# Joint Probabilistic Hypergraph Matching Labeled Multi-Bernoulli Filter for Rigid Target Tracking
## 3 JPHGM-LMB
The probability of association event \(\mathbf{b}_{l}\) given the measurement set \(\mathbf{Z}^{k}=\{\mathbf{Z}_{l}\}_{l=1}^{k}\) can be obtained via the total probability principle and Bayesian formulas  
\[ \begin{split} p(\mathbf{b}_{l}|\mathbf{Z}^{k})&=p(\mathbf{b}_{l} |\mathbf{Z}_{k},\mathbf{Z}^{k-1})\\ &=\frac{1}{c}p(\mathbf{Z}_{k}|\mathbf{b}_{l},\mathbf{Z}^{k-1})p(\mathbf{b}_{l}|\bm {Z}^{k-1}),\end{split}  \tag{24}\]  
where \(c\) is the normalization coefficient and of the form \(c=p(\mathbf{Z}_{k},\mathbf{Z}^{k-1})=\mathop{\sum}\limits_{l}^{H}p(\mathbf{Z}_{k},\mathbf{Z}^ {k-1}|\mathbf{b}_{l})p(\mathbf{b}_{l})\). For notational convenience to deduce \(p(\mathbf{Z}_{k}|\mathbf{b}_{l},\mathbf{Z}^{k-1})\) and \(p(\mathbf{b}_{l}|\mathbf{Z}^{k-1})\), two indicators are defined firstly. Target detection indicator  
\[\tau_{i}[\mathbf{b}_{l}]=\mathop{\sum}\limits_{j=1}^{M}\mathbf{b}_{l}^{i,j}\leq 1,i=1,...,N. \tag{25}\]  
False (unassociated) measurement indicator  
\[\varphi(\mathbf{b}_{l})=\mathop{\sum}\limits_{i=1}^{N}\left[1-\tau_{i}(\mathbf{b}_{l}) \right]=N-\mathop{\sum}\limits_{i=1}^{N}\tau_{i}(\mathbf{b}_{l}). \tag{26}\]  
From Equation (26), the number of undetected targets is \(\varphi(\mathbf{b}_{l})\), then  
\[p(\mathbf{b}_{l}|\varphi(\mathbf{b}_{l})=\{A_{N}^{N-\varphi(\mathbf{b}_{l})}\}^{-1}=\frac{ \varphi(\mathbf{b}_{l})!}{N!}, \tag{35}\]  
\[p(\varphi(\mathbf{b}_{l}))=p_{D}^{N-\varphi(\mathbf{b}_{l})}(1-p_{D})^{\varphi(\mathbf{b}_{ l})}. \tag{36}\]  
Substitute Equations (35) and (36) into Equation (34), then we can obtain  
\[p(\mathbf{b}_{l}|\mathbf{Z}^{k-1})=\frac{\phi(\mathbf{b}_{l})!}{N!}p_{D}^{N-\varphi(\mathbf{b} _{l})}(1-p_{D})^{\varphi(\mathbf{b}_{l})}. \tag{37}\]  
Substitute Equations (32), (33) and (37) into Equation (24), we have  
\[p(\mathbf{b}_{l}|\mathbf{Z}^{k})=\left\{ \begin{array}{ll}\frac{1}{c}\sum\limits_{i= 1}^{N}\left(\mathcal{N}[\mathbf{z}_{i}^{k};\mathbf{z}_{t_{i},k|k-1}^{k},\mathbf{S}_{it_{i}} ^{k}]\right)^{\tau_{i}[\mathbf{b}_{l}]}\frac{\varphi(\mathbf{b}_{l})!}{N!}p_{D}^{N- \varphi(\mathbf{b}_{l})}(1-p_{D})^{\varphi(\mathbf{b}_{l})},&\Delta>\alpha\\ \frac{1}{c}\sum\limits_{i=1}^{N}\left(\mathcal{N}[\mathbf{z}_{t_{i}}^{k};\mathbf{z}_{t _{i},k|k-1}^{k},\mathbf{S}_{it_{i}}^{k}]\cdot 10^{\varepsilon_{i}}\right)^{\tau_{i}[\mathbf{ b}_{l}]}\frac{\varphi(\mathbf{b}_{l})!}{N!}p_{D}^{N-\varphi(\mathbf{b}_{l})}(1-p_{D})^{ \varphi(\mathbf{b}_{l})},&\Delta\leq\alpha\end{array} \right. \tag{38}\]  
The marginal probability can be obtained  
\[p(b_{it_{i}}|\mathbf{Z}^{k})=\sum\limits_{\mathbf{b}}p(\mathbf{b}|\mathbf{Z}^{k})\hat{\omega}_ {it_{i}}(\mathbf{b}), \tag{39}\]  
where \(\hat{\omega}_{it_{i}}(\mathbf{b})=\left\{ \begin{array}{ll}1&if~{}~{}b_{jt}\in \mathbf{b}\\ 0&otherwise.\end{array} \right.\)  
At this point, the probability of each target associated with each measurement in Equation (39) is obtained by making full use of the structure information. This data association result mentioned above is integrated into LMB filter by revising the single target likelihood in Equation (15), the revised target likelihood is given below,  
\[g^{*}(z_{\theta}(\ell)|x,\ell)=g(z_{\theta}(\ell)|x,\ell)p(b_{\ell t_{\ell}}| \mathbf{Z}^{k}). \tag{40}\]  
The revised target likelihood \(g^{*}(z_{\theta}(\ell)|x,\ell)\) is integrated into the LMB filter by replacing \(g(z_{\theta}(\ell)|x,\ell)\), then JPGHM-LMB filter is obtained.