**5. Simulation**

In this section, the effectiveness of ET-GIW-PHD and anti-clutter ET-GIW-PHD were tested. Two scenarios with multiple targets were established. The surveillance area was set as [−1000 m, 1000 m] × [−<sup>1000</sup> m, 1000 <sup>m</sup>], then *ck* is 2.5 × <sup>10</sup>−<sup>7</sup> under the assumption that clutter is uniformly distributed over the surveillance area. We set totally 100 time steps and the sampling time is 1 s.

In the first scenario, four targets moving along different lines were generated:

$$\begin{aligned} \mathbf{x}\_0^{(1)} &= [-1000 \text{ m}, 1000 \text{ m}, 25 \text{ m/s}, -25 \text{ m/s}], t\_s^{(1)} = 5 \text{ s}, t\_\varepsilon^{(1)} = 45 \text{ s};\\ \mathbf{x}\_0^{(2)} &= [-1000 \text{ m}, -1000 \text{ m}, 25 \text{ m/s}, 25 \text{ m/s}], t\_s^{(2)} = 15 \text{ s}, t\_\varepsilon^{(2)} = 55 \text{ s};\\ \mathbf{x}\_0^{(3)} &= [1000 \text{ m}, -1000 \text{ m}, -25 \text{ m/s}, 25 \text{ m/s}], t\_s^{(3)} = 25 \text{ s}, t\_\varepsilon^{(3)} = 65 \text{ s};\\ \mathbf{x}\_0^{(4)} &= [1000 \text{ m}, 1000 \text{ m}, -25 \text{ m/s}, -25 \text{ m/s}], t\_s^{(4)} = 35 \text{ s}, t\_\varepsilon^{(4)} = 75 \text{ s}; \end{aligned} \tag{29}$$

where *x* (*j*) <sup>0</sup> is the initial state of jth target, *t* (*j*) *<sup>s</sup>* is the born time of jth target, *t* (*j*) *<sup>e</sup>* is the end time of jth target. The birth intensity in the first scenario is

$$D\_b(\mathfrak{F}) = \sum\_{j=1}^{4} w\_b \mathcal{N}(\mathbf{x}; \mathbf{x}\_0^{(j)}, \mathbf{P}\_b \otimes \mathbf{X}\_k) \mathcal{D} \mathcal{W}(\mathbf{X}\_k; v\_{b\prime}, \mathbf{V}\_b)\_{\prime} \tag{30}$$

where *wb* = 0.03, **P***<sup>b</sup>* = diag([100, 100]), *vb* = 10, **V***<sup>b</sup>* = diag([100, 100]).

In the second scenario, two targets were born at (−1000 m, 300 m) and (−1000 m, −300 m), respectively at *k* = 0 (*k* is time step). Next, they moved close gradually and then moved in parallel before separating. The birth intensity in the second scenario is

$$D\_b(\mathfrak{F}) = \sum\_{j=1}^{2} w\_b \mathcal{N}(\mathbf{x}; \mathfrak{m}\_j, \mathbf{P}\_b \otimes \mathfrak{X}\_k) \mathcal{T} \mathcal{W}(\mathbf{X}\_k; v\_{b\prime}, \mathbf{V}\_b)\_{\prime} \tag{31}$$

where *m*<sup>1</sup> = [−1000, 300, 25, −25], *m*<sup>2</sup> = [−1000, −300, 25, 25], *wb* = 0.03, **P***<sup>b</sup>* = diag([100, 100]), *vb* = 10, **V***<sup>b</sup>* = diag([100, 100]). The true trajectories of two scenario are shown in Figure 1.

**Figure 1.** True trajectories of two scenarios: (**a**) four targets move along different lines in scenario 1. (**b**) Two targets move closeer gradually and then move in parallel before separating in scenario 2.

The dynamic and measurement model are shown below. The target kinematic state is denoted as *x* = [*rx*,*ry*,*r*˙*x*,*r*˙*y*], where *rx* and *r*˙*<sup>x</sup>* is the position and velocity in the *x* direction, likewise of *y* direction. The time evolution of kinematic state given by

$$\mathbf{x}\_{k}^{(j)} = \mathbf{F}\_{k}^{(j)} \mathbf{x}\_{k-1}^{(j)} + \mathbf{w}\_{k}^{(j)} \, \, \, \tag{32}$$

*Sensors* **2019**, *19*, 5140

where *x* (*j*) *<sup>k</sup>* is the target state of jth target at time *<sup>k</sup>*, *<sup>w</sup><sup>k</sup>* is the process noise of jth target and is the Gaussian white noise with zero mean and covariance **<sup>Q</sup>**(*j*) *<sup>k</sup>* , **<sup>F</sup>**(*j*) *<sup>k</sup>* is the kinematic state transition matrix of jth target, given by

$$\mathbf{F}\_{k}^{(j)} = \begin{bmatrix} 1 & 0 & t & 0 \\ 0 & 1 & 0 & t \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix}, \mathbf{Q}\_{k}^{(j)} = \Omega^{2} \begin{bmatrix} \frac{t^{2}}{2} \\ \frac{t^{2}}{4} \end{bmatrix} \begin{bmatrix} \frac{t^{2}}{2} & t \end{bmatrix},\tag{33}$$

where *t* is the sampling time and Ω represents the acceleration error, *t* = 1 s and Ω = 5 m/s2 in this simulation.

In this simulation, the major and minor axes are 20 m and 15 m respectively for all targets. The major axis was aligned with the direction of motion of the target and the extent of these targets remained unchanged.

The measurement model can be expressed as

$$\mathbf{z}\_{k}^{(j)} = (\mathbf{H}\_{k} \odot \mathbf{I}\_{d})\mathbf{x}\_{k}^{(j)} + \mathbf{q}\_{k'} \tag{34}$$

where *z* (*j*) *<sup>k</sup>* is the measurements generated by the jth target at time *<sup>k</sup>*, *<sup>q</sup><sup>k</sup>* is the measurement noise and is the Gaussian white noise with zero mean and covariance **R***k*, **H***<sup>k</sup>* ⊗ **I***<sup>d</sup>* is the observation matrix, given by

$$\mathbf{H}\_k \otimes \mathbf{I}\_d = \left[ \begin{array}{cccc} 1 & 0 & 0 & 0 \\ & 0 & 1 & 0 \\ \end{array} \right], \mathbf{R}\_k = \left[ \begin{array}{cccc} 1 & 0 \\ 0 & 1 \\ \end{array} \right]. \tag{35}$$

where **H***<sup>k</sup>* = [1, 0], **I***<sup>d</sup>* = 1 0 0 1 .

In our experiment, the confidence coefficient *α* of anti-clutter ET-GIW-PHD is set to 0.99. A distance partition algorithm [17] is used for both filters, a measurement partition that contains several cells can be obtained for a given distance threshold. Clutter Poisson rate *λ<sup>c</sup>* is set to 35, then clutter density *<sup>λ</sup>cck* is 8.75 × <sup>10</sup>−<sup>6</sup> (the clutter density in this paper is higher than that of related references, such as [18,21]). The expected number of measurements generated by targets *γ* is set to 15. The probability of survival *ps* and the detection probability *pD* are assumed to be state independent and set to 0.99 and 0.98, respectively. The probability *pd* is set to 0.99.

Tracking results are evaluated using the optimal subpattern assignment metric (OSPA) [43], which is widely used to evaluate multiple-target tracking performance [39–42].

The OSPA distance is defined by

$$d\_p^c(\varkappa\_k, \mathfrak{H}\_k) = \left(\frac{1}{n} \left(\min\_{\pi \in \Pi\_n} \sum\_{i=1}^m d^c(\mathfrak{x}\_i, \mathfrak{x}\_{\pi(i)})^p + \mathbb{C}^p(n-m)\right)\right)^{1/p},\tag{36}$$

where *<sup>m</sup>* <sup>&</sup>lt; *<sup>n</sup>*, <sup>κ</sup>*<sup>k</sup>* <sup>=</sup> {*<sup>x</sup>* (1) *<sup>k</sup>* , *<sup>x</sup>* (2) *<sup>k</sup>* , ..., *<sup>x</sup>* (*m*) *<sup>k</sup>* }is the true RFS at time k, <sup>κ</sup><sup>ˆ</sup> <sup>=</sup> {*x*<sup>ˆ</sup> (1) *<sup>k</sup>* , *<sup>x</sup>*<sup>ˆ</sup> (2) *<sup>k</sup>* , ..., *<sup>x</sup>*<sup>ˆ</sup> (*n*) *<sup>k</sup>* } is the estimated RFS, <sup>∏</sup>*<sup>n</sup>* is the assignment results which assign <sup>κ</sup> to <sup>κ</sup><sup>ˆ</sup> , *<sup>p</sup>* means *<sup>p</sup>* <sup>−</sup> *norm*, *<sup>c</sup>* is the penalty cost for cardinality mismatch. In this simulation, *c* = 60 and *p* = 2.

ET-GIW-PHD and anti-clutter ET-GIW-PHD are applied to two scenarios mentioned above for performance evaluation. The trajectories generated by these two methods are presented in Figures 2 and 3.

**Figure 2.** The obtained trajectories of ET-GIW-PHD and anti-clutter ET-GIW-PHD in scenario 1: (**a**) ET-GIW-PHD. (**b**) Anti-clutter ET-GIW-PHD.

**Figure 3.** The obtained trajectories of ET-GIW-PHD and anti-clutter ET-GIW-PHD in scenario 2: (**a**) ET-GIW-PHD. (**b**) Anti-clutter ET-GIW-PHD.

From Figures 2 and 3 we can see that the trajectories of anti-clutter ET-GIW-PHD are almost identical to the true trajectories. Note that, in the results of ET-GIW-PHD, some peices of clutter are incorrectly considered as targets. However, our anti-clutter ET-GIW-PHD can deal with the clutter more correctly and achieves better performance.

To further verify the analysis in Section 3, the calculation of *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* in Equation (9) at *<sup>k</sup>* <sup>=</sup> <sup>40</sup> (*k* is time step) in scenario 1 is shown below. The partition result at *k* = 40 is given firstly in Figure 4.

**Figure 4.** The partition result at *k* = 40 in scenario 1.

From Figure 4 we can see that the measurements of four targets are correctly clustered, and two clutter (marked with arrows in Figure 4) are incorrectly partitioned into one cell.

*e*−*γ*(*j*) *γ*(*j*) *βFA*,*<sup>k</sup>* <sup>|</sup>*W*<sup>|</sup> *p* (*j*) *<sup>D</sup>* <sup>Λ</sup>(*j*,*W*) *<sup>k</sup> <sup>w</sup>*(*j*) *<sup>k</sup>*|*k*−<sup>1</sup> is denoted as *<sup>ψ</sup>j*,*<sup>W</sup>* for jth GIW component in the Wth cell, then

$$w\_{k|k}^{(j, \mathcal{W})} = \frac{\Psi\_{j, \mathcal{W}}}{\delta\_{|\mathcal{W}|, 1} + \sum\_{l=1}^{f\_{k|k-1}} \Psi\_{l, \mathcal{W}}}. \tag{37}$$

From simulation results, the number of components of predicted PHD is 14 at *k* = 40, then *Jk*|*k*−<sup>1</sup> = 14, *<sup>ψ</sup>j*,*<sup>W</sup>* of the clutter cell (marked with arrows in Figure 4) is obtained and shown in Table 4.

The likelihood Λ(*j*,*W*) *<sup>k</sup>* of each GIW component in this cell is very small since clutter does not obey the kinematic and extent model of target, therefore *ψj*,*<sup>W</sup>* achieve small value as shown in Table 4.


**Table 4.** The *ψj*,*<sup>W</sup>* of the clutter cell.

Because the number of measurement in this cell is two, Equation (37) is represent as

$$w\_{k|k}^{(j, \mathcal{W})} = \frac{\Psi\_{j, \mathcal{W}}}{\sum\_{l=1}^{k|k-1} \Psi\_{l, \mathcal{W}}}.\tag{38}$$

Equation (38) is a normalization process, *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* of the clutter cell is shown in Table 5.


**Table 5.** *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* of the clutter cell.

Although *<sup>ψ</sup>j*,*<sup>W</sup>* is small, *<sup>w</sup>*(*j*,*W*) *<sup>k</sup>*|*<sup>k</sup>* may achieve a large value (*w*(10,*W*) *<sup>k</sup>*|*<sup>k</sup>* <sup>=</sup> 0.99) and results in a ghost target. At *k* = 40, the estimated number of targets was 5 while true number is 4. That is, the number of targets was overestimated.

To test the influence of the clutter density on tracking performance, ET-GIW-PHD and anti-clutter ET-GIW-PHD were tested under different numbers of clutter modeled as Poisson distribution with Poisson rate *λc*. The clutter measurements are assumed to be uniformly distributed over the surveillance area. The OSPA distance of these two filters under different Poisson rate *λ<sup>c</sup>* is shown in Figures 5 and 6.

As we can see from Figures 5 and 6, when *λ<sup>c</sup>* is small, ET-GIW-PHD achieves good performance. However, as *λ<sup>c</sup>* increases, the performance of ET-GIW-PHD degrades significantly. In contrast, our anti-clutter ET-GIW-PHD achieves superior performance with varying *λc*, which demonstrates that anti-clutter ET-GIW-PHD is more robust to clutter than ET-GIW-PHD. The results of cardinality estimation are shown in Figures 7 and 8.

**Figure 5.** The optimal subpattern assignment metric (OSPA) distance of ET-GIW-PHD and anti-clutter ET-GIW-PHD under different Poisson rate of clutter in scenario 1: (**a**) Poisson rate *λ<sup>c</sup>* = 5. (**b**) Poisson rate *λ<sup>c</sup>* = 20. (**c**) Poisson rate *λ<sup>c</sup>* = 35. (**d**) Poisson rate *λ<sup>c</sup>* = 50.

**Figure 6.** The OSPA distance of ET-GIW-PHD and anti-clutter ET-GIW-PHD under different Poisson rate of clutter in scenario 2: (**a**) Poisson rate *λ<sup>c</sup>* = 5. (**b**) Poisson rate *λ<sup>c</sup>* = 20. (**c**) Poisson rate *λ<sup>c</sup>* = 35. (**d**) Poisson rate *λ<sup>c</sup>* = 50.

**Figure 7.** The cardinality estimation of ET-GIW-PHD and anti-clutter ET-GIW-PHD under different Poisson rate of clutter in scenario 1: (**a**) Poisson rate *λ<sup>c</sup>* = 5. (**b**) Poisson rate *λ<sup>c</sup>* = 20. (**c**) Poisson rate *λ<sup>c</sup>* = 35. (**d**) Poisson rate *λ<sup>c</sup>* = 50.

**Figure 8.** The cardinality estimation of ET-GIW-PHD and anti-clutter ET-GIW-PHD under different Poisson rate of clutter in scenario 2: (**a**) Poisson rate *λ<sup>c</sup>* = 5. (**b**) Poisson rate *λ<sup>c</sup>* = 20. (**c**) Poisson rate *λ<sup>c</sup>* = 35. (**d**) Poisson rate *λ<sup>c</sup>* = 50.

From Figures 7 and 8 we can see that the cardinality estimation error of ET-GIW-PHD increases as the *λ<sup>c</sup>* grows. That is, ET-GIW-PHD cannot avoid the overestimation of cardinality under high clutter density. When clutter density is small, the clutter spreads apart. Thus, it is unlikely to partition more than one clutter into one cell. In the presence of severe clutter, the probability that multiple clutter being partitioned into one cell increases, and thus ET-GIW-PHD could overestimate the cardinality. However, our anti-clutter ET-GIW-PHD uses not only the number of measurement, but also target state and spatial distribution of clutter for better cardinality estimation performance. Using hypothesis testing, the measurements can be distinguished more correctly. Therefore, a better tracking performance can be achieved. Extensive experiments have demonstrated the effectiveness of anti-clutter ET-GIW-PHD.
