*4.1. Target State Estimation*

The purpose of multi-target tracking is to jointly estimate the target cardinality and target states based on the observations. Multi-target tracking can be transformed into the recursive Bayesian estimation problem by modeling the state and measurement of multi-target using RFS. The RFS approach can effectively deal with the uncertainty of data association between the target and the measurement and the state probability density function of the set of targets. We use *π<sup>k</sup>* (·| *Z*1:*k*) to indicate the RFS posterior probability density of multi-target state; *fk*|*k*−<sup>1</sup> (· |·) to represent the multi-target transition density; *gk* (· |·) to represent the likelihood function. The posterior probability density of multiple targets is recursively calculated by the following prediction and update steps [11–14]:

$$
\pi\_{k\mid k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k-1} \right) = \int f\_{k\mid k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{X}\_{k-1} \right) \pi\_{k-1} \left( \mathbf{X}\_{k-1} \, \middle| \, \mathbf{Z}\_{1:k-1} \right) \delta \mathbf{X}\_{k-1}
$$

$$
\pi\_k \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k} \right) = \frac{\oint\_{\mathcal{X}} \left( \mathbf{Z}\_k \, \middle| \, \mathbf{X}\_k \right) \pi\_{k\mid k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k-1} \right)}{\int \int\_{\mathcal{X}} \left( \mathbf{Z}\_k \, \middle| \, \mathbf{X}\_k \right) \pi\_{k\mid k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k-1} \right) \delta \mathbf{X}\_k \, \middle|} \right)
$$

where the set integral on <sup>F</sup> (<sup>X</sup> <sup>×</sup> <sup>L</sup>) <sup>→</sup> <sup>R</sup> is defined as:

$$\int f\left(\mathbf{X}\right)\delta X = \sum\_{i=0}^{\infty} \frac{1}{i!} \int f\left(\left\{\mathbf{x}\_{1\prime}, \cdots, \mathbf{x}\_{i}\right\}\right) d\left(\mathbf{x}\_{1\prime}, \cdots, \mathbf{x}\_{i}\right) \dots$$

All information about the multiple targets states are included in the multi-target posterior, for example, the number and state of the target at the current time.

We experience with *q* pairs of sensors, therefore the above update step and prediction step can be rewritten as [59–61]:

$$
\pi\_{k|k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k-1}^{[1:\mathbf{Q}]} \right) = \int f\_{k|k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{X}\_{k-1} \right) \pi\_{k-1} \left( \mathbf{X}\_{k-1} \, \middle| \, \mathbf{Z}\_{1:k-1}^{[1:\mathbf{Q}]} \right) \delta \mathbf{X}\_{k-1} \tag{24}
$$

$$\pi\_k \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k-1}^{[1:Q]} \right) = \frac{\prod\_{q=1}^Q \mathcal{g}\_k \left( \mathbf{Z}\_k^{[q]} \, \middle| \, \mathbf{X}\_k \right) \pi\_{k|k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k-1}^{[1:Q]} \right)}{\int \prod\_{q=1}^Q \mathcal{g}\_k \left( \mathbf{Z}\_k^{[q]} \, \middle| \, \mathbf{X}\_k \right) \pi\_{k|k-1} \left( \mathbf{X}\_k \, \middle| \, \mathbf{Z}\_{1:k-1}^{[1:Q]} \right) \delta \mathbf{X}\_k} . \tag{25}$$

The recursive process is not easy to deal with exactly due to the non-linear of the observation equation. The sequential Monte Carlo(SMC) methods are a viable approach to approximate the integrals of interest using random samples.
