## 2 Background and Problem Formulation
### PMBM Filter  
The PMBM filter is a representative method in the RFS framework, and the relevant contents of random finite sets can be found by referring to [39]. A PMBM filter divides all targets \(\mathbf{X}^{all}\) into two disjoint sets: undetected targets \(\mathbf{X}^{u}\), characterized by the Poisson point process (PPP), and detected targets \(\mathbf{X}^{d}\), described by the multi-Bernoulli mixture (MBM) process. This process can be expressed as follows:  
\[f(\mathbf{X}^{all})=\sum_{\mathbf{X}^{u}\in\mathbf{X}^{l}= \mathbf{X}^{ul}}f^{u}(\mathbf{X}^{u})f^{d}(\mathbf{X}^{d}), \tag{5}\] \[f^{u}(\mathbf{X}^{u})=e^{-\langle D^{u};l\rangle}\prod_{ \mathbf{X}\in\mathbf{X}^{u}}D^{u}(\mathbf{X}),\] (6) \[f^{d}(\mathbf{X}^{d})=\sum_{j\in\mathbb{J}}\mathcal{W}^{j}f^{j}( \mathbf{X}^{d}),\] (7) \[f^{j}(\mathbf{X}^{d})=\sum_{\uplus_{j\in\mathbb{J}}\mathbf{X}^{ l}=\mathbf{X}^{l}}\prod_{i\in\mathbb{J}}f^{j,i}\Big{(}\mathbf{X}^{i}\Big{)}, \tag{8}\]  
where \(f(\mathbf{X}^{all})\), \(f^{u}(\mathbf{X}^{u})\), and \(f^{d}(\mathbf{X}^{d})\) are the density functions of all objects, undetected objects, and detected objects, respectively. \(\uplus\) represents a disjoint union. \(\langle a;b\rangle\) denotes the inner product of \(a\) and \(b\). \(D^{u}\) is the intensity function of undetected targets. \(f^{d}(\mathbf{X}^{d})\) is an MBM density, \(\mathbb{J}\) is the number of multi-Bernoulli (MB) mixtures, and the \(j\)-th MB is denoted as \(f^{j}(\mathbf{X}^{d})\), with a weight of \(\mathcal{W}\). \(\mathbb{J}\) is the target index set of the \(j\)-th MB. \(f^{j,i}(\mathbf{X}^{i})\) is the density of target \(i\) in the \(j\)-th MB. It was proved in [40] that the PMBM filter satisfies the conjugation property, so only the parameters describing the PPP and MBM need to be updated during the PMBM filter's recursion. The PMBM filter's recursion consists of two steps: a prediction step and an update step. At time \(k\), suppose that the posterior probability density of the target can be expressed with the following parameters:  
\[D^{u},\{\mathcal{W}^{j},\{r^{j,i},f^{j,i}\}_{i\in\mathbb{J}}\},j\in\mathbb{J}, \tag{9}\]  
where \(D^{u}\) denotes the intensity function of undetected targets, \(r^{j,i}\) and \(f^{j,i}\) are the existence probability and probability density of the \(i\)-th Bernoulli in the \(j\)-th MB RFS, and \(\mathcal{W}^{j}\) is the weight of the \(j\)-th MB. After the prediction step, we can obtain the parameterized PMBM probability density function as follows:  
\[D^{u}_{+},\{\mathcal{W}^{j}_{+},\{r^{j,i}_{+},f^{j,i}_{+}\}_{i\in\mathbb{J}}\},j\in\mathbb{J}, \tag{10}\]where  
\[D_{+}^{u} =D^{b}+\langle D^{u};P_{s}f_{k|k-1}\rangle, \tag{11}\] \[r_{+}^{j,i} =\langle f^{j,i};P_{s}\rangle r^{j,i},\] (12) \[f_{+}^{j,i} =\frac{\left\langle f^{j,i};P_{s}f_{k|k-1}\right\rangle}{\langle f ^{j,i};P_{s}\rangle},\] (13) \[\mathcal{W}_{+}^{j} =\mathcal{W}^{j}. \tag{14}\]  
The subscripts in (10) indicate the results of the prediction step. \(D^{b}\) denotes the intensity function of birth targets. \(P_{s}\) is the survival probability. \(f_{k|k-1}\) is the target transition density. The updated posterior density of the target can then be expressed using the predicted posterior density above and the resulting measurements \(\mathbf{Y}\):  
\[f^{u}(\mathbf{X}^{u})=e^{-\langle D^{u};l\rangle}\prod_{ \mathbf{X}\in\mathbf{X}^{u}}D^{u}(\mathbf{X}),\]0 \[f^{u}(\mathbf{X}^{u})=e^{-\langle D^{u};l\rangle}\prod_{ \mathbf{X}\in\mathbf{X}^{u}}D^{u}(\mathbf{X}),\]1 (16) \[f^{u}(\mathbf{X}^{u})=e^{-\langle D^{u};l\rangle}\prod_{ \mathbf{X}\in\mathbf{X}^{u}}D^{u}(\mathbf{X}),\]2  
where \(\mathbb{J}_{+}\) is the number of MB terms in the posterior density obtained at the prediction step. \(\mathcal{A}^{j}\) is the whole data association space for the \(j\)-th predicted MB, while \(A\) is one of the data associations in \(\mathcal{A}^{j}\). \(C\) is a nonempty disjoint cell index set; see [40] for more details.