## 2 AMTB Filter without Need for Clutter Density
### Establishment of Newborn Objects
The PDF of newborn object \(h\) is:  
\[N(\mathbf{x}_{h,k};\mathbf{m}_{h,k},\mathbf{P}_{h,k}) \tag{30}\]  
where  
\[\mathbf{m}_{h,k}=\mathbf{m}_{l,k|k-1}^{b}+A_{i}^{b}\cdot[\mathbf{z}_{\theta_{i},k}^{u}-\mathbf{ m}_{i,k|k-1}^{b}] \tag{31}\]  
\[\mathbf{P}_{h,k}=\mathbf{P}_{i,k|k-1}^{b}-A_{i}^{b}H_{k}\mathbf{P}_{i,k|k-1}^{b} \tag{32}\]  
\[A_{i}^{b}=\mathbf{P}_{i,k|k-1}^{b}(\mathbf{H}_{k})^{\mathrm{T}}[\mathbf{H}_{k}\mathbf{P}_{i,k|k-1 }^{b}(\mathbf{H}_{k})^{\mathrm{T}}+\mathbf{R}_{k}]^{-1} \tag{33}\]The track label of newborn object \(h\) and its existence probability are:  
\[\ell_{h,k}= \begin{bmatrix}k\\ h\end{bmatrix} ;\ r_{h,k}=1 \tag{34}\]  
The mean vectors of newborn object \(h\) at time steps \(k-2\) and \(k-1\) are given by:  
\[\mathbf{m}_{h,k-2}=\mathbf{m}_{i,k-2}^{h};\ \mathbf{m}_{h,k-1}=\mathbf{m}_{i,k-1}^{h} \tag{35}\]  
Since \(\mathbf{z}_{\theta,k}^{u}\) and \(\mathbf{z}_{i,k-1,k-1}^{u}\) are used to establish newborn object \(h\), they should be removed from sets \(\mathbf{Z}_{k}^{u}\) and \(\mathbf{Z}_{k-1}^{u}\) as:  
\[\mathbf{Z}_{k}^{u}=\mathbf{Z}_{k}^{u}\backslash\mathbf{z}_{\theta,k}^{u},\ \mathbf{Z}_{k-1}^{u}=\mathbf{Z}_{k-1}^{u} \backslash\mathbf{z}_{\epsilon_{i,k-1},k-1}^{u} \tag{36}\]  
Dealing with each potential birth object according to optimal solution \(\Theta\), we can acquire a set of newborn objects and the updated sets \(\mathbf{Z}_{k}^{u}\) and \(\mathbf{Z}_{k-1}^{u}\). The set of newborn objects are:  
\[\Big{\{}r_{h,k},N(\mathbf{x}_{h,k};\mathbf{m}_{h,k},\mathbf{P}_{h,k}),\ell_{h,k}\Big{\}}_{ h=1}^{N_{k}^{u}} \tag{37}\]  
where \(N_{k}^{u}\) is the number of newborn objects. We use the mean vectors of newborn objects and their track labels at time steps \(k-2\), \(k-1\) and \(k\) to form three sets \(\mathbf{X}_{k-2}^{b}\), \(\mathbf{X}_{k-1}^{b}\) and \(\mathbf{X}_{k}^{b}\). The three sets are given by:  
\[\mathbf{X}_{k-2}^{b}=\Big{\{}\mathbf{m}_{h,k-2},l_{h,k}\Big{\}}_{h=1}^{N_{k}^{u}},\ \mathbf{X}_{k-1}^{b}=\Big{\{}\mathbf{m}_{h,k-1},l_{h,k}\Big{\}}_{h=1}^{N_{k}^{u}},\ \mathbf{X}_{k}^{b}=\Big{\{}\mathbf{m}_{h,k},l_{h,k}\Big{\}}_{h=1}^{N_{k}^{u}} \tag{38}\]  
Algorithm 2 gives the pseudo-code for the establishment of newborn objects. ```
1:for\(i=1:N_{k-1}^{b}\)
2:for\(g=1:M_{k}^{u}\)
3:\(q_{ig}^{b}=(\mathbf{z}_{g,k}^{u}-\mathbf{z}_{i,k|k-1}^{b})^{\text{T}}\Big{(}\mathbf{S}_{ i,k}^{b}+\mathbf{R}_{k}\Big{)}^{-1}(\mathbf{z}_{g,k}^{u}-\mathbf{z}_{i,k|k-1}^{b})\). 4:endfor
5:endfor
6: Establish cost matrix \(\mathbf{C}\) according to (27) and (28). 7: Acquire an optimal solution \(\Theta=\Big{[}\theta_{1},\theta_{2},\cdots,\theta_{N_{k-1}^{b}}\Big{]}\) according to \(\mathbf{C}\). 8: Set \(h=0\), \(\mathbf{X}_{k-2}^{b}=\mathcal{O}\), \(\mathbf{X}_{k-1}^{b}=\mathcal{O}\) and \(\mathbf{X}_{k}^{b}=\mathcal{O}\). 9:for\(i=1:N_{k-1}^{b}\)
10:if\(1\leq\theta_{i}\leq M_{k}^{u}\)
11:\(h=h+1\), \(\mathbf{A}_{l}^{b}=P_{i,k|k-1}^{b}(\mathbf{H}_{k})^{\text{T}}[\mathbf{H}_{k}P_{i,k|k-1}^{b }(\mathbf{H}_{k})^{\text{T}}+\mathbf{R}_{k}]^{-1}\). 12:\(\mathbf{m}_{h,k}=\mathbf{m}_{i,k|k-1}^{b}+\mathbf{A}_{l}^{b}\cdot[\mathbf{z}_{\theta,k}^{u}- \mathbf{m}_{i,k|k-1}^{b}],\mathbf{P}_{h,k}=\mathbf{P}_{i,k|k-1}^{b}-\mathbf{A}_{l}^{b}\mathbf{H}_{ k}P_{i,k|k-1}^{b},r_{h,k}=1\),
13:\(\ell_{h,k}= \begin{bmatrix}k\\ h\end{bmatrix} \). 14:\(\mathbf{X}_{k-2}^{b}=\Big{[}\mathbf{X}_{k-2}^{b}\ [\mathbf{m}_{h,k-2}^{b}\ell_{h,k}]\Big{]},\mathbf{X}_{k-1}^{b}=\Big{[}\mathbf{X}_{k-1}^{b }\ [\mathbf{m}_{h,k-1}^{b};\ \ell_{h,k}]\Big{]}\),
15:\(\mathbf{X}_{k}^{b}=\Big{[}\mathbf{X}_{k}^{b}\ [\mathbf{m}_{h,k};\ \ell_{h,k}]\Big{]}\). 16: Remove \(\mathbf{z}_{\theta,k}^{u}\) and \(\mathbf{z}_{\epsilon_{i,k-1},k-1}^{u}\) from sets \(\mathbf{Z}_{k}^{u}\) and \(\mathbf{Z}_{k-1}^{u}\), respectively. 17:endif
18:endfor
19:\(N_{k}^{u}=h\). 20:Output:\(\Big{\{}r_{h,k^{\prime}},\mathbf{m}_{h,k^{\prime}},P_{h,k^{\prime}}\ell_{h,k}\Big{\}}_{ h=1}^{N_{k}^{u}},\ \mathbf{Z}_{k-1}^{u},\ \mathbf{Z}_{k}^{u},\mathbf{X}_{k-2}^{b},\ \mathbf{X}_{k-1}^{b},\ \mathbf{X}_{k}^{b}\). ```  
**Algorithm 2** Establishment of newborn objects