## 3 Prediction-Driven Measurement Sub-Partitioning
It makes our approach easily extended to other filters for tracking extended targets with the complicated spatial extent. To deal with the tracking scenario with clutters and unknown associates between measurements and targets, measurements and target states are modeled as random finite sets (RFS), respectively, i.e., \(X_{k-1}=\left\{x_{k-1}^{(i)}\right\}_{i=1}^{N_{x,k-1}}\), \(Z_{k}=\left\{z_{k}^{(i)}\right\}_{i=1}^{N_{x,k}}\). \(N_{x,k-1}\) is the number of targets estimated at time \(k-1\) and \(N_{z,k}\) is the number of measurements obtained at time \(k\). The corresponding dynamic model of each target state in the \(X_{k-1}\) and the measurement model of each measurement in \(Z_{k}\) are assumed as follows,

\[x_{k}^{(i)}=F_{k-1}x_{k-1}^{(i)}+w_{k-1} \tag{4}\]

\[z_{k}^{(i)}=H_{k}x_{k-1}^{(i)}+e_{k} \tag{5}\]

where \(F_{k-1}\) is the state transition matrix, \(w_{k-1}\) is the independent Gaussian white noise with the covariance \(Q_{k-1}\), \(H_{k}\) is the measurement matrix, and \(e_{k}\) is Gaussian white noise with the covariance \(R_{k}\). The proposed extended target tracking approach includes the prediction, measurement partitioning, and update, which are detailed as follows. ### Prediction

The predicted intensity \(D_{k|k-1}\)\((x)\), which contains the intensity for newly born targets, spawned targets and existing targets, can be given as follows,

\[D_{k|k-1}\)\((x)=\sum_{j=1}^{I_{k|k-1}}\omega_{k|k-1}^{(j)}\mathcal{N}\Big{(}x;m_{k| k-1}^{(j)},P_{k|k-1}^{(j)}\Big{)} \tag{6}\]

where \(J_{k|k-1}\) is the predicted number of Gaussian components, \(\omega_{k|k-1}^{(j)}\) is the predicted weight of the Gaussian component, \(m_{k|k-1}^{(j)}\) is the predicted mean of the Gaussian component, and \(P_{k|k-1}^{(j)}\) is the predicted covariance of the Gaussian component. Note that the proposed PMS only uses the predicted intensity for existing targets. Given the initial intensity \(D_{k-1}(x)\), the predicted intensity \(D_{S,k|k-1}\)\((x)\) for existing targets can be calculated as follows,

\[D_{S,k|k-1}\)\((x)=p_{S}\)\((x)\sum_{j=1}^{I_{k-1}}\omega_{k-1}^{(j)}\mathcal{N}\Big{(}x;m_{S,k|k-1}^{(j)},P_{S,k|k-1}^{(j)}\Big{)} \tag{7}\]

\[m_{S,k|k-1}^{(j)}=F_{k-1}m_{k-1}^{(j)} \tag{8}\]

\[P_{S,k|k-1}^{(j)}=Q_{k-1}+F_{k-1}P_{k-1}^{(j)}F_{k-1}^{\mathrm{T}} \tag{9}\]

where \(p_{S}\)\((x)\) is the survival probability, \(m_{S,k|k-1}^{(j)}\) is the transition mean of the Gaussian component, \(p_{S,k|k-1}^{(j)}\) is the transition covariance of the Gaussian component, \(\omega_{k-1}^{(j)}\) is the weight of the Gaussian component at time \(k-1\), \(m_{k-1}^{(j)}\) is the mean of the Gaussian component at time \(k-1\), and \(P_{k-1}^{(j)}\) is the covariance of the Gaussian component at time \(k-1\). ### Measurement Partitioning

The measurement partitioning is indispensable to MCETT and aims to make measurements (originated from the same target) divided into the same subset. As an integral part of ET-PHD filter, it would greatly affect the stability of the subsequent tracking algorithm and the estimation accuracy of both the target state and number. Besides, the computational burden of the measurements partitioning is also a key factor affecting the speed of multiple extended target tracking. Target predictions can be used to partition the measurements to obtain relatively accurate partitioning results. However, they are likely to cause measurements that are inaccurately divided when the cardinality underestimation problem occurs. There is a pressing need to modify it for improving the performance in the target number and state estimation. Based on PMS, the measurement set can be partitioned as accurately as possible. The range of K value is also narrowed, i.e., the lower and upper bounds of the interval \([K_{L},K_{U}]\) are set as \(K_{L}=J_{k-1}\) and \(K_{U}=J_{k-1}+J_{\theta,k}\times J_{k-1}+J_{\gamma,k}\), respectively. This strategy reduces the partition number and decreases the computational complexity greatly. Note that the predictive information is fully used to obtain the clustering centers.