*2.1. The Labeled RFS*

Throughout this article, we adhere to the following notations. The set exponential is denoted as [*h*(·)]*<sup>X</sup>* <sup>=</sup> <sup>∏</sup>*x*∈*<sup>X</sup> <sup>h</sup>*(*x*) while the inner product notation is denoted as *<sup>f</sup>* , *<sup>g</sup>* <sup>=</sup> \$ *<sup>f</sup>*(*x*)*g*(*x*)*dx*. The generalization of the Kronecker delta is denoted as follows:

$$\delta\_Y(X) = \begin{cases} 1 & X = Y \\ 0 & X \neq Y \end{cases}$$

The set inclusion function is written as follows:

$$\mathbf{1}\_Y(X) = \begin{cases} 1 & \text{\$X \subseteq \$ Y\$} \\ 0 & \text{otherwise} \end{cases}$$

**X** denotes the labeled set of objects, while **x** = (*x*, *l*) denotes a single labeled object, specifically, *<sup>x</sup>* <sup>∈</sup> <sup>X</sup> and *<sup>l</sup>* <sup>∈</sup> <sup>L</sup>, where <sup>X</sup> and <sup>L</sup> are respectively the kinematic state space and the discrete labels space at the current time step. L is a label extraction function, i.e., L(**x**) = *l* and F(**X**) denote sets of finite subsets of **X**. The "+" sign is used to indicate the next time step when applicable.

The Finite-Set Statistics (FISST) integration is defined as follows [54]:

$$f(X)\delta X = \sum\_{i=0}^{\infty} \frac{1}{i!} \int\_{\mathbb{X}^i} f(\{x\_1, \ldots, x\_i\}) d(x\_1, \ldots, x\_i)$$

In multi-object tracking problem, the cardinality of object sets varies when objects enter or leave the surveillance region. As RFS is a random set of points in the sense that the number of points in the set is random and the points themselves are also random and unordered [54], a set of random objects can be naturally characterized as a RFS. Being introduced systematically for the first time in Reference [13], the labeled RFS incorporates the identities of elements into the unlabeled counterpart. Precisely, with the state space X and marks space L, the labeled RFS is a marked simple point process whereas each realization has a distinct label [13,15]. The distinct label property is satisfied when **X** has the same cardinality as its labels L(**X**). Given this, the distinct label indicator can be written as follows [16]:

$$\Delta(\mathbf{X}) = \delta\_{|\mathbf{X}|}(|\mathcal{L}(\mathbf{X})|) \tag{1}$$

The introduction of labeled RFS to the multi-object tracking problem allows direct estimation of trajectories which cannot be done previously with conventional RFS without a separate labeling scheme.
