*2.2. Standard Multitarget Dynamic Model*

Given a multitarget state *X<sup>k</sup>* at time *k*, each state (*xk*, *<sup>k</sup>*) <sup>∈</sup> *<sup>X</sup><sup>k</sup>* can either exist with probability *PS*,*k*+1|*<sup>k</sup>* (*xk*) and evolve to a new state *<sup>x</sup>k*+<sup>1</sup> at next time step *<sup>k</sup>* <sup>+</sup> 1 with probability density *fk*<sup>+</sup>1|*<sup>k</sup>* (*xk*<sup>+</sup>1|*xk*, *<sup>k</sup>*) *δ<sup>k</sup>* (*<sup>k</sup>*+1) or disappear with probability 1 <sup>−</sup> *PS*,*k*+1|*<sup>k</sup>* (*xk*). Let *<sup>S</sup>k*+1|*k*(*x*) be the labeled Bernoulli RFS of the surviving target with state *x* from time *k* to time *k* + 1 and *Bk*+<sup>1</sup> be the labeled multi-Bernoulli RFS of the new-born targets at time *k* + 1, then the multitarget state *Xk*+<sup>1</sup> is the union of the surviving targets and the new-born ones,

$$\mathbf{X}\_{k+1} = \bigcup\_{\mathbf{x}\_{k} \in \mathbf{X}\_{k}} \mathbf{S}\_{k+1|k}(\mathbf{x}\_{k}) \cup \mathbf{B}\_{k+1'} \tag{4}$$

Following the convention in [9], in this work, the set *Bk*+<sup>1</sup> is distributed according to the labeled multi-Bernoulli (LMB) density. Furthermore, for simplicity, the subscript *k* for the current time is omitted, and the next time step *k* + 1 is indicated by the subscript - +- .

Assuming that the appearance, disappearance, and movement of each target are independent of the others, the multitarget transition density (The Mahler's Finite Set Statistics (FISST) notion of density is used in this paper for consistency with the probability density [42]) is [11,41]

$$f\left(\mathbf{X}\_{+}|\mathbf{X}\right) = f\_{\mathbb{S}}(\mathbf{X}\_{+} \cap (\mathbb{X} \times \mathbb{L})|\mathbf{X}) f\_{\mathbb{B},+}(\mathbf{X}\_{+} - (\mathbb{X} \times \mathbb{L})) \tag{5}$$

in which the distribution of new-born targets is given by

$$f\_{\mathbb{B},+}\left(\mathbb{B}\_{+}\right) = \Delta\left(\mathbb{B}\_{+}\right) \left[1\_{\mathbb{B}\_{+}}r\_{\mathbb{B},+}\right]^{\mathcal{L}(\mathbb{B}\_{+})} \left[1 - r\_{\mathbb{B},+}\right]^{\mathbb{B}\_{+} - \mathcal{L}(\mathbb{B}\_{+})} p\_{\mathbb{B},+}^{\mathbb{B}\_{+}}\tag{6}$$

where *rB*,+(-) is the birth probability of new target with new-born label -, and *pB*,+(·; -) is the distribution of its kinematic state [11]. The distribution of the survival targets is

$$f\_{\mathbb{S},+}\left(\mathbb{S}|X\right) = \Delta\left(\mathbb{S}\right)\Delta\left(X\right)1\_{\mathcal{L}\left(X\right)}\left(\mathcal{L}\left(\mathbb{S}\right)\right)\left[\mathbb{Y}\left(\mathbb{S};\cdot\right)\right]^{X}\tag{7}$$

$$\mathbb{Y}\left(\mathbb{S};x,\ell\right) = \sum\_{(\mathbf{x}\_{+},\ell\_{+}) \in \mathbb{S}} \delta\_{\ell}(\ell\_{+})P\_{\mathbb{S}}(\mathbf{x},\ell)f\_{\mathbb{S}}\left(\mathbf{x}\_{+}|\mathbf{x},\ell\right) + \left(1-1\_{\mathcal{L}\left(\mathbb{S}\right)}(\ell)\right)\left(1-P\_{\mathbb{S}}(\mathbf{x},\ell)\right).$$
