*2.3. Standard Multitarget Observation Model*

Assuming that there are *M* sensors, each state (*x*, -) <sup>∈</sup> *<sup>X</sup>* can be either detected by sensor *s*,*s* = 1, ... *M* with probability of detection *P*(*s*) *<sup>D</sup>* (*x*, -) and generate an observation *<sup>z</sup>*(*s*) <sup>∈</sup> *<sup>Z</sup>*(*s*) with likelihood *g* (*s*) *D <sup>z</sup>*(*s*)|*x*, - , or being miss detected with probability 1 <sup>−</sup> *<sup>P</sup>*(*s*) *<sup>D</sup>* (*x*, -). The set of multitarget observations collected by the *sth*-sensor at time *k* is *Z*(*s*) *<sup>k</sup>* = *z* (*s*) <sup>1</sup> ,..., *z* (*s*) *M* ∈ F (Z), with <sup>Z</sup> being the observation space. Note that, the *<sup>s</sup>th*−sensor can also receive spurious measurements or false alarms at each time step. Let *D*(*s*)(*x*) be the set of measurements generated by target with state *x* at time *k*, the multitarget observation at the current time *k* is the superposition of all observations of detected targets modeled by multi-Bernoulli RFS, i.e., *D*(*s*) (*X*) = & *<sup>x</sup>*∈*<sup>X</sup> <sup>D</sup>*(*s*) (*x*) and the clutter modeled by either Poisson or i.i.d. clutter RFS *C*(*s*).

$$Z^{(s)} = D^{(s)}\left(X\right) \cup \mathbb{C}^{(s)}\tag{8}$$

The likelihood function of a multitarget state **X** for sensor *s* is given as follows [9],

$$\operatorname{g}^{(s)}(Z^{(s)}|X) \propto \sum\_{\theta^{(s)} \in \Theta^{(s)}} \mathbf{1}\_{\Theta^{(s)}(\mathcal{L}(X))} \left(\theta^{(s)}\right) \left[\mathbf{Y}\_{Z^{(s)}}^{\left(s,\theta^{(s)}(\mathcal{L}(X))\right)}(\mathbf{x})\right]^{X} \tag{9}$$

where <sup>Θ</sup>(*s*) is the set of positive association map *<sup>θ</sup>*(*s*) at time *<sup>k</sup>*, *<sup>θ</sup>*(*s*) : <sup>L</sup> → {0, 1, ... , <sup>|</sup>*Z*(*s*)|}, such that - *θ*(*s*) (*i*) = *θ*(*s*) (*j*) <sup>⇒</sup> [*<sup>i</sup>* <sup>=</sup> *<sup>j</sup>*] (i.e., each observation in *<sup>Z</sup>*(*s*) is assigned to at most one target, then each target has a distinct label), Θ(*s*) (*J*) is the subset of Θ(*s*)with domain *J*, and

$$Y\_{Z^{(s)}}^{(s,j)}(\mathbf{x}) = \begin{cases} \frac{P\_D^{(s)}(\mathbf{x}) g^{(s)}(z\_j^{(s)} | \mathbf{x})}{\kappa^{(s)}(z\_j^{(s)})}, & j = 1:M^{(s)} \\ 1 - P\_D^{(s)}(\mathbf{x}) & j = 0. \end{cases} \tag{10}$$

Using the assumption that the sensors are conditionally independent (More concisely, the sensors do not interfere or influence each other while taking measurements or detections. The measurement noise, missed detections and clutter from each sensor in a multitarget scenario are, therefore, independent from the others), and let us define the following abbreviations

$$Z \triangleq (Z^{(1)}, \dots, Z^{(M)}),\tag{11}$$

$$
\Theta \triangleq \Theta^{(1)} \times \dots \times \Theta^{(M)},\tag{12}
$$

$$
\Theta\left(\left|\right> \triangleq \Theta^{(1)}\left(\left|\right>\right) \times \dots \times \Theta^{(M)}\left(\left|\right>\right),\tag{13}
$$

$$\theta \triangleq (\theta^{(1)}, \dots, \theta^{(M)}), \tag{14}$$

$$1\_{\Theta(I)}\left(\theta\right) \triangleq \prod\_{s=1}^{M} 1\_{\Theta^{(s)}(I)}(\theta^{(s)}),\tag{15}$$

$$\mathbf{Y}\_{Z}^{\left(\boldsymbol{j}^{(1)},...,\boldsymbol{j}^{(M)}\right)}\left(\mathbf{x},\boldsymbol{\ell}\right) \triangleq \prod\_{s=1}^{M} \mathbf{Y}\_{Z^{\left(s\right)}}^{\left(\boldsymbol{s},\boldsymbol{j}^{\left(s\right)}\right)}\left(\mathbf{x},\boldsymbol{\ell}\right),\tag{16}$$

then, the multi-sensor likelihood is written as

$$\begin{split} \log \left( Z | \mathbf{X} \right) &= \prod\_{s=1}^{M} \mathbf{g}^{(s)} \left( Z^{(s)} | \mathbf{X} \right) \\ &\propto \sum\_{\theta \in \Theta} \mathbf{1}\_{\Theta(\mathcal{L}(X))} \left( \theta \right) \left[ \mathbf{Y}\_{Z}^{\left( \theta(\mathcal{L}(\mathbf{x})) \right)} \left( \mathbf{x} \right) \right]^{\mathbf{X}} . \end{split} \tag{17}$$

Obviously, the form of the multi-sensor likelihood *<sup>g</sup>* (*Z*|*X*) in (17) and that of it its single-sensor counterpart in (9) are identical.
