A Multi-Active and Multi-Passive Sensor Fusion Algorithm for Multi-Target Tracking in Dense Group Clutter Environments

Abstract

Multi-target tracking (MTT) of multi-active and multi-passive sensor (MAMPS) systems in dense group clutter environments is facing significant challenges in measurement fusion. Due to the difference in measurement information characteristics in MAMPS fusion, it is difficult to effectively correlate and fuse different types of sensors’ measurements, leading to difficulty in taking full advantage of various types of sensors to improve target tracking accuracy. To this end, we present a novel MAMPS fusion algorithm, which is based on centralized measurement association fusion (MAF) and distributed deep neural network (DNN) track fusion, named the MAMPS-MAF-DNN algorithm. Firstly, to reduce the impact of the dense group clutter, a clutter pre-processing algorithm is elaborated, which combines the advantages of the CFDP (cluster by finding density peaks) and double threshold screening algorithms. Then, for the single-active and multi-passive sensor (SAMPS) system, a centralized MAF algorithm based on angle information is developed, called the SAMPS-MAF algorithm. Finally, the SAMPS-MAF algorithm is extended to the MAMPS system within the DNN framework, and the complete MAMPS-MAF-DNN algorithm is proposed. Experimental results indicate that, compared to the existing MAF and covariance intersection (CI) fusion algorithms, the proposed MAMPS-MAF-DNN algorithm can fully combine the advantages of multi-active and multi-passive sensors, efficiently reduce the computational complexity, and obviously improve the tracking accuracy.

1. Introduction

Multi-target tracking (MTT) algorithms appearing after target detection [1,2] are used to handle measurement uncertainties and estimate the target’s position, velocity, acceleration [3], etc. The classical MTT algorithms are mainly based on data association (DA), such as global nearest neighbor (GNN) [4], joint probabilistic data association (JPDA) [5], multiple hypothesis tracking (MHT) [6], etc. In recent years, in order to avoid DA, multi-target tracking algorithms based on random finite sets (RFS) [7] have been developed, such as probability hypothesis density (PHD) [8], cardinalized PHD (CPHD) [9], multi-target multi-Bernoulli (MeMBer) [10] and its improved version (cardinality-balanced MeMBer (CBMeMBer) [11]), generalized labeled multi-Bernoulli (GLMB) [12] and its simplified version (labeled multi-Bernoulli (LMB) [13]), Poisson multi-Bernoulli mixture (PMBM) [14] filters, etc. Additionally, in order to further enhance the tracking performance of RFS filtering algorithms, several innovative algorithms [15,16,17,18,19] have been proposed in our early work.

In general, the MTT algorithms mentioned above can achieve the effect of tracking multiple targets. However, these algorithms are not suitable for complex scenarios with high dense group clutter [20,21]. Compared to dense clutter, dense group clutter is a range of clutter distributed in a specific spatial area [22], such as forests, clouds, flocks, drones, etc., which can lead to a significant amount of computational complexity in data association. Moreover, when targets enter dense group clutter, it is difficult to separate them, which directly affects the tracking accuracy. Hence, to enhance the tracking accuracy, dense group clutter should be treated separately before filtering.

Recently, with the emergence of various types of sensors, multi-sensor fusion technology [23] has been widely applied in both military and civilian fields [24]. Meanwhile, multi-sensor MTT algorithms based on RFS have been proposed and mainly applied to the centralized fusion system. For the multi-sensor MTT based on RFS, Mahler proposed the general PHD (G-PHD) filter [25] but did not present its specific implementation. Subsequently, Nannuru et al. derived the general form of the G-PHD filter [26] and also gave its Gaussian mixture (GM) implementation. However, as the number of sensors and measurements increases, its computational complexity increases rapidly. Furthermore, Mahler also developed an approximate multi-sensor PHD filter, i.e., the iterated-corrector PHD (IC-PHD) [27] filter. It can avoid DA by using multiple single-sensor PHD filters in multi-sensor fusion, but its tracking accuracy is affected by the order of sensors. Subsequently, the multi-sensor joint GLMB (JGLMB) and multi-sensor LMB were proposed with good results [12].

Those multi-sensor MTT algorithms mentioned above can solve the multi-target measurement fusion problem of the same type of sensors. However, they are not suitable for different types of sensors, such as active and passive sensors [28,29,30,31]. Active sensors mainly include ultrasonic sensors, active radar, LiDAR, etc., and can obtain high-dimensional measurements, including position and angle information for targets, while passive sensors mainly include infrared sensors, passive radar, sonar, etc. and can only obtain lower-dimensional measurements, including angle information for targets [32,33,34]. There are differences in the dimensions and information properties of measurements obtained from different types of sensors, which can cause measurement association and fusion to become bigger obstacles to those algorithms.

In order to solve the above multi-source sensor fusion problem, for the MAMPS system, inspired by the potential of DNN and MAF theory, this work develops a hybrid fusion algorithm based on the centralized MAF and distributed DNN track fusion, called the MAMPS-MAF-DNN algorithm, which is an extension our previous work [18]. Compared to the algorithm presented by [18], the proposed MAMPS-MAF-DNN algorithm in this work provides a more detailed version and adds innovative double threshold screening and centralized MAF algorithms. In addition, the innovative SAMPS-MAF algorithm is extended to the MAMPS system within the DNN framework, and the complete MAMPS-MAF-DNN algorithm is proposed in this work. The main contributions of our work are summarized as follows.

• A pre-processing of clutter is carried out by combining the CFDP and double threshold screening algorithms before measurement fusion to eliminate the effect of the dense group clutter.
• A MAF algorithm for every SAMPS system, i.e., the SAMPS-MAF algorithm, is presented to achieve the MAMPS-MAF-DNN algorithm, which includes the following three steps. Firstly, the measurements’ initial and secondary screenings, based on the extended Kalman filter (EKF) and statistics, are designed using the angle information of measurements that both two types of sensors have, leading to shrinkage of the measurement association group. Secondly, measurement fusion based on angle information is completed via the least squares (LS) algorithm [35], obtaining plenty of measurement points. Thirdly, these measurement points are corrected by using the unique position measurements of the active sensor, which can eliminate the wrong ones and obtain the final measurement points available for filter tracking.
• MAMPS track fusion based on DNN is proposed, which can be considered as a distributed fusion of track results from multiple SAMPS systems. The commonly used track fusion algorithms are covariance intersection (CI) fusion and covariance union (CU) fusion [36,37,38,39,40], which are essentially a weighted fusion of the local track. Based on the powerful feature extraction ability of the DNN, in this work, we adopt it for track fusion to achieve the purpose of training each local track weight.

This paper is organized as follows. First of all, the necessary background knowledge about target tracking and DNN is reviewed in Section 2. Then, Section 3 presents the pre-processing of clutter, and the SAMPS-MAF algorithm is proposed in Section 4. For the MAMPS system, the MAMPS-MAF-DNN algorithm is developed in Section 5. Finally, simulation results and theoretical analysis are performed in Section 6, and Section 7 concludes this work.

2. Background

2.1. Target Motion Model

The target motion model can be depicted by a discrete-time model. In the state space, the state transition equation is modeled as

$${\mathit{x}}_k=f_{k|k-1}({\mathit{x}}_{k-1},{\mathit{w}}_{k-1})$$

(1)

where ${\mathit{x}}_k$ is the state vector at time $k$, $f_{k|k-1}(\cdot )$ is the state transition function, and ${\mathit{w}}_{k-1}$ represents zero mean Gaussian process noise with the covariance ${\mathit{Q}}_{k-1}$.

At time $k$, the target state can be denoted as ${\mathit{x}}_k={[x_k,v_{x,k},y_k,v_{y,k},z_k,v_{z,k},{\omega }_k]}^T$ in the constant turn (CT) model, where $x_k$, $y_k$, and $z_k$ represent the position of a target; $v_{x,k}$, $v_{y,k}$, and $v_{z,k}$ represent the corresponding velocity; and ${\omega }_k$ is the constant angular velocity. Thus, under the non-linear assumption, the state transition matrix of the CT model can be written as

$$\mathit{F}=\left[\begin{array}{ccccccc}1& {\displaystyle \frac{\sin {\omega }_{k}T}{{\omega }_k}}& 0& -{\displaystyle \frac{1-\cos {\omega }_{k}T}{{\omega }_k}}& 0& 0& 0\\ 0& \cos {\omega }_{k}T& 0& -\sin {\omega }_{k}T& 0& 0& 0\\ 0& {\displaystyle \frac{1-\cos {\omega }_{k}T}{{\omega }_k}}& 1& {\displaystyle \frac{\sin {\omega }_{k}T}{{\omega }_k}}& 0& 0& 0\\ 0& \sin {\omega }_{k}T& 0& \cos {\omega }_{k}T& 0& 0& 0\\ 0& 0& 0& 0& 1& T& 0\\ 0& 0& 0& 0& 0& 1& 0\\ 0& 0& 0& 0& 0& 0& 1\end{array}\right]$$

(2)

where $T$ is the sampling interval.

It is worth noting that this work mainly studies maneuvering target tracking, where what differs from the CT model is that ${\omega }_k$ is not a constant any longer.

2.2. Sensor Observation Model

In this work, the observation equation is modeled as

$${\mathit{z}}_k=h_k({\mathit{x}}_k,{\mathit{v}}_k)$$

(3)

where $h_k(\cdot )$ is the observation function, ${\mathit{v}}_k$ is zero-mean Gaussian observation noise with the covariance ${\mathit{R}}_k$, and ${\mathit{z}}_k$, which refers to the measurement vector at time $k$, has different forms for different types of sensors.

Active radar can receive and transmit signals in general, so it can get a complete measurement that includes the pitch angle $\beta$, azimuth angle $\alpha$, and position coordinates. Therefore, at time $k$, the measurement obtained by active radar can be expressed as ${\mathit{z}}_k={[{\alpha }_k,{\beta }_k,x_k,y_k,z_k]}^T$.

Compared to active radar, passive radar can only receive signals reflected from targets, so it can only get the measurement that includes the azimuth angle $\alpha$ and pitch angle $\beta$. Therefore, at time $k$, the measurement obtained by passive radar can be expressed as ${\mathit{z}}_k={[{\alpha }_k,{\beta }_k]}^T$.

For all active and passive sensors, we define ${\tilde{x}}_i$, ${\tilde{y}}_i$ and ${\tilde{z}}_i$ as the 3-dimensional (3D) position coordinates of the $i^{th}$ sensor $S_i$, then $\alpha$ and $\beta$ can be calculated by

$$\alpha =\arctan \left({\displaystyle \frac{y-{\tilde{y}}_i}{x-{\tilde{x}}_i}}\right)+v_{\alpha }$$

(4)

$$\beta =\arctan \left({\displaystyle \frac{z-{\tilde{z}}_i}{\sqrt{{(x-{\tilde{x}}_i)}^2+{(y-{\tilde{y}}_i)}^2}}}\right)+v_{\beta }$$

(5)

where $x$, $y$, and $z$ are the 3D position coordinates of the target; and $v_{\alpha }$ and $v_{\beta }$ are observation noise of the azimuth angle $\alpha$ and pitch angle $\beta$, respectively.

3. Pre-Processing for Clutter

In the actual tracking scenario, dense group clutter has the characteristics of being dense in the middle and sparse elsewhere. Therefore, this type of clutter can be removed via the clustering algorithm. In this section, in order to obtain the cluster’s center of clutter, the CFDP algorithm is first introduced in Section 3.1. Then, according to the cluster centers obtained, we establish double thresholds to screen dense group clutter in Section 3.2.

3.1. CFDP

In MTT algorithms based on RFS, the measurement set ${\mathit{Z}}_k$ received by the sensor can be expressed as

$${\mathit{Z}}_k={\mathit{K}}_k\cup \left(\underset{{\mathit{x}}_k\in {\mathit{X}}_k}{\cup }{\mathbf{\Theta }}_k({\mathit{x}}_k)\right)$$

(6)

where ${\mathit{K}}_k$ and ${\mathbf{\Theta }}_k$ are measurement sets from clutter and real targets, respectively. The actual model form of ${\mathit{K}}_k$ depends on the specific clutter characteristics and is usually modeled as a Poisson RFS with intensity ${\kappa }_k(z)={\lambda }_{c,k}{c}_k(z)$, where ${\lambda }_{c,k}$ and $c_k(z)$ are clutter rate and clutter density, respectively.

Since the clutter density in the real target tracking environment is difficult to effectively estimate, most MTT algorithms assume that the clutter is uniformly distributed. However, the actual target tracking environment is complex and variable, including not only uniformly distributed clutter but also the dense group clutter with unknown distribution, as shown in Figure 1. At this moment, it is difficult to effectively describe the current clutter distribution characteristics using the single uniform clutter model; therefore, the clutter needs to be reformulated.

According to Figure 1, the new clutter intensity can be expressed as

$${\kappa }_k(\mathit{z})={\kappa }_{u,k}({\mathit{z}}_u)+{\kappa }_{d,k}({\mathit{z}}_d)$$

(7)

where ${\kappa }_{u,k}({\mathit{z}}_u)$ and ${\kappa }_{d,k}({\mathit{z}}_d)$ denote the intensity of the uniform clutter and dense group clutter, respectively. ${\kappa }_{u,k}({\mathit{z}}_u)$ can be calculated by ${\kappa }_{u,k}({\mathit{z}}_u)={\lambda }_{u,k}/V$, $V$ is the area of the sensor’s detection field, and ${\lambda }_{u,k}$ is the clutter rate of uniform clutter.

It can be easily determined from Figure 1 that cluster centers are surrounded by neighbor points with lower local density and they are at a relatively large distance from any points with a higher local density, which conforms to Gaussian distribution. Accordingly, dense group clutter can be characterized by a two-dimensional Gaussian distribution, i.e.,

$${\kappa }_{d,k}({\mathit{z}}_d)={\displaystyle \sum _{j=1}^{J_{c,k}}{\displaystyle \sum _{i=1}^{{\lambda }_{j,k}}{w}_k^{(i,j)}\mathcal{N}({\mathit{z}}_d;{\mathit{m}}_k^{(i,j)},{\mathit{P}}_k^{(i,j)})}}$$

(8)

where $J_{c,k}$ is the number of clutter groups; ${\lambda }_{j,k}$ is the clutter rate of the $j^{th}$ clutter group; and $\mathit{m}$, $\mathit{P}$, and $w$ are the mean, covariance, and weight of the Gaussian component of the $j^{th}$ clutter group, respectively.

Therefore, the new clutter model can be represented as

$${\kappa }_k(\mathit{z})={\lambda }_{u,k}{c}_{u,k}({\mathit{z}}_u)+{\displaystyle \sum _{j=1}^{J_{c,k}}{\displaystyle \sum _{i=1}^{{\lambda }_{j,k}}{w}_k^{(i,j)}\mathcal{N}({\mathit{z}}_d;{\mathit{m}}_k^{(i,j)},{\mathit{P}}_k^{(i,j)})}}$$

(9)

Thus, depending on the new clutter model and clutter characteristics, for each measurement point ${\mathit{z}}_k^i$, the local density ${\rho }_i$ and distance ${\delta }_i$ from points with higher density will be calculated and only depend on the Euclidean distance $d_{ij}$ between ${\mathit{z}}_k^i$ and ${\mathit{z}}_k^j$. Measurements obtained by the active sensor are represented as the set $E={\left\{{\mathit{z}}_k^i\right\}}_{i=1}^{M_k}$, $I_e=\{1,2,\dots ,M_k\}$ is the corresponding set of indicators, and $M_k$ is the measurement number, where ${\mathit{z}}_k^i$ belongs the measurement set of targets or clutter. Thus, for each ${\mathit{z}}_k^i$ in $E$, the corresponding local density ${\rho }_i$ can be written as

$${\rho }_i={\displaystyle \sum _{j\in I_e}\chi (d_{ij}-d_c)}$$

(10)

where

$$\chi (x)=\{\begin{array}{c}1,x<0\\ 0,x\ge 0\end{array}$$

(11)

and $d_c$ is the cutoff distance.

Remark: If the value of $d_c$ is too large, the value of ${\rho }_i$ will be so large that a cluster center’s distinguishing degree is not high; if the value of $d_c$ is too small, the same cluster may be split into several clusters. In other words, when $d_c>\max \left\{d_{ij}\right\}$, all measurements will belong to the same cluster; when $d_c<\min \left\{d_{ij}\right\}$, each measurement will become a separate cluster. If the value of $d_c$ is a constant, it will mainly depend on the specific scenario, and different problems will have significant differences in magnitude. Therefore, in this work, to determine $d_c$, a ratio is considered. Thus, for each measurement, one can choose $d_c$ to make the average number of neighbors about 1% to 2% of the total number of $d_{ij}$.

Additionally, in this work, ${\delta }_i$ is defined as the minimum distance between $z_k^i$ and any other point with higher density, which can be computed as

$${\delta }_i=\{\begin{array}{c}\underset{j\in I_e^i}{\min }\{d_{ij}\},I_e^i\ne \varnothing \\ \underset{j\in I_e}{\max }\{d_{ij}\},I_e^i=\varnothing \end{array}$$

(12)

where $I_e^i=\{j\in I_e|{\rho }_j>{\rho }_i\}$.

Combining (10) and (12), we can observe that the larger ${\rho }_i$ and ${\delta }_i$, the more likely ${\mathit{z}}_k^i$ is the cluster center. Thus, a measuring standard ${\gamma }_i$ of the cluster center can be defined by combining ${\rho }_i$ and ${\delta }_i$, which can be computed as

$${\gamma }_i={\rho }_i{\delta }_i,i\in I_e$$

(13)

In order to present the defined measuring standard more clearly, we first normalize $\gamma$, then draw its distribution graph, as shown in Figure 2. We can draw a conclusion that non-cluster centers’ measuring standards are relatively smooth and a significant leap will be observed when transitioning from non-cluster centers to cluster centers, where the leap can be determined by numerical detection. In this work, if ${\gamma }_i>{\gamma }_{leap}$, ${\mathit{z}}_k^i$ is regarded as a cluster center, where ${\gamma }_{leap}$ is the leaping point.

For completeness, we summarize the key steps of Section 3.1 in Algorithm 1.

Algorithm 1. Pseudocode for CFDP algorithm.
Input: ${\left\{{\mathit{z}}_k^i\right\}}_{i=1}^{M_k}$
Output: $nu{m}_k$, $in{d}_k$, $cluster\_cente{r}_k$
1	Initialize: $nu{m}_k=0$; $in{d}_k=\left\{\right\}$; $cluster\_cente{r}_k=\left\{\right\}$;
2	for $i,j=1,2,\dots ,M_k$ do
3	$dist(i,j)\leftarrow |{\mathit{z}}_k^i-{\mathit{z}}_k^j|$;
4	end
5	$pos\leftarrow round(0.5\ast M_k\ast (M_k-1)\ast 0.02)$;
6	$dist\_asc\leftarrow ascend dist\left(i,j\right)$;
7	$d_c\leftarrow dist\_asc(pos)$;
8	for $i=1,2,\dots ,M_k$ do
9	Compute ${\rho }_i$, ${\delta }_i$, ${\gamma }_i$;
10	end
11	Plots ${\gamma }_i-i$ to find ${\gamma }_{leap}$;
12	for $i=1,2,\dots ,M_k$ do
13	If ${\gamma }_i>{\gamma }_{leap}$ then
14	$nu{m}_k\leftarrow nu{m}_k+1$; append $i$ to $in{d}_k$; append ${\mathit{z}}_k^i$ to $cluster\_cente{r}_k$
15	end
16	end
17	return $nu{m}_k$, $in{d}_k$, $cluster\_cente{r}_k$

3.2. Clutter Screening via Double Thresholds

The previously obtained cluster centers are all in the form of measurements from the active sensor, so the $j^{th}$ cluster center at time $k$ can be described as ${\mathit{z}}_{ac,k}^j={[{\alpha }_{ac,k}^j,{\beta }_{ac,k}^j,x_{ac,k}^j,y_{ac,k}^j,z_{ac,k}^j]}^T$ by (3), (4), and (5), where the subscript “$ac$” represents the active sensor’s clustering center. Therefore, the distance threshold $d_k^j$ is established by the 3$\sigma$ rule, and all measurements within the threshold are regarded as coming from the $j^{th}$ clutter group and need to be screened, which can be described by

$$\Vert {\mathit{z}}_{a,k}^i-{\mathit{z}}_{ac,k}^j\Vert <d_k^j=3{\sigma }_k^j$$

(14)

where ${\mathit{z}}_{a,k}^i$ denotes the $i^{th}$ measurement derived from the active sensor, and ${\sigma }_k^j$ is the Gaussian distribution covariance of the $j^{th}$ clutter group.

Furthermore, it is not possible to establish a distance threshold directly due to the lack of position information for passive sensors. Therefore, it is necessary to use the angle information to set the angle threshold. However, passive sensors have a large number and different position coordinates, so they are different from each cluster center of the azimuth angle. Thus, the measurement form of each cluster center relative to each passive sensor needs to be calculated by (4) and (5), so the $j^{th}$ cluster center is written as ${\mathit{z}}_{pc,k}^{l,j}={[{\alpha }_{pc,k}^{l,j},{\beta }_{pc,k}^{l,j}]}^T$ under the $l^{th}$ passive sensor at time $k$, where the subscript “$pc$” represents the passive sensor’s clustering center. We define $X_{c,k}^j=[x_{ac,k}^j,y_{ac,k}^j,z_{ac,k}^j]$ and $S_l=[{\tilde{x}}_l,{\tilde{y}}_l,{\tilde{z}}_l]$ as the position coordinates of the $j^{th}$ cluster center at time $k$ and the $l^{th}$ passive sensor, respectively.

As shown in Figure 3, by making a tangent line across the sensor location to the circle where the dense group clutter is located, the maximum offset $\Delta {\alpha }_k^{l,j}$ of the azimuth angle can be obtained by

$$\Delta {\alpha }_k^{l,j}=\arcsin \left({\displaystyle \frac{3{\sigma }_k^j}{\sqrt{{(x_{ac,k}^j-{\tilde{x}}_l)}^2+{(y_{ac,k}^j-{\tilde{y}}_l)}^2+{(z_{ac,k}^j-{\tilde{z}}_l)}^2}}}\right)$$

(15)

The angle threshold ${\theta }_k^{l,j}$ can be established by

$$\Vert {\alpha }_{p,k}^l-{\alpha }_{pc,k}^{l,j}\Vert \le {\theta }_k^{l,j}=\Delta {\alpha }_k^{l,j}$$

(16)

where ${\alpha }_{p,k}^l$ is the azimuth angle of measurement derived from the $l^{th}$ passive sensor. All measurements of the $l^{th}$ passive sensor within the threshold ${\theta }_k^{l,j}$ can be regarded as coming from the $j^{th}$ dense group clutter and need to be sieved.

4. SAMPS-MAF

The influence of dense group clutter on the tracking results has been greatly limited in Section 3. To associate and fuse the remaining measurements from the SAMPS system, this section proposes the SAMPS-MAF algorithm, including the following items. Firstly, a measurement initial screening based on EKF to exclude the uniform clutters is given in Section 4.1. Secondly, Section 4.2 builds statistics based on the azimuth angle to exclude the wrong measurement association groups. Finally, measurement fusion based on LS and the correction of measurement points used for filtering are introduced in Section 4.3.

4.1. Measurement Initial Screening Based on EKF

The clutter pre-processing in Section 3 is effective in dealing with dense group clutter; however, uniform clutter still exists, which will cause a significant number of invalid measurement association groups. Thus, a measurement initial screening is required before association and fusion. A primary measurement screening algorithm based on association flare angle was proposed in the [37]; however, its mathematical derivation is too complex, and its screening efficiency is also not high. Therefore, this part proposes another measurement initial screening algorithm based on EKF.

Suppose ${\mathit{x}}_{k-1}$ is the target state at time $k-1$, the predicted state at time $k$ can be represented by

$${\mathit{x}}_{k|k-1}=f_{k|k-1}({\mathit{x}}_{k-1})$$

(17)

Thus, the predicted azimuth angle ${\alpha }_{k|k-1}^n$ of ${\mathit{x}}_{k|k-1}$ for $n^{th}$ sensor $S_n$ can be calculated by

$${\alpha }_{k|k-1}^n=\arctan \left({\displaystyle \frac{y_{k|k-1}-{\tilde{y}}_n}{x_{k|k-1}-{\tilde{x}}_n}}\right)$$

(18)

where $x_{k|k-1}$ and $y_{k|k-1}$ are position components of ${\mathit{x}}_{k|k-1}$.

Suppose ${\left\{{\mathit{z}}_{k,n}^i\right\}}_{i=1}^{{\tilde{M}}_{n,k}}$ represents the set of measurements of the sensor $S_n$ after pre-processing for clutter, and ${\tilde{M}}_{n,k}$ is the number of remaining measurements. Combining (3) and (4), the azimuth angle ${\alpha }_{k,n}^i$ of ${\mathit{z}}_{k,n}^i$ from the sensor $S_n$ can be obtained at time $k$. According to EKF, the covariance of the angle residual $v_{k|k-1}^{i,n}={\alpha }_{k,n}^i-{\alpha }_{k|k-1}^n$ can be computed as

$${\mathit{S}}_k={\mathit{H}}_k{\mathit{P}}_{k|k-1}{\mathit{H}}_k^T+{\mathit{R}}_k$$

(19)

where

$${\mathit{P}}_{k|k-1}={\mathit{F}}_k{\mathit{P}}_{k-1}{\mathit{F}}_k^T+{\mathit{Q}}_{k-1}$$

(20)

$${\mathit{H}}_k={\displaystyle \frac{\partial }{\partial \mathit{x}}}{h}_k(\mathit{x}){|}_{\mathit{x}={\mathit{x}}_{k|k-1}}$$

(21)

$${\mathit{F}}_k={\displaystyle \frac{\partial }{\partial \mathit{x}}}{f}_{k|k-1}(\mathit{x}){|}_{\mathit{x}={\mathit{x}}_{k-1}}$$

(22)

Combining $v_{k|k-1}^{i,n}$ and ${\mathit{S}}_k$, for each sensor, an angled gate can be established for measurement screening at time $k$, i.e.,

$$V_k(\epsilon )\triangleq v_{k|k-1}^{i,n}{\mathit{S}}_k^{-1}{(v_{k|k-1}^{i,n})}^T\le \epsilon$$

(23)

where $\epsilon$ is decided by the chi-squared distribution corresponding to the measurement dimension.

If ${\alpha }_{k,n}^i$ of the measurement from the sensor $S_n$ satisfies (23), the measurement ${\mathit{z}}_{k,n}^i$ is considered as an effective target measurement from the sensor $S_n$. A set of effective measurements of the sensor $S_n$ can be obtained by iterating ${\tilde{M}}_{n,k}$ measurements at time $k$.

4.2. Statistics-Based Secondary Screening

The number of measurements to be associated is significantly reduced after the related processing in Section 3 and Section 4.1. However, the initial screening algorithm alone does not completely exclude the wrong association groups. To this end, this section performs secondary screening of effective measurements for each sensor by constructing statistics.

Suppose that there are $N(N>2)$ sensors in the observation space, where the $N^{th}$ sensor $S_N$ is the active sensor and the rest are passive sensors, i.e., the SAMPS system. Each measurement association group can be described as $\{({\alpha }_{1,r},{\beta }_{1,r}),({\alpha }_{2,s},{\beta }_{2,s}),\dots ,({\alpha }_{N-1,t},{\beta }_{N-1,t}),({\alpha }_{N,u},{\beta }_{N,u})\}$, where subscript $(N,u)$ represents the $u^{th}$ $(\alpha ,\beta )$ pair of $S_N$. Since disparate straight lines may not intersect in 3D space, statistics will be constructed in 2D space.

As shown in Figure 4, we define $M_{j,m}(x_{M,jm},y_{M,jm})$ as the intersection of lines where ${\alpha }_{1,r}$ and ${\alpha }_{j,m}(1<j<N,m\in \{s,\dots ,t\})$ are located, and ${\alpha }_{M,jm}$ is the azimuth angle of $M_{j,m}$ from sensor $S_N$. It is obvious that $M_{j,m}$ and ${\alpha }_{M,jm}$ will be generated for each measurement association group.

Combining ${\alpha }_{1,r}$, ${\alpha }_{j,m}$, and position coordinates $({\tilde{x}}_n,{\tilde{y}}_n),1\le n\le N$ of the sensor $S_n$, $M_{j,m}(x_{M,jm},y_{M,jm})$ and ${\alpha }_{M,jm}$ can be calculated by

$$\tan {\alpha }_{1,r}={\displaystyle \frac{y_{M,jm}-{\tilde{y}}_1}{x_{M,jm}-{\tilde{x}}_1}},\tan {\alpha }_{j,m}={\displaystyle \frac{y_{M,jm}-{\tilde{y}}_j}{x_{M,jm}-{\tilde{x}}_j}}$$

(24)

$$\{\begin{array}{l}{x}_{M,jm}={\displaystyle \frac{{\tilde{x}}_1\tan {\alpha }_{1,r}-{\tilde{x}}_j\tan {\alpha }_{j,m}+{\tilde{y}}_j-{\tilde{y}}_1}{\tan {\alpha }_{1,r}-\tan {\alpha }_{j,m}}}\\ y_{M,jm}={\displaystyle \frac{{\tilde{y}}_j\tan {\alpha }_{1,r}-{\tilde{y}}_1\tan {\alpha }_{j,m}+({\tilde{x}}_1-{\tilde{x}}_j)\tan {\alpha }_{1,r}\tan {\alpha }_{j,m}}{\tan {\alpha }_{1,r}-\tan {\alpha }_{j,m}}}\end{array}$$

(25)

$$\tan {\alpha }_{M,jm}={\displaystyle \frac{y_{M,jm}-{\tilde{y}}_N}{x_{M,jm}-{\tilde{x}}_N}}$$

(26)

It is clear that ${\alpha }_{M,jm}={\alpha }_{N,u}$ holds in an ideal tracking scenario without measurement noise. In the practical tracking scenario, however, there exists measurement noise, so ${\Delta }_{a,jm}={\alpha }_{M,jm}-{\alpha }_{N,u}$ is needed as the statistics. In this work, $8.47\times {10}^6$ sample data are carried out in 50 Monte Carlo (MC) trials for the distribution of ${\Delta }_{a,jm}$. As shown in the probability density function curve of Figure 5, ${\Delta }_{a,jm}$ approximately follows the standard Gaussian distribution, i.e., ${\Delta }_{\alpha ,jm}\sim \mathcal{N}(0,{\tau }_{\Delta }^2)$, where ${\tau }_{\Delta }^2$ is the variance of ${\Delta }_{a,jm}$, which can be obtained by

$${\tau }_{\Delta }^2={\left|{\displaystyle \frac{\partial {\Delta }_{\alpha ,jm}}{\partial {\alpha }_{1,r}}}{\tau }_{\alpha ,1}\right|}^2+{\left|{\displaystyle \frac{\partial {\Delta }_{\alpha ,jm}}{\partial {\alpha }_{j,m}}}{\tau }_{\alpha ,j}\right|}^2+{\left|{\displaystyle \frac{\partial {\Delta }_{\alpha ,jm}}{\partial {\alpha }_{N,u}}}{\tau }_{\alpha ,N}\right|}^2$$

(27)

where ${\tau }_{\alpha ,1}$, ${\tau }_{\alpha ,j}$, and ${\tau }_{\alpha ,N}$ are the standard deviation of each sensor measurement error.

In this work, the confidence interval is set to $(-3{\tau }_{\Delta },3{\tau }_{\Delta })$ in accordance with the properties of standard Gaussian distribution, whose corresponding probability is 99.73%. Throughout all ${\Delta }_{a,jm}$, if $-3{\tau }_{\Delta }<{\Delta }_{a,jm}<3{\tau }_{\Delta }$ for all ${\Delta }_{a,jm}$, the present measurement association group $\{({\alpha }_{1,r},{\beta }_{1,r}),({\alpha }_{2,s},{\beta }_{2,s}),\mathrm{\dots },({\alpha }_{N-1,t},{\beta }_{N-1,t}),({\alpha }_{N,u},{\beta }_{N,u})\}$ is considered to belong to the same target.

The above algorithm of constructing statistics is based on a SASMPS system. However, when only one passive sensor is available, namely, the single-active and single-passive sensors (SASPS) system, this algorithm is no longer applicable. Therefore, the statistics need to be reconstructed for the SASPS system.

Suppose that $S_1$ and $S_2$ are the passive and active sensors, respectively. Each measurement association group can be described as $\{({\alpha }_{1,r},{\beta }_{1,r}),({\alpha }_{2,u},{\beta }_{2,u})\}$, and $C(x_c,y_c)$ is defined as the intersection of lines where ${\alpha }_{1,r}$ and ${\alpha }_{2,u}$ are located. Thus, combining ${\alpha }_{1,r}$, ${\alpha }_{2,u}$, and the position coordinates $({\tilde{x}}_n,{\tilde{y}}_n),n=1,2$ of the sensor $S_n$, $C(x_c,y_c)$ can be computed by (25). $D(x_{2,u},y_{2,u})$ is defined as the location information of measurement from $S_2$. It is clear that, in the ideal tracking scenario without measurement noise, $C(x_c,y_c)=D(x_{2,u},y_{2,u})$ holds. In the practical tracking scenario, however, measurement noise exists. Therefore, we can set association thresholds for $D$ by

$$\{\begin{array}{c}\left|x_{2,u}-x_c\right|<3{\sigma }_x\\ \left|y_{2,u}-y_c\right|<3{\sigma }_y\end{array}$$

(28)

where ${\sigma }_l(l=x,y)$ is the standard deviation of the measurement noise of $S_2$.

Averaging coordinates of all $C_j$ satisfying (28) and $D$ and combining the $z_{2,u}$ of measurements from $S_2$ can obtain the final measurement before tracking.

4.3. Measurement Fusion and Correction

The statistics-based secondary screening can screen a large number of wrong measurement association groups, making it possible that measurement fusion can be achieved with smaller association-group numbers. Therefore, measurement fusion and correction of each measurement association group will be introduced in this section.

4.3.1. Measurements Fusion Based LS

As explained in Section 4.2, each sensor’s $(\alpha ,\beta )$ pair can establish a location line. In this section, in order to represent $N$ location lines corresponding to the measurement association group $\{({\alpha }_{1,r},{\beta }_{1,r}),({\alpha }_{2,s},{\beta }_{2,s}),\dots ,({\alpha }_{N-1,t},{\beta }_{N-1,t}),({\alpha }_{N,u},{\beta }_{N,u})\}$, we define set $\{L_1,L_2,\dots ,L_N\}$, where $N$ is the sensor number. In the ideal tracking scenario without measurement noise, $N$ location lines from the same measurement association group should intersect at one point, namely, the corresponding target’s location. Nevertheless, due to the presence of measurement noise, these lines will not intersect at any point. Even worse, these lines may not intersect with other lines at all. In order to find the relatively accurate target position, the point with the shortest distance to all lines is taken as the target’s measurement point, namely, $T(x_T,y_T,z_T)$ in Figure 6 (taking $N=4$ as an example).

The direction cosine of the location line $L_i$$(i=1,2,\dots ,N)$ is given by

$$l_i=\cos {\beta }_i\cos {\alpha }_i,m_i=\cos {\beta }_i\sin {\alpha }_i,n_i=\sin {\beta }_i$$

(29)

Subsequently, the location line can be calculated as

$${\displaystyle \frac{x_T-x_i}{l_i}}={\displaystyle \frac{y_T-y_i}{m_i}}={\displaystyle \frac{z_T-z_i}{n_i}}$$

(30)

The distance between $T$ and $L_i$ is obtained by

$$d_i={\displaystyle \frac{\left|\begin{array}{ccc}\overrightarrow{i}& \overrightarrow{j}& \overrightarrow{k}\\ x_T-x_i& y_T-y_i& z_T-z_i\\ l_i& m_i& n_i\end{array}\right|}{\sqrt{l_i^2+m_i^2+n_i^2}}}$$

(31)

Thus, the squared distance sum from $T$ to $N$ location lines can be represented as

$$d^2={\displaystyle \sum _{i=1}^N{d}_i^2}={\displaystyle \sum _{i=1}^N(u_i^2+v_i^2+w_i^2)}$$

(32)

where

$$\{\begin{array}{l}{u}_i=(y_T-y_i)n_i-(z_T-z_i)m_i\\ v_i=(z_T-z_i)l_i-(x_T-x_i)n_i\\ w_i=(x_T-x_i)m_i-(y_T-y_i)l_i\end{array}$$

(33)

Considering LS, let ${\displaystyle \frac{\partial d^2}{\partial x_T}}=0$, ${\displaystyle \frac{\partial d^2}{\partial y_T}}=0$, and ${\displaystyle \frac{\partial d^2}{\partial z_T}}=0$, and we can obtain

$$\left[\begin{array}{ccc}L& T& S\\ T& M& R\\ S& R& O\end{array}\right]\left[\begin{array}{c}{x}_T\\ y_T\\ z_T\end{array}\right]=\left[\begin{array}{c}E\\ F\\ G\end{array}\right]$$

(34)

where

$$L={\displaystyle \sum _{i=1}^N(m_i^2+n_i^2)},M={\displaystyle \sum _{i=1}^N(n_i^2+l_i^2)},O={\displaystyle \sum _{i=1}^N(l_i^2+m_i^2)},$$

(35)

$$R=-{\displaystyle \sum _{i=1}^N(m_i{n}_i)},S=-{\displaystyle \sum _{i=1}^N(l_i{n}_i)},T=-{\displaystyle \sum _{i=1}^N(l_i{m}_i)}$$

(36)

$$\begin{array}{l}E={\displaystyle \sum _{i=1}^N\left[(m_i^2+n_i^2)x_i-l_i{m}_i{y}_i-l_i{n}_i{z}_i\right]}\\ F={\displaystyle \sum _{i=1}^N\left[-l_i{m}_i{x}_i+(n_i^2+l_i^2)y_i-m_i{n}_i{z}_i\right]}\\ G={\displaystyle \sum _{i=1}^N\left[-l_i{n}_i{x}_i-m_i{n}_i{y}_i+(m_i^2+l_i^2)z_i\right]}\end{array}$$

(37)

Finally, the solution to (34) is obtained, namely, the measurement point positions that we need, which can be calculated as

$$\{\begin{array}{l}{x}_T={\displaystyle \frac{EMO+FRS+GRT-GMS-FTO-E{R}^2}{Q}}\\ y_T={\displaystyle \frac{ERS+FLO+GST-GLR-ETO-F{S}^2}{Q}}\\ z_T={\displaystyle \frac{ERT+FST+GLM-EMS-FLR-G{T}^2}{Q}}\end{array}$$

(38)

where

$$Q=\left|\begin{array}{ccc}L& T& S\\ T& M& R\\ S& R& O\end{array}\right|$$

(39)

4.3.2. Correction of Measurement Points

After obtaining all measurement points, we can start using an appropriate filter for tracking. Nevertheless, these measurement points may originate from the incorrect association groups, directly affecting the tracking results. Hence, correction of the measurement points needs to be carried out to eliminate the negative effect.

Without measurement noise, the squared distance sums from the measurement point $T$ to the three corresponding location lines should meet

$$d^2={\displaystyle \sum _{i=1}^N{d}_i^2}=0$$

(40)

However, in practice, due to the presence of measurement noise, (40) is no longer valid, namely,

$$d^2={\displaystyle \sum _{i=1}^N{d}_i^2}\ne 0$$

(41)

Obviously, by (41), we can find that $d^2$ corresponding to correct points is much smaller than that corresponding to any other incorrect points. Hence, we can determine whether the measurement point has been corrected by $d^2$.

Assuming the position of target $A$ is $(x_A,y_A,z_A)$, the measurement points’ error obtained by LS is usually within a small range of $(-{\sigma }_l,{\sigma }_l)$, where ${\sigma }_l(l=x,y,z)$ is the standard deviation of the active-sensor measurement error $v_l\sim \mathcal{N}(0,{\sigma }_l^2)$, and then the position of the measurement point $P$ with the maximum error can be written as

$$\{\begin{array}{l}{x}_P=x_A\pm {\sigma }_x\\ y_P=y_A\pm {\sigma }_y\\ z_P=z_A\pm {\sigma }_z\end{array}$$

(42)

As shown in Figure 7, $d$ is the distance from $P$ to the location line L. It is clear that we have $\overline{PA}\ge d$ in a right triangle $△PAB$, where $B$ is the perpendicular line’s foot from $P$ to $L$. In Figure 7, two dashed lines represent the location lines with measurement error. Thus, within the error of $3{\sigma }_l$, $\overline{PA}\ge d$ is available.

The length of $\overline{PA}$ is given by

$$\overline{PA}=\sqrt{{\sigma }_x^2+{\sigma }_y^2+{\sigma }_z^2}$$

(43)

Furthermore, using (43), the approximate gate limit $\lambda$ of $d^2$ can be obtained by

$$\lambda =3({\sigma }_x^2+{\sigma }_y^2+{\sigma }_z^2)$$

(44)

If $d^2\le \lambda$, the corresponding measurement point is considered as originating from the target or clutter. Otherwise, it is an incorrect measurement point.

As for active sensors, position information from measurements can be used to eliminate incorrect measurement points. If the distance between the measurement point and the corresponding active sensor’s position measurement is long enough, this measurement point is considered as an incorrect point.

Using (42), we can observe that the measurement point acquired by LS satisfies

$$\{\begin{array}{l}{\widehat{x}}_T=x_A+a{\sigma }_x\\ {\widehat{y}}_T=y_A+b{\sigma }_y\\ {\widehat{z}}_T=z_A+c{\sigma }_z\end{array}$$

(45)

where $a,b,c\in (-1,1)$.

In general, the active sensor’s position measurement can be represented as

$$\{\begin{array}{l}{x}_{N,u}=x_A+e{\sigma }_x\\ y_{N,u}=y_A+f{\sigma }_y\\ z_{N,u}=z_A+g{\sigma }_z\end{array}$$

(46)

Hence, the method of determining whether one position measurement originates from one target is to set a confidence interval $(-3{\sigma }_l,3{\sigma }_l)$, so that $e,f,g\in (-3,3)$. According to (45) and (46), we can obtain

$$\{\begin{array}{l}\left|{\widehat{x}}_T-x_{N,u}\right|=\left|a-e\right|{\sigma }_x\\ \left|{\widehat{y}}_T-{\tilde{y}}_{N,u}\right|=\left|b-f\right|{\sigma }_y\\ \left|{\widehat{z}}_T-z_{N,u}\right|=\left|c-g\right|{\sigma }_z\end{array}$$

(47)

Then, the maximum gate limits can be defined as

$$\begin{array}{l}{\lambda }_x\triangleq 4{\sigma }_x,\\ {\lambda }_y\triangleq 4{\sigma }_y,\\ {\lambda }_z\triangleq 4{\sigma }_z\end{array}$$

(48)

If one measurement point follows $\left|{\widehat{x}}_T-x_{N,u}\right|\le 4{\sigma }_x$, $\left|{\widehat{y}}_T-y_{N,u}\right|\le 4{\sigma }_y$ and $\left|{\widehat{z}}_T-z_{N,u}\right|\le 4{\sigma }_z$, this point is considered as one correct measurement point. Finally, more than 90% of the incorrect measurement points can be eliminated by the above operations.

5. MAMPS-MAF-DNN

The MAMPS system can be seen as a combination of multiple SAMPS systems, and its track fusion can be achieved using distributed track fusion algorithms. Hence, the proposed hybrid MAMPS-MAF-DNN fusion algorithm can be implemented through combining the centralized SAMPS-MAF algorithm and distributed DNN track fusion framework. The detailed implementation process of the algorithm is shown in Figure 8. During the training phase, the preprocessed measurement set from each active sensor is first filtered separately using its corresponding SAMPS-MAF algorithm. The filtered estimation is then normalized as the input to the DNN, and targets’ real tracks are finally used to train targets. During the tracking phase, the target tracks’ fusion prediction is performed by feeding the normalized filtering estimation states into the trained DNN.

5.1. Data Sets

Data sets used for training DNNs are generated through software simulation, and the detailed execution steps are as follows:

Step 1: Generate the real target track set $X_{truth}=\{{\mathit{x}}_1,\dots ,{\mathit{x}}_k\}$ and measurement set $Z$ according to (1) and (3), respectively.

Step 2: First, the measurement set $Z$ obtained in Step 1 is preprocessed using the clutter pre-processing algorithm in Section 3. Then, using the SAMPS-MAF algorithm from Section 4 for each active sensor corresponding to the SAMPS system, the track set estimated $X_e=\{{\overline{X}}_1,\dots ,{\overline{X}}_n\}$ is obtained, where $n$ is the number of active sensors, ${\overline{X}}_i=\{{\overline{\mathit{x}}}_1,\dots ,{\overline{\mathit{x}}}_k\}(1\le i\le n)$, and ${\overline{\mathit{x}}}_k={[{\overline{x}}_k,{\overline{v}}_{x,k},{\overline{y}}_k,{\overline{v}}_{y,k},{\overline{z}}_k,{\overline{v}}_{z,k},{\overline{w}}_k]}^T$.

Step 3: Combine the track set estimated $X_e$ from Step 2 with the corresponding real target’s track set $X_{truth}$ from Step 1 as data sets, where $X_e$ and $X_{truth}$ are, respectively, used as the input and output.

Step 4: Perform normalized preprocessing of input data to prevent network saturation in the first few layers, i.e.,

$${\mathit{x}}_k^{input}={\displaystyle \frac{{\overline{\mathit{x}}}_k-{\overline{\mathit{x}}}_{\min }}{{\overline{\mathit{x}}}_{\max }-{\overline{\mathit{x}}}_{\min }}}$$

(49)

Remark: The actual data sets of input and output are, respectively, $X_{input}=\{{\mathit{x}}_1^{input},\dots ,{\mathit{x}}_k^{input}\}$ and $X_{output}=X_{truth}$. However, in this work, in order to obtain more accurate network training during 100 trials, approximately 5000 target track points gained by 10 MC experiments are used as data sets for input and output.

5.2. Training DNNs

In the proposed algorithm, a DNN with two hidden layers and 64 nodes per layer is trained on 90% of the data set, and the remaining 10% is used for testing. In this work, three active sensors and three passive sensors are used as the MAMPS system, thereby the dimension for each input and output is, respectively, set to $21\times 1$ and $7\times 1$. In our network, the initial learning rate is set to 0.01, and the activation function is chosen as the sigmoid function. The training platform is Matlab, the computer CPU is Intel Core i5-12400, and the computer system is Win7. As shown in Figure 9, after 30 epochs, the training loss reaches 0.05338, while the testing loss reaches 0.04409.

6. Numerical Experiments

6.1. Scenario and Parameter Setting

In simulations, the CT model is adopted to represent the motion model of the maneuvering target. Over the observation region $[-3500\mathrm{m},3500\mathrm{m}]\times [-3500\mathrm{m},3500\mathrm{m}]$ at an altitude of $500\mathrm{m}$, a 3D multi-target tracking scenario is considered. Targets’ initial states, birth times, and death times are listed in

Table 1. Targets’ initial states, birth times, and death times.

Target	Initial State	Birth Time	Death Time
1	${[-1000,30,2000,-10,500,0,-\pi /360]}^T$	1	75
2	${[1500,-18,-2000,25,500,0,\pi /270]}^T$	10	100
3	${[-1000,40,-2000,12,500,0,\pi /180]}^T$	16	83
4	${[1500,13,-2000,26,500,0,0]}^T$	20	100
5	${[-1000,-25,2000,-30,500,0,\pi /360]}^T$	25	100
6	${[-2000,17,-1500,37,500,0,-\pi /180]}^T$	30	100

. Targets’ ground truths are shown in Figure 10, where $○$ and $△$ denote the start and end of each trajectory, respectively.

We set eight sensors, where $S_{6-8}$ are active sensors and the rest are passive sensors. The parameters of the sensors are listed in

Table 2. Parameters of sensors.

Sensor	Position (m)
1	${[0,-5000,0]}^T$
2	${[5000,0,0]}^T$
3	${[-5000,0,0]}^T$
4	${[2500,-2500,0]}^T$
5	${[-2500,-2500,0]}^T$
6	${[0,5000,0]}^T$
7	${[2500,2500,0]}^T$
8	${[-2500,2500,0]}^T$

.

In this work, the PHD and LMB filters are chosen as tracking filters to validate the effectiveness of the proposed algorithm. The parameters of the filters are listed in

Table 3. Parameters of filters.

Parameter	Value
Sampling period $\Delta T$	1 s
Standard deviation ${\sigma }_{\alpha }$ of error of $\alpha$	1 mrad
Standard deviation ${\sigma }_{\beta }$ of error of $\beta$	1 mrad
Standard deviation ${\sigma }_x$ of error of $x$	10 m
Standard deviation ${\sigma }_y$ of error of $y$	10 m
Standard deviation ${\sigma }_z$ of error of $z$	10 m
Survival probability	0.99
Detection probability	0.99

.

In simulations, we adopt the optimal sub-pattern assignment (OSPA) distance to evaluate the multi-target tracking performance, i.e.,

$${\overline{d}}_p^{(c)}(X,Y)=\{\begin{array}{cc}0,& m=n=0\hfill \\ {\left({\displaystyle \frac{1}{n}}\left(\underset{\pi \in {\prod }_n}{\min }{\displaystyle \sum _{i=1}^m{d}^{(c)}{(x_i,y_{\pi (i)})}^p+c^p(n-m)}\right)\right)}^{1/p},& m\le n\hfill \\ {\overline{d}}_p^{(c)}(Y,X),& m>n\hfill \end{array}$$

(50)

where $X=\{x_1,\dots ,x_m\}$ and $Y=\{y_1,\dots ,y_n\}$ are arbitrary finite subsets, $1\le p<\infty$, and $c>0$. In our simulations, they are set to $p=1$ and $c=100$, respectively. Further, the OSPA distance can be divided into the location error and cardinality error, i.e.,

$${\overline{e}}_{p,loc}^{(c)}(X,Y)=\{\begin{array}{cc}{\left({\displaystyle \frac{1}{n}}\left(\underset{\pi \in {\prod }_n}{\min }{\displaystyle \sum _{i=1}^m{d}^{(c)}{(x_i,y_{\pi (i)})}^p}\right)\right)}^{1/p},& m\le n\hfill \\ {\overline{e}}_{p,loc}^{(c)}(Y,X),& m>n\hfill \end{array}$$

(51)

$${\overline{e}}_{p,card}^{(c)}(X,Y)={\left({\displaystyle \frac{1}{\max (m,n)}}\left(c^p(n-m)\right)\right)}^{1/p}$$

(52)

6.2. Experiment 1: Analysis of the Effectiveness of Pre-Processing on Clutter

6.2.1. Analysis of Clustering Effectiveness

In the simulation, clutter measurements are uniformly distributed over the observation region and can be represented by a Poisson RFS, with the intensity being

$${\kappa }_{u,k}({\mathit{z}}_u)={\displaystyle \frac{\mathcal{U}({\mathit{z}}_u)}{{\displaystyle \int \mathcal{U}({\mathit{z}}_u)}d{\mathit{z}}_u}}{\lambda }_{u,k}$$

(53)

where ${\lambda }_{u,k}$ is the clutter rate, $\mathcal{U}(\cdot )$ is a uniform density within the surveillance region, and ${\lambda }_{u,k}=10$ per scan in simulations.

The dense group clutter can be generated by 2D Gaussian distribution, and its number follows the Poisson distribution with a mean of ${\lambda }_{j,k}$. In simulations, we take ${\lambda }_{j,k}=30$ and suppose that there are two groups of dense clutter in the scenario. The model parameters are listed in

Table 4. Parameters of the dense group clutter.

Clutter Group	Mean (m)	Covariance
1	${[-900,-1945]}^T$	$[1000,0;0,1000]$
2	${[1670,-203]}^T$	$[1000,0;0,1000]$

.

We conduct three MC experiments with 100 trials and average these results for every MC experiment. The results of cluster centers obtained by CFDP in Section 3.1 are listed in

Table 5. Simulation results of cluster centers.

MC	1	2	3
Cluster centers (m)	${[-894.6,-1943.0]}^T$	${[-893.4,-1943.3]}^T$	${[-899.9,-1944.8]}^T$
	${[1672.5,-205.7]}^T$	${[1667.4,-202.1]}^T$	${[1663.7,-207.8]}^T$

.

It can be seen from Table 5 that dense clutter groups’ estimated numbers are the same as the preset values, and cluster centers are estimated as closer to the preset dense clutter groups’ mean. This implies that the CFDP algorithm provides an accurate estimate of dense clutter groups’ numbers and cluster centers’ positions.

6.2.2. Analysis of Double Threshold Screening Effectiveness

In this section, we compare measurement association groups’ numbers before and after double threshold screening to validate the effectiveness of the clutter screening introduced in Section 3.2. Thus, data are chosen from random continuous times of $k-1$, $k$, and $k+1$. The comparison results are shown in

Table 6. Comparison of the number of measurement association groups before and after screening.

Time	Before	After
$k-1$	132,651	12,650
$k$	39,304	4352
$k+1$	85,184	7581

.

It can be observed from Table 6 that the effectiveness of the proposed clutter screening algorithm is pretty remarkable, which can effectively reduce the computational burden of data fusion. This implies that the pre-processing algorithm presented in Section 3 can significantly reduce the impact of dense group clutter.

6.3. Experiment 2: Analysis of the Effectiveness of Initial Screening Based on EKF

This section mainly explains the effectiveness of the proposed algorithm developed in Section 4.1 by comparing measurement association groups’ numbers before the initial screening and after separately running the initial screening algorithm based on angle association (Algorithm 2) [37] and EKF (Algorithm 3) in this work. Data are chosen from random continuous times of $k-1$, $k$, and $k+1$. The results are shown in

Table 7. Comparison of the number of measurement association groups before and after initial screening.

Time	Before	After (Algorithm 2)	After (Algorithm 3)
$k-1$	1728	932	167
$k$	2744	1474	243
$k+1$	4096	2120	560

.

As can be seen from Table 7, after using the algorithm introduced in Section 4.1, measurement association groups’ numbers can be decreased significantly compared with the algorithm presented in [37]. Further, the proposed algorithm can obviously reduce the computational complexity.

6.4. Experiment 3: Simulation Results

6.4.1. SASPS System

In this section, to verify the SAMPS-MAF algorithm presented in Section 4, 100 MC trials are run by PHD and LMB filters with measurements from the SASPS system, and a comparison by running 100 MC LMB filter trials with measurements from the single-active sensor (SAS) system consisting of sensors $S_1$ and $S_6$ is made. The simulation results are shown in Figure 11, where the cardinality mean is drawn in Figure 11a, while OSPA distance, OSPA location error, and OSPA cardinality error are, respectively, drawn in Figure 11b–d.

It is clear from Figure 11a,d that the cardinality means of PHD and LMB filters under the SASPS and SAS systems are close to the true ones, which shows the LMB filter has higher accuracy in the estimation of target numbers than that of the PHD filter under the SASPS system. According to Figure 11c, the location error under the SASPS system is smaller than the SAS system at any sampling time. According to Figure 11b, to sum up, the proposed SAMPS-MAF algorithm has a better effect for multi-target tracking under the SASPS system than the SAS system.

6.4.2. SAMPS System

To validate the multi-target tracking effectiveness of the SAMPS-MAF algorithm, 100 MC trials by PHD and LMB filters are run in this section. Compared to the SASPS system, we increase the number of passive sensors and use the passive sensors $S_{1-5}$ and active sensor $S_6$ in the SAMPS system. The simulation results are shown in Figure 12, where the cardinality mean is drawn in Figure 12a, while OSPA distance, OSPA location error, and OSPA cardinality error are, respectively, drawn in Figure 12b–d.

From Figure 12a,d, it is clear that the estimation of targets’ numbers for PHD and LMB filters under the SAMPS system are close to the true ones, but they have a larger error than under the SAS system. Additionally, according to Figure 11c and Figure 12c, the location error under the SAMPS system is not only much smaller than the SAS system at any sampling time but it also has a significant improvement compared to the SASPS system. From Figure 11b and Figure 12b, the proposed SAMPS-MAF algorithm can effectively estimate targets’ numbers and locations and has a better effect on the multi-target tracking accuracy under the SAMPS system than the SASPS and SAS systems.

6.4.3. MAMPS System

In this section, we use passive sensors $S_{1-3}$ and active sensors $S_{6-8}$ as the MAMPS system. Since the PHD filter cannot provide the target track label information, the track association can be performed based on the track label information provided by the LMB filter. Thus, to verify the multi-target tracking effectiveness of the proposed MAMPS-MAF-DNN algorithm, 100 MC trials using only the LMB filter are run. Besides the MAMPS-MAF-DNN algorithm, the SAMPS-MAF algorithm and MAMPS-MAF-CI algorithm combining the SAMPS-MAF algorithm and CI track fusion algorithm for the MAMPS system are used to make a comparison. The simulation results are shown in Figure 13, where the cardinality mean is drawn in Figure 13a, while OSPA distance, OSPA location error, and OSPA cardinality error are, respectively, drawn in Figure 13b–d.

It can be clearly seen from Figure 13a,d that estimations of the targets’ numbers for the MAMPS-MAF-DNN fusion are very close to the true values in most cases and have the smallest error among all comparison algorithms. In addition, according to Figure 13c, the location error of the MAMPS-MAF-DNN fusion algorithm is not only much smaller than that of the MAMPS-MAF-CI fusion algorithm but also has a significant improvement compared to the SAMPS system. Moreover, as shown in Figure 13b, the proposed MAMPS-MAF-DNN fusion algorithm can effectively estimate targets’ numbers and locations and has a better effect on the multi-target tracking accuracy.

7. Conclusions

For multi-target tracking using the MAMPS with dense group clutter, the clutter pre-processing algorithm of CFDP with double threshold screening included and the measurement initial screening algorithm based on EKF are proposed to solve the problems with clutter and high computational complexity due to an explosion in measurement association group numbers. Simultaneously, the MAMPS-MAF-DNN algorithm is proposed to solve the problem of ineffective fusion of multi-source sensor information. Experimental results validate that the proposed algorithm can estimate targets’ numbers and states in complex tracking scenarios with high-density clutter and outperform the popular CI fusion algorithm in tracking accuracy. The following conclusions can be extracted from this work.

• The clutter pre-processing algorithm can effectively reduce the number of measurement association groups on the premise of ensuring tracking accuracy. In other words, the amount of clutter can be effectively reduced to achieve the purpose of reducing computational complexity.
• The SAMPS-MAF algorithm can fully utilize measurement information from various sensors to achieve an effective fusion of multi-source sensors, and the tracking accuracy also increases with the growth of the number of sensors.
• The MAMPS-MAF-DNN algorithm provides a complete solution for achieving multi-target tracking with multiple active and passive sensors and fully exploits the advantages of DNNs to achieve the improvement of track fusion accuracy compared with traditional track fusion algorithms.
• This work only applies deep learning to track fusion and still uses RFS-based filtering algorithms. Future work will use deep learning to achieve target tracking, such as long short-term memory (LSTM) networks, transformer networks, etc.