# Joint Probabilistic Hypergraph Matching Labeled Multi-Bernoulli Filter for Rigid Target Tracking
## 4 Simulation  
In this section, the performance of proposed method is evaluated on two simulated scenarios with missed detections and clutter. In the first scenario, a 2D rigid target with four scattering centers is tracked, these four scattering centers are considered as four targets and their relative position are assumed to be unchanged, the effect of detection probability and maneuverability on the tracking performance is analyzed. In the second scenario, the validation of proposed method is extended to more complex case, two 2D rigid targets with four scattering centers are tracked, the effect of measurement errors and occlusions are considered. In our experiment, the target state \(\mathbf{x}=[r_{x},r_{y},\dot{r}_{x},\dot{r}_{y}]\) consists of the position and velocity components, where \(r_{x}\) and \(\dot{r}_{x}\) are the position and velocity in the \(x\) direction, likewise of \(y\) direction. The discrete-time version of the CV model is employed, the time evolution of target state is given  
\[\mathbf{x}_{k}^{(j)}=\mathbf{F}_{k}^{(j)}\mathbf{x}_{k-1}^{(j)}+\mathbf{w}_{k}^{(j)}, \tag{41}\]where \(\mathbf{x}_{k}^{(j)}\) is the target state of \(j\)-th target at time \(k\), \(\mathbf{F}_{k}^{(j)}\) is the state transition matrix, \(\mathbf{w}_{k}^{(j)}\) is the process noise which is modeled as Gaussian white noise with zero mean and covariance \(\mathbf{Q}_{k}^{(j)}\),  
\[\mathbf{F}_{k}^{(j)}=\left[ \begin{array}{cccc}1&0&t&0\\ 0&1&0&t\\ 0&0&1&0\\ 0&0&0&1\end{array} \right],\mathbf{Q}_{k}^{(j)}=\Omega^{2}\left[ \begin{array}[ ]{c}\frac{t^{2}}{\tau}\\ \frac{t}{\tau}\\ t\\ t\end{array} \right]\left[ \begin{array}{cc}\frac{t^{2}}{\tau}&\frac{t^{2}}{ \tau}&t\\ \end{array} \right], \tag{42}\]  
where \(\Omega\) represents the acceleration error and \(t\) is the sampling time. For simplicity, the sensor is assumed to measure the position directly, and the measurement model is  
\[\mathbf{z}_{k}^{(j)}=\mathbf{H}_{k}\mathbf{x}_{k}^{(j)}+\mathbf{q}_{k}, \tag{43}\]  
where \(\mathbf{z}_{k}^{(j)}\) is the measurement generated by \(j\)-th target at time \(k\), \(\mathbf{q}_{k}\) is the measurement noise which is modeled as Gaussian white noise with zero mean and covariance \(\mathbf{R}_{k}\), and  
\[\mathbf{H}_{k}=\left[ \begin{array}{cccc}1&0&0&0\\ 0&1&0&0\end{array} \right],\mathbf{R}_{k}=\left[ \begin{array}{cc}\sigma_{m}^ {2}&0\\ 0&\sigma_{m}^{2}\end{array} \right]. \tag{44}\]  
where \(\sigma_{m}\) is the standard deviation of measurement noise, then the single likelihood in Equation (15) is \(g(z_{\theta}(\ell)|x,\ell)=\mathcal{N}(z_{\theta}(\ell);\mathbf{H}_{k}x_{ \ell},\mathbf{R}_{k})\). In this paper, optimal subpattern assignment metric (OSPA) [35] and clear multiple object tracking (mot) metrics [36] are used for performance evaluation over 100 Monte Carlo runs. The OSPA distance is defined by  
\[d_{p}^{c}(\mathbf{\sigma}_{k},\hat{\mathbf{\sigma}}_{k})=\left(\frac{1}{n}\left(\min_{ \pi\in\Pi_{n}}\sum_{i=1}^{m}d^{c}(\mathbf{x}_{i},\hat{\mathbf{x}}_{\pi(i)})^{p}+C^{p}( n-m)\right)\right)^{1/p}, \tag{45}\]  
where \(m<n\), \(\varkappa_{k}=\{\mathbf{x}_{k}^{(1)},\mathbf{x}_{k}^{(2)},...,\mathbf{x}_{k}^{(m)}\}\) is the true RFS at time k, \(\hat{\varkappa}=\{\hat{\mathbf{x}}_{k}^{(1)},\hat{\mathbf{x}}_{k}^{(2)},...,\hat{\mathbf{ x}}_{k}^{(n)}\}\) is the estimated RFS, \(\prod_{n}\) is the assignment results which assign \(\varkappa\) to \(\hat{\varkappa}\), \(p\) means \(p-norm\), \(c\) is the penalty cost for cardinality mismatch. In this simulation, \(c=5\) and \(p=2\). Clear mot metrics contains two metrics including the multiple object tracking precision (MOTP) (which measures the performance of target states estimation, smaller value means better performance) and the multiple object tracking accuracy (MOTA) (which takes misses, false positives, mismatches into account). In MTT system, targets which have no estimates are considered as misses, the estimates for which no real target exists are considered as false positives. If the label of a target changed compared to previous frame, a mismatch happened.