## 2 AMTB Filter without Need for Clutter Density
### Establishment of Newborn Objects  
In this step, we associate the unused observations at step \(k\) with the potential birth objects. A newborn object is established if a measurement is assigned to a potential birth object. Let \(Z^{u}_{k}=\left\{z^{u}_{g,k}\right\}_{g=1}^{M^{u}_{k}}\) denote a set of unused measurements where \(M^{u}_{k}\) is the number of unused measurements. The Mahalanobis distance between unused measurement \(z^{u}_{g,k}\) and potential birth object \(i\) is:  
\[q^{b}_{ig}=(z^{u}_{g,k}-z^{b}_{i,k|k-1})^{\mathrm{T}}\Big{(}S^{b}_{i,k}+R_{k} \Big{)}^{-1}(z^{u}_{g,k}-z^{b}_{i,k|k-1}) \tag{26}\]  
Identical to Section 2.2, we use the 2-dimensional assignment to associate the measurement with the potential birth object. The cost matrix for the 2-dimensional assignment is:  
\[\mathbf{C}= \begin{bmatrix}\mathbf{C}_{1}&\mathbf{C}_{2}\end{bmatrix}  \tag{27}\]  
where\[\mathbf{C}_{1}=\left[q_{ig}^{b}\right]_{N_{k-1}^{b}\times M_{k}^{a}}=\left[ \begin{array}{ cccc}q_{11}^{b}&q_{12}^{b}&\cdots&q_{1,M_{k}^{a}}^{b}\\ q_{21}^{b}&q_{22}^{b}&\cdots&q_{2,M_{k}^{a}}^{b}\\ \vdots&\vdots&\ddots&\vdots\\ q_{N_{k-1}^{b},1}^{b}&q_{N_{k-1}^{b},2}^{b}&\cdots&q_{N_{k-1}^{b},M_{k}^{a}}^{b }\end{array} \right],\ \mathbf{C}_{2}=\left[ \begin{array}{cccc}q_{\alpha}&\infty&\cdots&\infty\\ \infty&q_{\alpha}&\cdots&\infty\\ \vdots&\vdots&\ddots&\vdots\\ \infty&\infty&\cdots&q_{\alpha}\end{array} \right]_{N_{k-1}^{b}\times N_{k-1} ^{b}} \tag{28}\]  
```
1:for\(i=1:N_{k-1}^{e}\)
2:for\(j=1:M_{k}\)
3:\(q_{ij}^{e}=(z_{j,k}-z_{i,k|k-1}^{e})^{\mathrm{T}}\big{(}S_{i,k}^{e}+R_{k} \big{)}^{-1}(z_{j,k}-z_{i,k|k-1}^{e}).\)
4:endfor
5:endfor
6: Form cost matrix \(\mathbf{C}\) in terms of (12) and (13). 7: Acquire an optimal solution \(\Theta=\left[\theta_{1},\theta_{2},\cdots,\theta_{N_{k-1}^{e}}\right]\) according to \(\mathbf{C}\). 8: Set \(\mathbf{Z}_{k}^{a}=\)\(\mathcal{O}\); and \(g_{j,k}=0\) for \(j=1,2,\cdots,M_{k}\). 9:for\(i=1:N_{k-1}^{e}\)
10:if\(1\leq\theta_{i}\leq M_{k}\)
11:\(j=\theta_{i}\), \(r_{ik}^{e}=1\), \(A_{i}^{e}=\mathbf{P}_{i,k|k-1}^{e}[\mathbf{H}_{k}\mathbf{P}_{i,k|k-1}^{e}(\mathbf{H}_{k})^{ \mathrm{T}}+\mathbf{R}_{k}]^{-1}\),
12:\(\mathbf{m}_{ij,k}^{e}=\mathbf{m}_{i,k|k-1}^{e}+A_{i}^{e}\cdot[\mathbf{z}_{j,k}-\mathbf{m}_{i,k |k-1}^{e}]\), \(\mathbf{P}_{i,k}^{e}=\mathbf{P}_{i,k|k-1}^{e}-A_{i}^{e}\mathbf{H}_{k}\mathbf{P}_{i,k|k-1}^{e} \cdot g_{j,k}=1\). 13:else
14:\(\mathbf{m}_{i,k}^{e}=\mathbf{m}_{i,k|k-1}^{e},\mathbf{P}_{i,k}^{e}=\mathbf{P}_{i,k|k-1}^{e},r _{i,k}^{e}=(1-p_{D})r_{i,k|k-1}^{e}\). 15:endif
16:\(\ell_{i,k}^{e}=\ell_{i,k|k-1}^{e}\)
17:endfor
18:for\(j=1:M_{k}\)
19:if\(g_{j,k}=0\)
20:\(\mathbf{Z}_{k}^{u}=[\mathbf{Z}_{k}^{u}\mathbf{z}_{j,k}]\)
21:endif
22:endfor
23:Output:\(\left\{r_{i,k}^{e},m_{i,k}^{e},\mathbf{P}_{i,k}^{e},\ell_{i,k}^{e}\right\}_{i=1}^{N_{k-1} ^{e}}\). ```  
**Algorithm 1** Update of existing objects  
In terms of \(\mathbf{C}\), we can acquire an optimal solution with the minimum cost by using the optimized Murty algorithm [24]. The optimal solution can be given as:  
\[\Theta=\left[\theta_{1},\theta_{2},\cdots,\theta_{N_{k-1}^{b}}\right] \tag{29}\]  
where \(\theta_{i}\in\left\{1,2,\cdots,M_{k}^{u}+N_{k-1}^{b}\right\}\) for \(i=1,2,\cdots,N_{k-1}^{b}\). \(1\leq\theta_{i}\leq M_{k}^{u}\) demonstrates that \(\mathbf{z}_{\theta_{i},k}^{u}\) is assigned to potential birth object \(i\). A newborn object is established if a measurement is assigned to a potential birth object. Since \(1\leq\theta_{i}\leq M_{k}^{u}\) demonstrates that \(\mathbf{z}_{\theta_{i},k}^{u}\) is assigned to potential birth object \(i\), a newborn object \(h\) is established according to potential birth object \(i\) and measurement \(\mathbf{z}_{\theta_{i},k}^{u}\).