# Multiple Extended Target Tracking Algorithm Based on Spatio-Temporal Correlation
## 2 Mathematical Modeling  
Multiple extended target tracking is used to estimate the number and motion state of multiple extended targets based on sensor observation data in a noisy environment. The number of targets and measurements changes over time with the birth, derivation, and death of targets and missed detection of sensors. [PERSON] proposed a multi-extended target tracking framework based on a Poisson multi-target model. In the RFS multi-extended filtering framework, the state and measurement of the target are, respectively, expressed as \(X_{k}=\left\{x_{k}^{1},\cdots,x_{k}^{N_{k,k}}\right\}\in\text{X}\) and \(Z_{k}=\left\{z_{k}^{1},\cdots,z_{k}^{N_{k,k}}\right\}\in\text{Z}\), where \(x^{N_{k,k}}\) is the state of a single target and \(N_{x,k}\) is the number of targets at time \(k\). \(z_{k}^{N_{k,k}}\) is a single measurement value and \(N_{z,k}\) is the number of measurements at time \(k\). Since the prediction formula of ET-GM-PHD is the same as that of the standard GM-PHD filter, only the update formula of the ET-GM-PHD filter is considered here. The measurement update equation can be expressed as follows:  
\[D_{k|k}(x|Z)=L_{Z_{k}}(x)D_{k|k-1}(x|Z) \tag{1}\]  
where \(L_{Z_{k}}(x)\) represents the measurement pseudo-likelihood function and is defined as follows:  
\[ \begin{split} L_{Z_{k}}(x)&\overset{\Delta}{=}1- \left(1-e^{-\gamma(x)}\right)p_{D}(x)+e^{-\gamma(x)}p_{D}(x)\\ &\times\sum_{P\angle Z_{k}}\omega_{P}\sum_{W\in P}\frac{\gamma(x )^{|W|}}{d_{W}}\cdot\prod_{z_{k}\in W}\frac{\phi_{z_{k}}(x)}{\lambda_{k}c_{k} (z_{k})}\end{split}  \tag{2}\]  
where \(P\angle Z_{k}\) denotes that the measurement set \(Z_{k}\) is divided into non-empty measurement units \(W\); \(\phi_{z_{k}}(x)=p(z_{k}|x)\) represents the measurement likelihood function generated by a single target; \(\gamma(x)\) represents the expected number of target measurements; \(p_{D}(x)\) represents the probability of object detection; \(\lambda_{k}\) denotes the average number of clutter \begin{table}
\begin{tabular}{c c} \hline
**Measurement Partition Method** & **Main Characteristic** \\ \hline Traditional method & Updating requires iterating over all possible subsets of measurement partitions \\ \hline Algorithm based on distance partition [6; 14] & The computational complexity is high, and it is difficult to \\ Algorithm based on distance partition [6; 14] & accurately divide the measurement sub-set corresponding to the target \\  & in the case of a large number of targets and lacks robustness. \\ \hline K-means++ clustering [15] & The center point needs to be found iteratively. \\  & When the number of targets is large and the distance is close, it is difficult to \\  & distinguish the measurement sets of different targets. \\ \hline Spectral clustering [16] & This method is based on adjacency matrix propagation, which can divide the \\  & measurement set more accurately. However, this method is \\  & very sensitive to clutter, and it needs to eliminate the clutter in addition, \\  & and it is complex to deal with dense clutter scenes. \\ \hline Expectation maximization (EM) [6] & The EM measurement partitioning algorithm converges to \\  & the nearest local maximum of the likelihood function. \\ Expectation maximization (EM) [6] & Since the EM partitioning is initialized using the predicted GIW components, \\  & the partitioning performance is sensitive to \\  & the maneuverability of the motion model and prediction results. \\ \hline Spatio-temporal correlation (proposed) & The measurement set partitions are hierarchically processed simultaneously. \\  & The computational efficiency and accuracy are higher. \\ \hline \end{tabular}
\end{table} 
Table 1: Extended target tracking algorithm based on different measurement partition methods. measurements; and \(c_{k}(z_{k})\) is the uniform clutter density in the surveillance area. For more details of the ET-GM-PHD algorithm, we can refer to reference [14].