1. Introduction
Bearings-only multi-target tracking (MTT) [
1,
2,
3] with clutter and missed detections is a challenging nonlinear problem. The filter for such a problem should not only deal with the nonlinearity of measurements, but also solve the measurement origin uncertainty of multiple targets. Bearings-only measurements [
4] are often obtained by passive sensors, such as sonars, and contain a low level of information about detected targets, leading to the observability problem [
5]. For bearings-only single-target tracking, the observable level can be evaluated by the Cramer–Rao lower bound (CRLB) [
6,
7]. However, it is difficult to evaluate the level of observability in a theoretical context for the bearings-only multi-target tracking with clutter and missed detections [
8]. In order to overcome the non-observable problem, the sensor needs to outmaneuver the targets [
5,
9]. As the origin of a measurement is not known for target tracking in clutter, the tracking filter needs to have a proper track-to-measurement-association method [
1] or something equivalent, such as symmetric measurement equations [
10] and random finite sets [
11].
For tracking targets using nonlinear measurements, the most popular method is the extended Kalman filter (EKF) [
12]. It linearizes the nonlinear measurement function around the predicted target state under the assumption that the true target state is close enough to the predicted state. Then, the measurement update of the filter can be performed using the Kalman filter (KF) [
12], which has a good performance for linear systems. The unscented Kalman filter (UKF) [
13] applies the unscented transform instead of linearization to the nonlinear measurement function. The probability density functions (pdfs) are represented and propagated using several sigma points. The particle filter (PF) [
14] represents the nonlinear pdfs by considering the amount of particles and can obtain better performances than the EKF and the UKF. However, particle filters have a heavy computational load. Apart from these nonlinear filters, many other filters exist for nonlinear systems, such as the cubature Kalman filter (CKF) [
15], the shifted Rayleigh filter (SRF) [
16] and the ensemble Kalman filter (EnKF) [
17].
To handle the measurement origin uncertainty problem in MTT with clutters, many techniques have been developed. Many of these methods belong to the following categories: joint probability data association (JPDA) [
18], multiple hypothesis tracking (MHT) [
19,
20] and random finite set (RFS) [
11] based methods. The JPDA filters come from combing the probability data association (PDA) [
1] filter with joint events and can only track fixed and known numbers of targets. To accommodate varied and unknown numbers of targets, the joint integrated PDA (JIPDA) [
21] filter, which is able to estimate the probability of target existence, is proposed. To improve the tracking accuracy, a multi-scan multi-target tracking algorithm, the joint integrated track splitting (JITS), was developed in [
22]. As the JIPDA and the JITS suffer from heavy computational load when tracking targets in mutual proximity, the linear multi-target IPDA (LMIPDA) and the linear multi-target ITS (LMITS) were proposed in [
22,
23], respectively. Besides, the iterative JIPDA (iJIPDA) [
24] tracker is also a computationally-efficient algorithm for the MTT. The MHT filters attempt to maintain and evaluate a set of measurement hypotheses with high track scores. There are many versions of the MHT filter, and most of them can be grouped into two classes: the track-oriented class [
25] and the measurement-oriented class [
19]. The JPDA and MHT approaches are formulated via data association, which takes a large proportion of computational resources, while the RFS approach is an emerging paradigm and is established without data association. The RFS-based filters [
26,
27,
28] treat multi-target states and measurements as the state finite set and the measurement finite set, respectively. The RFS-based filters try to estimate the target set based on the measurement set. In this way, the RFS-based filters are performed in a computational efficient way for the multi-target tracking.
Among RFS-based filters, the probability hypothesis density (PHD) [
26] filter is a first moment approximation to the multi-target predicted and posterior densities. It propagates the target intensities without considering data associations between targets and measurements. As there is no close form to the PHD filter, a sequential Monte Carlo (SMC) or Gaussian mixture (GM) technique was implemented, which resulted in the SMC-PHD [
29,
30] filter or the GM-PHD [
31] filter. For bearings-only MTT, it is difficult to apply the SMC-PHD filter because of the problem of extracting estimated target states. For this reason, the GM-PHD filter is considered in this paper. The GM-PHD filter associated with the EKF and the UKF are termed as GM-PHD-EKF and GM-PHD-UKF, respectively.
In the GM-PHD-EKF and the GM-PHD-UKF, the predicted intensity is modeled by a Gaussian mixture, and the likelihood function [
12] is approximated using a single Gaussian distribution. The updated intensity of each filter is obtained after the predicted intensity is updated by the measurements. As bearings-only measurements suffer from severe nonlinearity, the likelihood function approximated by a single Gaussian is not accurate enough. Furthermore, the accuracy of the updated intensity cannot get enough improvements after the predicted intensity is updated by the inaccurate likelihood function of the measurements. In this paper, the Gaussian mixture measurements-PHD (GMM-PHD) filter is proposed to address bearings-only MTT with clutter and missed detections. To improve the tracking accuracy, the target intensities are approximated by Gaussian mixtures, and the likelihood function is also modeled by a Gaussian mixture in the GMM-PHD filter. In this way, the updated intensity of the GMM-PHD is much more accurate than those of the GM-PHD-EKF and the GM-PHD-UKF after the predicted intensity is updated by a more accurate likelihood function, which is modeled by Gaussian mixtures. The proposed filter is a nonlinear MTT algorithm that can address not only bearings-only measurements, but also other nonlinear measurements. For bearings-only MTT, the targets may appear at any place in the measurement space. In order to reduce the number of selected parameters and avoid the undesirable effect of poor parameter selection [
8], the derivation of the GMM-PHD filter is processed with a partially-uniform target birth model [
8,
32], but not a Gaussian mixture birth model. In the simulation experiment, the performance of the GMM-PHD is compared with the GM-PHD-EKF and the GM-PHD-UKF in terms of the optimal subpattern assignment (OSPA) [
33] distance, the OSPA localization, the OSPA cardinality and the average CPU time.
The rest of the paper is arranged as follows. The bearings-only MTT problem is presented in
Section 2. The proposed GMM-PHD filter is derived in
Section 3. A simulation study is given in
Section 4. Finally, a conclusion is proposed in
Section 5.
2. The Bearings-Only MTT Problem
In this paper, bearings-only MTT with one passive sensor on a maneuvering platform is studied. The targets obey the continuous white noise acceleration (CWNA) motion model [
1,
12]. The target motion model and the sensor measurement model in the 2D Cartesian coordinates case are considered in this section. For the target
t, the target state
with position
and velocity
at time
k is expressed as:
An RFS
of target states can contain any number
of targets at time
k and is given by:
Let χ denote the single target state space and denote the set of all finite subsets of χ, i.e., and .
As the target
t is assumed to follow the CWNA motion model, its dynamic model can be expressed as:
where the state propagation matrix
Φ is time-invariant,
is a sequence of zero mean, white Gaussian process noises with covariance:
T is the sampling time,
is the
identity matrix and
q is the power spectral density (PSD) [
1]. The sensor state
is given as:
Each target
t can be detected with the probability
at time
k. The sensor can generate a measurement when the target is detected. We use
to denote the target measurement at time
k for target
t. The measurement from the radar is:
where:
is zero mean, white Gaussian measurement noise with covariance:
that it is uncorrelated with
. Therefore, each target
t can generate a measurement RFS
, which can be either
(the target
t is detectable) or ∅ (the target
t is not detected). In addition, the sensor also produces false measurements, which form an RFS
at each time
k. Then, the MTT measurement RFS
at time
k can be expressed as:
3. The Proposed GMM-PHD Filter
In RFS-based methods, the Bayesian recursion of multi-target posterior density is propagated in time as:
where
and
denote the multi-target transition density and likelihood function, respectively. Here,
denotes the multi-target posterior, density and
is an appropriate reference measure on
[
31].
To reduce the computational intractability, a first moment approximation (the PHD filter) of the recursion is proposed in [
26]. The general form of the PHD filter recursion (without target spawning) is given by:
where
and
denote the predicted intensity from time
to
k and the updated intensity, respectively,
is the survival probability, meaning the target still survives at time
k,
represents the single target transition density from time
to
k,
denotes the prior intensity of spontaneous target births at time
k,
is the single target measurement likelihood function and
is the clutter intensity.
3.1. Intensity Prediction
According to the Gaussian mixture assumption, the posterior intensity at time
can be expressed as:
Let
denote the bearing, range, course and speed in polar coordinates, respectively. The function
represents the transformation of the density
from polar coordinates to Cartesian coordinates. Then, the predicted intensity at time
k is given by:
where:
In Equation (
17), the survived probability
is assumed to be independent of the target state and is given as
. In Equation (
18),
is the expected number of targets appearing at time
k,
is the uniform distribution in
θ over the region
,
is the prior mean of the range and
is its prior variance. Similarly,
is the prior mean of the speed associated with its prior variance
, and
is the prior course variance. The predicted estimates
and
are obtained via the Kalman prediction:
The target state is augmented by a binary variable
β, so we may distinguish between the surviving components and the birth components:
where
. As noted in [
34], the surviving and birth components are separated to avoid biasing the cardinality estimates.
3.2. GMM Likelihood Approximation
In GM-PHD filters, the likelihood function is always modeled as a single Gaussian distribution. However, it is approximated by a Gaussian mixture in the proposed GMM-PHD filter.
Though the measurement noise is assumed to be Gaussian, the measurement uncertainty is non-Gaussian (non-ellipse) in Cartesian coordinates. The GMM measurement presentation first divides the non-elliptical measurement uncertainty into several segments. Then, each segment is modeled by one Gaussian distribution, and the whole measurement uncertainty can be approximated by a Gaussian mixture.
Suppose the measurement uncertainty in Cartesian coordinates is determined by the range interval
and the measurement
with standard deviation
. The range interval is divided into
subintervals, given by [
35,
36]:
where:
Then, each segment
a can be determined by
in polar coordinates. Let
and
, then the segment
a is approximated by a Gaussian distribution with mean
and covariance
in Cartesian coordinates:
where:
Obviously, the area of each segment is different, and the weight of the segment
a is proportional to its area, given by [
35]:
and
Then, the measurement likelihood function in Cartesian coordinate
is approximated as a Gaussian mixture:
where,
is the observation matrix and the constant
is calculated as [
37]:
Since the measurement noise is modeled as Gaussian, the likelihood function in polar coordinates
is expressed as:
Figure 1 gives an example of the GMM presentation. In the figure, the measurement uncertainty of the GMM-PHD is approximated by six measurement components (shown as the solid ellipses), and that of the GM-PHD-EKF is modeled by a single Gaussian distribution (shown as the dashed ellipse). Actually, the measurement uncertainty of the GM-PHD-UKF is also approximated by one Gaussian distribution. The true target is displayed by a cross. It is obvious that the measurement likelihood function approximated using Gaussian mixtures in the GMM-PHD is much more accurate than the approximation with one Gaussian in the GM-PHD-EKF. Thus, the GMM-PHD can perform with a much higher tracking accuracy than the GM-PHD-EKF and the GM-PHD-UKF.
3.3. Intensity Update
New targets are assumed to be always detected at their time of birth. Furthermore, we also assume that the detection probability for a surviving target is independent of their state and:
According to Equation (
14), the posterior intensity at time
k is given by:
and:
In the first step of Equation (
34), the likelihood function
is expressed by a Gaussian distribution Equation (
31), but not a Gaussian mixture Equation (
29), as the target birth model
is given in polar coordinates (uniform across the bearing space and Gaussian in the range and velocity). As both of them are expressed in the same polar coordinates, we have:
In the above equation,
where
is the indicator function of the bearing space region
and
is the volume of
, and we use the approximation:
which is reasonable in practice by assuming that
is small compared to the region
. The transformation function
transforms the function
from polar coordinates to Cartesian coordinates, given by:
where the weight
is given by Equation (
27), the position part of the mean
and the covariance
are given by Equations (24) and (25), respectively, and the velocity part is calculated by the approximation [
38,
39]:
where:
In the denominator of Equations (33) and (34), we have the following:
where
is the likelihood of measurement component
against the component
i with the innovation covariance:
and the updated target states and corresponding covariance are given by:
with the Kalman gain:
The following approximation for the posterior intensity can be obtained:
and:
where:
3.4. Component Management and State Extraction
Just like the GM-PHD filter, the number of Gaussian components of the updated intensity exponentially increases in time. To reduce the computational load, same techniques (component merging and pruning) in [
27,
31] are also implemented in the GMM-PHD filter. If the weight of a Gaussian component is lower than the preset threshold, it will be discarded. When some of components are close enough, they will be merged into one Gaussian component. If the total number of Gaussian components is bigger than the maximum value
, only
components with high probability will be retained. More details about component management can be found in [
31]. The state extraction is also the same as the method in [
31]. If the weights are bigger than a threshold, the corresponding states are extracted as the outputs of the filter.