50\) s. At \(k=67\) s, there is a target newborn and a target spawned from the existing target. Especially when a new target is spawned from the existing target, the locations of these two targets are very close. It is another typical form of tracking closely spaced targets. In this scenario, the surveillance area is \([-1000\ 1000]\times[-1000\ 1000]\). The target state contains the position and velocity, i.e., \(X_{k}=[x_{k},y_{k},\dot{x}_{k},\dot{y}_{k}]^{\mathsf{T}}\), where \((x_{k},y_{k})\) and \((\dot{x}_{k},\dot{y}_{k})\) are the position and velocity in the two-dimensional Cartesian coordinate system, respectively. It is assumed that each target generates measurements with the Poisson distribution (the Poisson ratio is \(\gamma^{(i)}=10,\forall i\)). The clutter measurements are also generated with \(\lambda=10\) at each time step. The observed measurement is denoted as \(z_{k}^{(j)}=\left[x_{k}^{(j)},y_{k}^{(j)}\right]^{\mathsf{T}}\). The probability of detection and survival are set to \(p_{S}=0.99\) and \(p_{D}=0.99\), respectively. \[\mathrm{K}_{k}^{(j)}=P_{k|k-1}^{(j)}H_{K}^{\mathrm{T}}(R_{K}+H_{K}P_{k|k-1}^{( j)}H_{K}^{\mathrm{T}})^{-1} \tag{26}\]0 True tracks and measurements with clutter. Green square is the beginning point of the track; red circle is the end point of the track. Each extended target is supposed to perform a nearly constant velocity motion model with the sampling interval \(T=1\) s, and the dynamic and measurements models are as follows,

\[\mathrm{K}_{k}^{(j)}=P_{k|k-1}^{(j)}H_{K}^{\mathrm{T}}(R_{K}+H_{K}P_{k|k-1}^{( j)}H_{K}^{\mathrm{T}})^{-1} \tag{26}\]1

\[\mathrm{K}_{k}^{(j)}=P_{k|k-1}^{(j)}H_{K}^{\mathrm{T}}(R_{K}+H_{K}P_{k|k-1}^{( j)}H_{K}^{\mathrm{T}})^{-1} \tag{26}\]2

where \(N\left(x;m,P\right)\) is a Gaussian density with mean \(m\) and covariance \(P\). Here, \(F\) is the state transition matrix, \(Q\) is the process noise covariance, \(H\) is the measurement matrix, and \(R_{k}\) is the observation noise covariance. They have the following forms,

\[F=\left[ \begin{array}{cccc}1&0&T&0\\ 0&1&0&T\\ 0&0&1&0\\ 0&0&0&1\end{array} \right],Q=\sigma_{W}^{2}\left[ \begin{array}{cccc}\frac{T ^{2}}{2}&0&T&0\\ 0&\frac{T^{2}}{2}&0&T\end{array} \right]^{\mathrm{T}}\]

\[H=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\end{array} \right],\ R_{k}=\sigma_{e}^{2}1_{2}\]

where \(\sigma_{W}=2\) m/s\({}^{2}\) is the standard deviation of process noise and \(\sigma_{e}=20\) m is the standard deviation of measurement noise. The birth intensity for the existing targets is

\[D_{b}(x)=0.1\mathcal{N}(x;m_{b}^{(1)},P_{b})+0.1\mathcal{N}(x;m_{b}^{(2)},P_{ b}) \tag{36}\]

where \(m_{b}^{(1)}=\left[250,250,0,0\right]^{T}\), \(m_{b}^{(2)}=\left[-250,-250,0,0\right]^{T}\), and \(P_{b}=\text{diag}\left(\left[100,100,25,25\right]\right)\). Figure 4 gives the comparison results of target number estimation. It can be seen that PMS has a more accurate estimation of target number than DP, PP and SP especially when a new target is spawned from the existing target at \(k=67\) s. This is because PMS finds relatively accurate alternative points to replace unreasonable target predictions according to the geometry of a regular polygon, which facilitates partitioning the measurements more accurately to avoid the target number underestimation. For further evaluating the tracking performance of PMS, we compare it with DP, PP, and SP by using the OSPA distance. As shown in Figure 5, PMS outperforms DP, especially when targets are spatially close (i.e., targets are crossed at \(k=50\) s and a new target is spawned at \(k=67\) s).