## 2 AMTB Filter without Need for Clutter Density
### Update of Existing Objects  
In this step, we associate the observations at step \(k\) with the existing targets. The existence of an object is confirmed and its state is updated if a measurement is assigned to it. An object is not detected if no measurement is assigned to it, and its state is given by its predicted state. The Mahalanobis distance is used to measure the correlation between the target and the measurement, and denoting the measurement set at time step \(k\) by \(\mathbf{y}_{k}=\left\{\mathbf{z}_{j,k}\right\}_{j=1}^{M_{k}}\) where \(M_{k}\) is the observation number. The Mahalanobis distance between measurement \(\mathbf{z}_{j,k}\) and existing object \(i\) may be given by:  
\[q_{ij}^{e}=(\mathbf{z}_{j,k}-\mathbf{z}_{i,k|k-1}^{e})^{T}\Big{(}\mathbf{S}_{i,k}^{e}+\mathbf{ R}_{k}\Big{)}^{-1}(\mathbf{z}_{j,k}-\mathbf{z}_{i,k|k-1}^{e}) \tag{11}\]  
\(q_{ij}^{e}\) follows a chi-square distribution, and its degree of freedom equals the dimension of observation \(\mathbf{z}_{j,k}\). An acceptance threshold \(q_{\alpha}\) may be determined in terms of the chi-square distribution table after giving a confidence level \(\alpha\). If \(q_{ij}^{e}<q_{\alpha}\), we confirm that \(\mathbf{z}_{j,k}\) falls in the acceptance gate of existing target \(i\). To avoid the track splitting, we use the 2-dimensional assignment to associate the measurement with the existing target. The cost matrix for the 2-dimensional assignment is:  
\[\mathbf{C}= \begin{bmatrix}\mathbf{C}_{1}&\mathbf{C}_{2}\end{bmatrix}  \tag{12}\]  
where  
\[\mathbf{C}_{1}= \begin{bmatrix}q_{ij}^{e}\end{bmatrix} _{N_{k-1}^{e}\times M_{k}}= \begin{bmatrix}q_{11}^{e}&q_{12}^{e}&\cdots&q_{1,M_{k}}^{e}\\ q_{21}^{e}&q_{22}^{e}&\cdots&q_{2,M_{k}}^{e}\\ \vdots&\vdots&\ddots&\vdots\\ q_{N_{k-1}^{e},1}^{e}&q_{N_{k-1}^{e},2}^{e}&\cdots&q_{N_{k-1}^{e},M_{k}}^{e} \end{bmatrix} ,\ \mathbf{C}_{2}= \begin{bmatrix}q_{\alpha}&\infty&\cdots&\infty\\ \infty&q_{\alpha}&\cdots&\infty\\ \vdots&\vdots&\ddots&\vdots\\ \infty&\infty&\cdots&q_{\alpha}\end{bmatrix} _{N_{k-1}^{e}\times N_{k-1}^{e}} \tag{13}\]  
According to \(\mathbf{C}\), we acquire an optimal solution with the minimum cost by using the optimized Murty algorithm [24]. The optimal solution can be given as:  
\[\Theta= \begin{bmatrix}\theta_{1},\theta_{2},\cdots,\theta_{N_{k-1}^{e}}\end{bmatrix}  \tag{14}\]  
where \(\theta_{i}\in\left\{1,2,\cdots,M_{k}+N_{k-1}^{e}\right\}\) for \(i=1,2,\cdots,N_{k-1}^{e}\). \(1\leq\theta_{i}\leq M_{k}\) demonstrates that \(\mathbf{z}_{\theta_{i},k}\) is assigned to existing object \(i\), while \(\theta_{i}>M_{k}\) reveals that no measurement is assigned to existing object \(i\). To detect the potential birth objects and form newborn tracks, the proposed filter needs a set of unused measurements. We use binary variable \(g_{j,k}\) to denote whether measurement \(\mathbf{z}_{j,k}\) is used or not, and set \(g_{j,k}=0\) for \(j=1,2,\cdots,M_{k}\). If \(1\leq\theta_{i}\leq M_{k}\), we set \(j=\theta_{i}\) and use observation \(\mathbf{z}_{j,k}\) to update the predicted PDF of object \(i\). The PDF of object \(i\) at time step \(k\) can be given by:  
\[N(\mathbf{x}_{i,k};\mathbf{m}_{i,k}^{e},\mathbf{P}_{i,k}^{e})=N(\mathbf{x}_{i,k};\mathbf{m}_{ij}^{e },\mathbf{P}_{ij}^{e}) \tag{15}\]where  
\[m^{e}_{ij}=m^{e}_{i,k|k-1}+A^{e}_{i}\cdot[z_{j,k}-m^{e}_{i,k|k-1}] \tag{16}\]  
\[P^{e}_{ij}=P^{e}_{i,k|k-1}-A^{e}_{i}R_{k}P^{e}_{i,k|k-1} \tag{17}\]  
\[A^{e}_{i}=P^{e}_{i,k|k-1}(H_{k})^{\mathrm{T}}[H_{k}P^{e}_{i,k|k-1}(H_{k})^{ \mathrm{T}}+R_{k}]^{-1} \tag{18}\]  
In this case, its existence probability is:  
\[r^{e}_{i,k}=1 \tag{19}\]  
Since measurement \(z_{j,k}\) is used, \(g_{j,k}\) is updated by:  
\[g_{j,k}=1 \tag{20}\]  
If \(\theta_{i}>M_{k}\), object \(i\) is not detected because no observation is assigned to it.