In order to demonstrate the performance of the proposed algorithm, we implement the proposed measurement partition method, the distance partition, and the K-means++ method under the framework of ET-GM-PHD filter [10]. The two examples are implemented on Dell computer with Intel(R) Core(TM) CPU i5 3470, 3.2 GHz and 4GB RAM (Dell Inc., Round Rock, TX, USA).

Two different metrics are used for evaluating the algorithm performance. The first is the statistics of the target number estimates, and the second is the optimal subpattern assignment (OSPA) distance [27], which is recently developed and defined as

{\overline{d}}_{p}^{\left(c\right)}\left(X,Y\right)={\left(\frac{1}{n}\left(\underset{\pi \in {\prod}_{n}}{\mathrm{min}}{\displaystyle \sum _{i=1}^{m}{d}^{\left(c\right)}{\left({\mathbf{x}}_{i},{\mathbf{y}}_{\pi \left(i\right)}\right)}^{p}+{c}^{p}\left(n-m\right)}\right)\right)}^{1/p}

(10)

where *X* = {**x**_{1}, ⋯, **x**_{
m
}} and *Y* = {**y**_{1}, ⋯, **y**_{
n
}} are arbitrary finite subsets, 1 ≤ *p* < ∞, *c* > 0, *m*, *n* ∈ *N*_{
o
} = {0, 1, 2, ⋯}. If *m* > *n*, then {\overline{d}}_{p}^{\left(c\right)}\left(X,Y\right)={\overline{d}}_{p}^{\left(c\right)}\left(Y,X\right). In the simulation, the parameters of OSPA distance are set as *p* = 2 and *c* = 60. 100 Monte Carlo runs are performed.

### 4.1 Example 1: crossing extended target tracking

Assume that *Χ*_{
k
} = [*x*_{
k
}, *y*_{
k
}, *v*_{x,k}, *v*_{y,k}]^{T} denotes the extended target state at time *k*, where (*x*_{
k
}, *y*_{
k
}) denotes the target position and (*v*_{x,k}, *v*_{y,k}) denotes the velocity. {z}_{k}^{\left(j\right)}=\left[{x}_{k}^{\left(j\right)},{y}_{k}^{\left(j\right)}\right] is the measurement of the *j* th sensor. Each extended target follows a linear Gaussian dynamical model, and sensor has a linear Gaussian measurement model, i.e.,

{f}_{k|k-1}\left({x}_{k}|{x}_{k-1}\right)=N\left({x}_{k};F{x}_{k-1},{Q}_{k-1}\right)

{g}_{k}\left({z}_{k}|{x}_{k}\right)=N\left({z}_{k};H{x}_{k},{R}_{k}\right)

where *N*(⋅; *m*, *P*) denotes a Gaussian density with mean *m* and covariance *P*. F=\left[\begin{array}{cc}\hfill {I}_{2}\hfill & \hfill \mathit{\Delta t}{I}_{2}\hfill \\ \hfill {0}_{2}\hfill & \hfill {I}_{2}\hfill \end{array}\right] is the state transition matrix, and {Q}_{k-1}={\sigma}_{v}^{2}\left[\begin{array}{cc}\hfill \frac{\Delta {t}^{4}}{4}{I}_{2}\hfill & \hfill \frac{\Delta {t}^{3}}{2}{I}_{2}\hfill \\ \hfill \frac{\Delta {t}^{3}}{2}{I}_{2}\hfill & \hfill \Delta {t}^{2}{I}_{2}\hfill \end{array}\right] is the process noise covariance, where *I*_{2} and 0_{2} denote the 2 × 2 identity and zero matrices, respectively. *σ*_{
v
} = 2 is the standard deviation of the process noise, and *Δt* = 1*s* is the sample interval. {H}_{k}=\left[\begin{array}{cc}\hfill {I}_{2}\hfill & \hfill {0}_{2}\hfill \end{array}\right] denotes the measurement matrix, {R}_{k}={\sigma}_{\u03f5}^{2}{I}_{2} is the covariance of the measurement noise, and *σ*_{
ϵ
} = 20 is the standard deviation of the measurement noise. Let the probabilities of target survival and detection be *P*_{
s
} = 0.99 and *P*_{
D
} = 0.99, respectively. The clutter is modeled as a Poisson RFS with the mean *λ* =10 over the observation space.

The birth intensity is described as

\begin{array}{ll}\phantom{\rule{1em}{0ex}}{D}_{b}\left(x\right)=& 0.1N\left(x;{m}_{b}^{\left(1\right)},{P}_{b}\right)+0.1N\left(x;{m}_{b}^{\left(2\right)},{P}_{b}\right)\\ +\phantom{\rule{0.25em}{0ex}}0.1N\left(x;{m}_{b}^{\left(3\right)},{P}_{b}\right)+0.1N\left(x;{m}_{b}^{\left(4\right)},{P}_{b}\right)\end{array}

where {m}_{b}^{\left(1\right)}={\left[\u2012800\phantom{\rule{0.25em}{0ex}}\u2012800\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0\right]}^{T}, {m}_{b}^{\left(2\right)}={\left[\u2012800\phantom{\rule{0.25em}{0ex}}\u2012300\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0\right]}^{T}, {m}_{b}^{\left(3\right)}={\left[\u2012492\phantom{\rule{0.25em}{0ex}}230\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0\right]}^{T}, {m}_{b}^{\left(4\right)}={\left[\phantom{\rule{0.25em}{0ex}}\u2012654\phantom{\rule{0.25em}{0ex}}409\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0\right]}^{T}, and *P*_{
b
} = *diag*([100, 100.25, 25]).Figure 2 shows the real tracks of the extended targets with cluttered measurements in x- and y-coordinates. Figure 3 shows the number estimate of the extended targets, and Figure 4 shows the OSPA distance. As can be seen, the proposed method has an accuracy similar to that of the distance partition method, but better than the K-means++ method, which is mainly because the K-means++ method is sensitive to the clutter and needs a good initial cluster center.

Figure 5 shows the partition number of the three different methods, and Figure 6 shows the average run time of the three methods. It is clear that the proposed method has the smallest partition number and least computational cost, followed by the distance partition method, and the K-means++ method has the biggest partition number and highest computational cost. The reason is that the clutter measurements are removed from the measurements by the density analysis technique, and the neighbor propagation technique is introduced to initially partition the measurements. However, for the K-means++ method, the way to set the k value is a problem, so the target number traversal technique is used. In this simulation, we also set *K* ∈ (*K*_{
L
}, *K*_{
U
}). Notice that the partition number sharply decreases at time 50 s; the reason is that the targets make a cross with each other, and thus the measurements mix and cannot be separated.

### 4.2 Example 2: close spaced extended targets tracking

Assume that two close spaced extended targets make a parallel motion 100 m apart in this scenario, and the birth intensity is described as {D}_{b}\left(x\right)=0.1N\left(x;{m}_{b}^{\left(1\right)},{P}_{b}\right)+0.1N\left(x;{m}_{b}^{\left(2\right)},{P}_{b}\right), where, {m}_{b}^{\left(1\right)}={\left[-800\phantom{\rule{0.25em}{0ex}}-600\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0\right]}^{T}, {m}_{b}^{\left(2\right)}={\left[-700\phantom{\rule{0.25em}{0ex}}-500\phantom{\rule{0.25em}{0ex}}0\phantom{\rule{0.25em}{0ex}}0\right]}^{T}, and *P*_{
b
} = *diag*([100, 100.25, 25]). The other parameters are the same as those of example 1.Figure 7 shows the real tracks of the extended targets with cluttered measurements in x- and y-coordinates. Figure 8 shows the number estimate of the extended targets, and Figure 9 shows the OSPA distance. Figure 10 shows the partition number of the three different methods, and Figure 11 shows the average run time of the three methods.

It is clear that the proposed method has the smallest partition number and least computational cost although its accuracy is similar to that of the distance partition and obviously better than that of the K-means++ method.