*2.1. Multitarget States*

As mentioned in Section 1, algorithms using non-labeled RFS cannot produce trajectories without using heuristic techniques [10]. The Labeled RFS framework, introduced in [11,41], is a principled approach to produce target tracks. Moreover, it is the only method that can produce trajectories from the filtering density [10]. In the labeled RFS frame work, a labeled target at time *k* is represented by a kinematic target state vector *xk* in state space X and its unique label *<sup>k</sup>* in the (discrete) label space L, and hence *x* = (*x*, -) <sup>∈</sup> <sup>X</sup> <sup>×</sup> <sup>L</sup>. This unique label is characterized by two parameters: time of target birth *τ* and the index of individual targets born at the same time *ρ*, i.e., *<sup>k</sup>* = (*τ*, *<sup>ρ</sup>*) <sup>∈</sup> <sup>L</sup> [11]. Hence, formally, a trajectory of each target is a sequence of consecutive labeled states with the same label [11]. Note that the label space for all targets born up to time *<sup>k</sup>* is the disjoint union <sup>L</sup>*<sup>k</sup>* <sup>=</sup> <sup>L</sup>*k*−<sup>1</sup> / B*<sup>k</sup>* where B*<sup>k</sup>* is the label space for targets born at time *<sup>k</sup>*, and <sup>L</sup>*k*−<sup>1</sup> is the label space of the targets born prior to time *k*. To distinguish the unlabeled states from labeled ones, the normal and bold letters (e.g., *x*, *X*, *x***,** *X*) are used, respectively. Suppose that at time *k*, there are *N* targets with corresponding states *xk*,1, ... , *xk*,*N*, then the multitarget state can be represented as follows:

$$\mathbf{X}\_k = \left\{ \mathbf{x}\_{k,1}, \dots, \mathbf{x}\_{k,N} \right\} \in \mathcal{F} \left( \mathbb{X} \times \mathbb{L}\_k \right) \tag{2}$$

**Definition 1.** *[11] Let* <sup>L</sup> : <sup>X</sup> <sup>×</sup> <sup>L</sup> <sup>→</sup> <sup>L</sup> *be the projection* <sup>L</sup>(*x*; -) = -*, and hence* <sup>L</sup>(*X*) = {L (*x*) : *<sup>x</sup>* <sup>∈</sup> *<sup>X</sup>*} *is the set of labels of <sup>X</sup>. A labeled RFS with space* <sup>X</sup> *and (discrete) label space* <sup>L</sup> *is an RFS on* <sup>X</sup> <sup>×</sup> <sup>L</sup> *such that each realization <sup>X</sup> has distinct labels, i.e.,* |L(*X*)<sup>|</sup> <sup>=</sup> <sup>|</sup>*X*|*.*

Since each target in a multitarget state has a distict label, *<sup>δ</sup>*|*X*|(|L(*X*)|) = 1, the distinct label indicator can be defined as follows [11]

$$
\Delta(X) \triangleq \delta\_{|X|}(|\mathcal{L}(X)|). \tag{3}
$$
