*4.2. Particle Filter Implementation*

Since the multi-target posterior probability density recursion requires the calculation of multi-set integral (24) and (25), its computational complexity is much larger than that of the single-target filtering process [16,61,62]. By the SMC method, the weighted particles can be estimated by recursive approximation to estimate the posterior probability density.

At the current time *k*, the particles are sampled by SMC, obtained from the spatial distribution of the target.

$$\bar{X}\_k^{(i)} \sim p\left(\cdot \middle| X\_{k-1'}^{(i)}, Z\_k\right) \tag{26}$$

 *ω*(*i*) *<sup>k</sup>*−1, *<sup>X</sup>*(*i*) *k*−1 *<sup>N</sup> <sup>i</sup>*=<sup>1</sup> represents the set of importance weighted particles at time *<sup>k</sup>* <sup>−</sup> 1 and the multi-target posterior probability density can be expressed as:

$$\text{Tr}\_{k-1\mid k-1} \left( X\_k \, \vert Z\_{1:k-1} \right) \approx \sum\_{i=1}^{N} \omega\_{k-1}^{(i)} \delta\_{X\_{k-1}^{(i)}} \left( X\_{k-1} \right) \tag{27}$$
