*3.1. GLMB Filter*

**Definition 2.** *A GLMB density is a labeled multitarget density given as follows [11]*

$$
\pi \left( \mathbf{X} \right) = \Delta \left( \mathbf{X} \right) \sum\_{\boldsymbol{\varrho} \in \Xi, \boldsymbol{l} \subseteq \mathbf{L}} \sum\_{\boldsymbol{\varrho} \in \mathbb{L}, \boldsymbol{l} \subseteq \mathbf{L}} \omega^{\left( \boldsymbol{l}, \boldsymbol{\varrho} \right)} \delta\_{\boldsymbol{l}} \left[ \mathcal{L} \left( \mathbf{X} \right) \right] \left[ p^{\left( \boldsymbol{\varrho} \right)} \right]^{\mathbf{X}} \tag{21}
$$

*where the discrete space* Ξ *is the space of association map histories* Θ0:*<sup>k</sup>* **-** Θ<sup>0</sup> × ... × Θ*<sup>k</sup> , each* = (*θ*1:*k*) ∈ Ξ *represents a history of the (multisensor) positive 1-1 map, the weight ω*(*J*,) *and multitarget exponential* - *p*() *X satisfy*

$$\sum\_{\mathcal{G}\in\Xi} \sum\_{J\in\mathbb{L}} \omega^{(J,\mathfrak{e})} \delta\_{\mathfrak{l}} \left[ \mathcal{L}\left(\mathbf{X}\right) \right] = 1,\tag{22}$$

$$\int p^{(\varrho)}\left(\mathbf{x},\ell\right)d\mathbf{x} = 1.\tag{23}$$

Noting that, in Equation (21), while *<sup>ω</sup>*(*J*,) (<sup>L</sup> (*X*)) is a function of only the labels of the multitarget state *<sup>X</sup>*, whereas - *p*() *X* depends on entire set *X*.

The cardinality distribution *Pr* (|*X*<sup>|</sup> <sup>=</sup> *<sup>n</sup>*), existence probability *<sup>r</sup>* (-) and probability density *p* (*x*, -) of a track -<sup>∈</sup> <sup>L</sup> are given as follows [11]:

$$\Pr\left(|\mathbf{X}|=n\right) = \sum\_{\emptyset \in \Sigma} \sum\_{I \in \mathcal{L}} \delta\_{\mathbf{n}}\left[|I|\right] \omega^{\left(I,\varrho\right)}\tag{24}$$

$$r\left(\ell\right) = \sum\_{\emptyset \in \mathbb{Z}} \sum\_{f \in \mathbb{L}} \mathbf{1}\_f\left(\ell\right) \omega^{\left(l,\varrho\right)}\tag{25}$$

$$p\left(\mathbf{x},\ell\right) = \frac{1}{r\left(\ell\right)}\sum\_{\boldsymbol{\varrho}\in\mathfrak{L}}\sum\_{\boldsymbol{\varrho}\in\mathbb{L}}\mathbf{1}\_{\int}\left(\ell\right)\omega^{\left(\left[\ell,\boldsymbol{\varrho}\right)}p^{\left(\boldsymbol{\varrho}\right)}\left(\mathbf{x},\ell\right)\tag{26}$$
