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Sensors 2016, 16(5), 653; https://doi.org/10.3390/s16050653
Article
Cubature Information SMCPHD for MultiTarget Tracking
^{1}
School of Electronic and Information Engineering, Beihang University, Beijing 100191, China
^{2}
School of Information and Communication Engineering, North University of China, Taiyuan 030051, China
^{3}
Collaborative Innovation Center of Geospatial Technology, 129 Luoyu Road, Wuhan 430079, China
^{*}
Author to whom correspondence should be addressed.
Academic Editors:
Felipe Gonzalez Toro
and
Antonios Tsourdos
Received: 6 March 2016 / Accepted: 1 May 2016 / Published: 9 May 2016
Abstract
:In multitarget tracking, the key problem lies in estimating the number and states of individual targets, in which the challenge is the timevarying multitarget numbers and states. Recently, several multitarget tracking approaches, based on the sequential Monte Carlo probability hypothesis density (SMCPHD) filter, have been presented to solve such a problem. However, most of these approaches select the transition density as the importance sampling (IS) function, which is inefficient in a nonlinear scenario. To enhance the performance of the conventional SMCPHD filter, we propose in this paper two approaches using the cubature information filter (CIF) for multitarget tracking. More specifically, we first apply the posterior intensity as the IS function. Then, we propose to utilize the CIF algorithm with a gating method to calculate the IS function, namely CISMCPHD approach. Meanwhile, a fast implementation of the CISMCPHD approach is proposed, which clusters the particles into several groups according to the Gaussian mixture components. With the constructed components, the IS function is approximated instead of particles. As a result, the computational complexity of the CISMCPHD approach can be significantly reduced. The simulation results demonstrate the effectiveness of our approaches.
Keywords:
Sequential monte carlo; probability hypothesis density; importance sampling; cubature information filter; Gaussian mixturePACS:
J01011. Introduction
1.1. Background
Multitarget tracking refers to sequential approximation of the states (positions, velocities, etc.), and the number of individual targets. It has been widely used in groundmovingtarget tracking [1], visual tracking [2], and distribution fusion [3]. In multitarget tracking, both the state and observation sets of targets are timevarying. In practice, the associations between state and observation sets are always unknown, thus posing a challenge for multitarget tracking. The conventional approaches, such as the nearestneighbour Kalman filter (NNKF) [4], Extended Kalman Filter (EKF) [5,6], multiple hypothesis tracking (MHT) [7], and joint probabilistic data association (JPDA) [8], are used to formulate the explicit associations between states and observations of targets. However, caused by the targets appearing and disappearing, the state and observation sets of the multitarget are uncertain. Such uncertainty costs high complexity in the conventional approaches on constructing the association between them. Several approaches have emerged to improve the performance of conventional approaches in terms of tracking accuracy [9,10,11,12,13].
Recently, filters based on random finite sets (RFS) have been developed as alternative frameworks of the traditional algorithms [14,15,16,17] to estimate the multitarget number and states. In RFS formulations, both the multitarget state and observation sets are modelled as random finite sets. Given such models, the multitarget tracking problem is formulated under the Bayesian framework [14]. Since the optimal RFSbased Bayesian filter is computationally intractable, the probability hypothesis density (PHD) filter has been proposed as a suboptimal Bayesian filter for multitarget tracking [15,16,17]. By propagating the first order moment (namely the intensity), the PHD filter can save much computational complexity in comparison to the optimal RFSbased Bayesian filter. Moreover, it avoids the combinatorial problem arising from data association, which is the bottleneck for conventional multitarget tracking approaches to estimate multitarget number and states.
However, the difficulty of the PHD filter is that it is intractable to derive closeform solutions to the PHD filter formulations. To solve such difficulties, Vo et al. [18] proposed the bootstrap SMCPHD (BSMCPHD) filter, which selects the transition density as the IS function. The modified versions of BSMCPHD filter have emerged in [19,20,21]. As for those BSMCPHD filters, the IS function selection simplifies the weight computation in the prediction stage of the SMCPHD filter. In practice, such a selection leads to a few particles with largevalued weights in the update stage, when targets have nonlinear motion [22].
To improve the tracking performance of the conventional SMCPHD filter, several efficient approaches have been presented. Morelande et al. proposed the RaoBlackwellised SMCPHD (RBSMCPHD) filter in [23]. The RBSMCPHD filter utilizes several auxiliary variables to define the IS function, thus making it suitable for the intensity approximation of the SMCPHD filter with conditionally linear Gaussian models. Motivated by the auxiliary particle filter, Whiteley et al. proposed the auxiliary particle PHD (APHD) filter in [24], in which the auxiliary variable is preselected to minimize the variance of the IS weights. Although the APHD filter enhances the efficiency of the SMCPHD filter in nonlinear scenario, it yields poor performance in case of severe nonlinearities or high process noise [22]. Inspired by the unscented particle filter, Yoon et al. [25] utilized the unscented information filter (UIF) to design the IS function of SMCPHD (called USMCPHD filter). Since such a design takes the current observations of targets into account, it is more stable than the BSMCPHD filter. However, the drawback of this filter is that its performance has been influenced by the selection of the sigmapoints of UIF. Ristic et al. [20] proposed a novel state estimation method (called IBSMCPHD), rather than partitioning particles in adhoc manner. Their method groups the particles in the update stage, thus enjoying the computational efficiency.
1.2. Our Work and Contributions
In this paper, we propose a cubature information SMCPHD (CISMCPHD) and its fast implementation (FCISMCPHD) approaches, which can be used to estimate the timevarying number and states of multitarget. As aforementioned, the disadvantage of conventional SMCPHD filters is the tracking inefficiency in nonlinear scenario. To avoid such inefficiency, our CISMCPHD approach applies the posterior intensity as the IS function. With such a selection, the current observations can be incorporated into the IS function design. Then, we utilize the cubature information filter (CIF) [26] with a gating method to calculate the IS function. Benefitting from tracking accuracy of CIF in high dimensional nonlinear case, the CISMCPHD approach is capable of estimating the timevarying number and states of targets. To avoid initializing birth intensity in whole state space, a birth intensity initialization method is proposed for our CISMCPHD approach. At last, we present the FCISMCPHD approach to reduce the computational complexity by considering groups of particles as Gaussian mixture components. These components are applied to approximate the IS functions instead of particles of the CISMCPHD approach. Since the number of Gaussian components is much less than that of particles, the computational complexity can be greatly reduced. The main contributions of our work are listed as follows.
 (1)
 We propose a novel IS function approximating method, which utilizes the CIF with a gating method to enhance the estimation accuracy of the SMCPHD filter. Specifically, first, the posterior intensity is applied as the IS function of our CISMCPHD approach, in order to incorporate the current observation set into the IS function approximation. Then, the gating method is integrated into the update step of the CIF for approximating the IS function. Benefitting from the most recent success of CIF in nonlinear state estimation, the tracking performance of the proposed CISMCPHD approach is significantly enhanced.
 (2)
 We develop a method to initialize the birth intensity for the next tracking recursion. Since the current estimated targets (i.e., current survival targets) are not possible to be the birth targets at the next recursion, the observations of estimated targets are removed from the current observation set. Then, using an unbiased model, the remaining observations are mapped to state space for the birth intensity initialization. As such, the birth intensity can be adaptively initialized, making the target tracking more accurate and stable.
 (3)
 We develop a fast version of the CISMCPHD approach (namely FCISMCPHD). We first consider each group of particles as a Gaussian mixture component. Then these components are used to approximate the IS functions of the CISMCPHD approach. With the approximated IS functions, the particles can be sampled from these components for the intensity prediction and update steps. As a result, the computational complexity of the proposed CISMCPHD approach can be significantly reduced.
The rest of this paper is organized as follows. In Section 2, a brief overview of the SMCPHD filter is provided. Section 3 proposes our CISMCPHD approach. A fast implementation of CISMCPHD approach is presented in Section 4. Simulation results are demonstrated in Section 5, and Section 6 concludes this paper.
2. A Brief Overview of The SMCPHD Filter
In this section, we review the basic idea of the SMCPHD filter in detail. The main notations used in this section are defined as follows.
${\mathbf{x}}_{k}$  The state of a dynamic target at time k 
${\mathbf{z}}_{k}$  The observation of a dynamic target at time k 
${\gamma}_{k}(\xb7)$  The intensity of birth target at time k 
${L}_{k}$  The number of the survival particles at time k 
${J}_{k}$  The number of the birth particles at time k 
${p}_{s}(\xb7)$  The survival probability of target 
${p}_{d}(\xb7)$  The detected probability of target 
$\pi (\xb7,\xb7)$  The IS function of birth intensity 
$q(\xb7\xb7)$  The IS function of survival intensity 
The SMCPHD filter, motivated by the particle filter, is a sequential implementation of the PHD filter. In the SMCPHD filter, the posterior intensity can be represented by a set of random samples of state vector ${\mathbf{x}}_{k}$ with associated weights, which are usually called particles. By substituting these particles into the recursion of the PHD filter, the multidimensional integrals can be replaced by summations of the particles, which is computationally tractable.
More specifically, we define the particle set at time $k1$ as ${\{{\mathbf{x}}_{k1}^{\left(i\right)},{w}_{k1k1}^{\left(i\right)}\}}_{i=1}^{{L}_{k1}}$, where ${\mathbf{x}}_{k1}^{\left(i\right)}$ and ${w}_{k1k1}^{\left(i\right)}$ are the state and weight of the ith particle at time $k1$, respectively. The posterior intensity at time $k1$ can be modelled by
where ${\mathbf{Z}}_{1:k1}$ is the multitarget observations from time 1 to $k1$, and $\delta (\xb7)$ is the Dirac Delta function. Notice that ${\mathbf{Z}}_{k}=\{{\mathbf{z}}_{k,1},{\mathbf{z}}_{k,2},\dots ,{\mathbf{z}}_{k,M}\}$. Given ${D}_{k1k1}\left({\mathbf{x}}_{k1}\right{\mathbf{Z}}_{1:k1})$, the implementation of the SMCPHD filter consists of the prediction and update stages.
$${D}_{k1k1}\left({\mathbf{x}}_{k1}\right{\mathbf{Z}}_{1:k1})=\sum _{i=1}^{{L}_{k1}}{w}_{k1k1}^{\left(i\right)}\xb7\delta (\mathbf{x}{\mathbf{x}}_{k1}^{\left(i\right)})$$
Prediction: We first denote IS functions for the survival and birth targets as $q\left({\mathbf{x}}_{k}^{\left(i\right)}\right{\mathbf{x}}_{k1}^{\left(i\right)},{\mathbf{Z}}_{k})$ and $\pi ({\mathbf{x}}_{k}^{\left(i\right)},{\mathbf{Z}}_{k})$, respectively. Then, according to [18], the predicted intensity can be formulated by these IS functions,
where
$${D}_{kk1}\left({\mathbf{x}}_{k}\right{\mathbf{Z}}_{1:k1})=\sum _{i=1}^{{L}_{k1}+{J}_{k}}{w}_{kk1}^{\left(i\right)}\xb7\delta (\mathbf{x}{\mathbf{x}}_{k}^{\left(i\right)})$$
$${\mathbf{x}}_{k}^{\left(i\right)}\sim \left\{\begin{array}{cc}q\left({\mathbf{x}}_{k}^{\left(i\right)}\right{\mathbf{x}}_{k1}^{\left(i\right)},{\mathbf{Z}}_{k})\hfill & i=1,\dots ,{L}_{k1}\hfill \\ \pi ({\mathbf{x}}_{k}^{\left(i\right)},{\mathbf{Z}}_{k})\hfill & i={L}_{k1}+1,\dots ,{L}_{k1}+{J}_{k}\hfill \end{array}\right.$$
$${w}_{kk1}^{\left(i\right)}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\left\{\begin{array}{cc}\frac{{p}_{s}\left({\mathbf{x}}_{k1}^{\left(i\right)}\right)\xb7f\left({\mathbf{x}}_{k}^{\left(i\right)}\right{\mathbf{x}}_{k1}^{\left(i\right)})}{q\left({\mathbf{x}}_{k}^{\left(i\right)}\right{\mathbf{x}}_{k1}^{\left(i\right)},{\mathbf{Z}}_{k})}\phantom{\rule{0.166667em}{0ex}}\hfill & \phantom{\rule{0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}1,2,\dots ,{L}_{k1}\hfill \\ \frac{\gamma \left({\mathbf{x}}_{k}^{\left(i\right)}\right)}{{J}_{k}\xb7\pi ({\mathbf{x}}_{k}^{\left(i\right)},{\mathbf{Z}}_{k})}\phantom{\rule{0.166667em}{0ex}}\hfill & \phantom{\rule{0.166667em}{0ex}}i\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}{L}_{k1}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}1,\dots ,{L}_{k1}\phantom{\rule{0.166667em}{0ex}}+\phantom{\rule{0.166667em}{0ex}}{J}_{k}\hfill \end{array}\right.$$
In Equation (4), ${J}_{k}$ is calculated by ${J}_{k}=\nu \int \gamma \left(\mathbf{x}\right)d\mathbf{x}$, where ν is the particle number of each birth target. Equations (2)–(4) can be then used to predict the states and weights of the particles.
Update: In this stage, we obtain the posterior intensity by updating Equation (2). Then, we have the following update strategy,
where
$${D}_{kk}\left({\mathbf{x}}_{k}\right{\mathbf{Z}}_{1:k})=\sum _{i=1}^{{L}_{k1}+{J}_{k}}{w}_{kk}^{\left(i\right)}\delta (\mathbf{x}{\mathbf{x}}_{k}^{\left(i\right)})$$
$${w}_{kk}^{\left(i\right)}=\left(1{p}_{d}\left({\mathbf{x}}_{k}^{\left(i\right)}\right)+\sum _{\mathbf{z}\in {\mathbf{Z}}_{k}}\frac{{p}_{d}\left({\mathbf{x}}_{k}^{\left(i\right)}\right){g}_{k}\left(\mathbf{z}\right{\mathbf{x}}_{k}^{\left(i\right)})}{\kappa \left(\mathbf{z}\right)+{C}_{z}}\right){w}_{kk1}^{\left(i\right)}$$
$\kappa (\xb7)$ denotes the clutter intensity, and
$${C}_{z}=\sum _{i=1}^{{L}_{k1}+{J}_{k}}{p}_{d}\left({\mathbf{x}}_{k}^{\left(i\right)}\right){g}_{k}\left(\mathbf{z}\right{\mathbf{x}}_{k,i}){w}_{kk1}$$
Equations (2)–(6) include the main procedure of SMCPHD at one recursion. Commonly, to avoid the degeneracy of particles, the resampling strategy is utilized to resample particle set ${\{{\mathbf{x}}_{k}^{\left(i\right)},{w}_{kk}^{\left(i\right)}\}}_{i=1}^{{L}_{k1}+{J}_{k}}$. After resampling, we can use the clustering methods to extract number and states of targets.
3. The CISMCPHD Approach for MultiTarget Tracking
In this section, we present our CISMCPHD approach. To be more specific, we propose a novel IS function approximation algorithm incorporating the CIF and a gating method in Section 3.1. Then, Section 3.2 develops a method to initialize the birth intensity. Finally, Section 3.3 introduces the state extraction method for state estimation.
The nonlinear dynamic model of the target with state ${\mathbf{x}}_{k}$ at time k is given as follows,
where $\varphi (\xb7)$ is the state transition function, and $\phi (\xb7)$ denotes the relationship between state and observation. ${\mathbf{v}}_{k1}$ and ${\mathbf{w}}_{k}$ are the process and observation noises at time $k1$ and k, respectively. Both ${\mathbf{v}}_{k1}$ and ${\mathbf{w}}_{k}$ are assumed to be Gaussian noises with zero means, and their covariances are denoted as ${\mathbf{Q}}_{k1}$ and ${\mathbf{R}}_{k}$. According to this model, the transition density $f\left({\mathbf{x}}_{k}\right{\mathbf{x}}_{k1})$ and likelihood ${g}_{k}\left({\mathbf{z}}_{k}\right{\mathbf{x}}_{k})$ are subject to Gaussian distributions.
$$\begin{array}{cc}\hfill \mathrm{Process}\mathrm{model}:& {\mathbf{x}}_{k}=\varphi \left({\mathbf{x}}_{k1}\right)+{\mathbf{v}}_{k1}\end{array}$$
$$\begin{array}{cc}\hfill \mathrm{Observation}\mathrm{model}:& {\mathbf{z}}_{k}=\phi \left({\mathbf{x}}_{k}\right)+{\mathbf{w}}_{k}\end{array}$$
3.1. The IS Function Approximation Algorithm
As mentioned in Section 2, most of the conventional SMCPHD filters utilize the transition density function as the IS functions, resulting in great tracking error for targets with nonlinear dynamics. A novel IS function approximation algorithm, incorporating the CIF with a gating method, is presented to improve the tracking accuracy.
In our approach, we select the IS functions of Equations (3) and (4) as
where ${\mathbf{m}}_{k,s}^{(\xb7)}$ and ${\mathbf{m}}_{k,b}^{(\xb7)}$ are means of the survival and birth particles, respectively. ${\mathbf{P}}_{k,s}^{(\xb7)}$ and ${\mathbf{P}}_{k,b}^{(\xb7)}$ denotes the corresponding covariances of them.
$$\begin{array}{cc}\hfill \text{Survival IS}:& q\left({\mathbf{x}}_{k}^{\left(i\right)}\right{\mathbf{x}}_{k1}^{\left(i\right)},{\mathbf{Z}}_{k})=N({\mathbf{x}}_{k}^{\left(i\right)};{\mathbf{m}}_{k,s}^{\left(i\right)},{\mathbf{P}}_{k,s}^{\left(i\right)})\end{array}$$
$$\begin{array}{cc}\hfill \text{Birth IS}:& \pi \left({\mathbf{x}}_{k}^{\left(i\right)}\right{\mathbf{Z}}_{k})=N({\mathbf{x}}_{k}^{\left(i\right)};{\mathbf{m}}_{k,b}^{\left(i\right)},{\mathbf{P}}_{k,b}^{\left(i\right)})\end{array}$$
Then, the problem of IS function design can be reduced to calculate ${\mathbf{m}}_{k}^{\left(i\right)}$ and ${\mathbf{P}}_{k}^{\left(i\right)}$. Now, we discuss on how to calculate them. For simplicity, they are replaced by ${\mathbf{m}}_{k}$ and ${\mathbf{P}}_{k}$, respectively. Here we use the CIF and gating methods to estimate them.
Before introducing the CIF method, we review the cubature rules [27]. The cubature rules are used to approximate the Gaussian weight integral. Assuming $c\left(\mathbf{x}\right)$ is a function on the ndimension ${\mathbb{R}}^{n}$, its Gaussian weight integral can be approximated by
where
and ${\left[\mathbf{1}\right]}_{j}$ is the jth vector of the set
$${\mathbf{I}}_{N}\left(c\right)={\int}_{{\mathbb{R}}^{n}}c\left(\mathbf{x}\right)N(\mathbf{x};\mathbf{m},\mathbf{P})\approx \frac{1}{2n}\sum _{j=1}^{2n}c(\mathbf{m}+\sqrt{\mathbf{P}}{\mathbf{\alpha}}_{j})$$
$$\begin{array}{ccc}{\mathbf{\alpha}}_{j}\hfill & =\hfill & \sqrt{n}{\left[\mathbf{1}\right]}_{j}\hfill \end{array},j=1,2,\dots ,2n$$
$$\left\{\left[\begin{array}{c}1\hfill \\ 0\hfill \\ \vdots \hfill \\ 0\hfill \end{array}\right],\cdots ,\left[\begin{array}{c}0\hfill \\ 0\hfill \\ \vdots \hfill \\ 1\hfill \end{array}\right],\left[\begin{array}{c}1\hfill \\ 0\hfill \\ \vdots \hfill \\ 0\hfill \end{array}\right],\left[\begin{array}{c}0\hfill \\ 0\hfill \\ \vdots \hfill \\ 1\hfill \end{array}\right]\right\}$$
According to Equation (12), the cubature rules can be used to compute the multidimension integrals in the prediction and update steps of the CIF method.
Prediction: In this step, we first predict the state ${\mathbf{m}}_{kk1}$ and covariance ${\mathbf{P}}_{kk1}$ according to cubature rules. Then, the predicted information state vector ${\mathbf{y}}_{kk1}$ and matrix ${\mathbf{Y}}_{kk1}$ are estimated for the update step.
Let ${\mathbf{m}}_{k1}$ and ${\mathbf{P}}_{k1}$ be the previous state and covariance, respectively. According to Equations (12) and (13), the jth cubature point ${\mathbf{\chi}}_{k1,j}$ can be estimated by
$${\mathbf{\chi}}_{k1,j}=\sqrt{{\mathbf{P}}_{k1}}{\mathbf{\alpha}}_{j}+{\mathbf{m}}_{k1}$$
Then, we can calculate ${\mathbf{m}}_{kk1}$ and ${\mathbf{P}}_{kk1}$ using the following formulations:
where ${(\xb7)}^{T}$ is the transpose operator, and
$$\begin{array}{ccc}\hfill {\mathbf{m}}_{kk1}& =& \frac{1}{2n}\sum _{j=1}^{2n}{\mathbf{\chi}}_{k1,j}^{*}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {\mathbf{P}}_{kk1}& =& \frac{1}{2n}\sum _{j=1}^{2n}{\mathbf{\chi}}_{k1,j}^{*}{\left({\mathbf{\chi}}_{k1,j}^{*}\right)}^{T}{\mathbf{m}}_{kk1}{\left({\mathbf{m}}_{kk1}\right)}^{T}\hfill \\ & & +{\mathbf{Q}}_{k1}\hfill \end{array}$$
$${\mathbf{\chi}}_{k1,j}^{*}=\varphi \left({\mathbf{\chi}}_{k1,j}\right)$$
Given Equations (15) and (16), the information forms of ${\mathbf{m}}_{kk1}$ and ${\mathbf{P}}_{kk1}$ are represented [28] by
and
where ${\mathbf{y}}_{kk1}$ and ${\mathbf{Y}}_{kk1}$ are the information state and matrix, respectively.
$${\mathbf{y}}_{kk1}={\mathbf{Y}}_{kk1}{\mathbf{m}}_{kk1}$$
$${\mathbf{Y}}_{kk1}={\left({\mathbf{P}}_{kk1}\right)}^{1}$$
Update: We use the observation set ${\mathbf{Z}}_{k}$ to update the predicted ${\mathbf{y}}_{kk1}$ and ${\mathbf{Y}}_{kk1}$ in the current step. In order to construct the associations between the observation set ${\mathbf{Z}}_{k}$ and predicted observation ${\mathbf{z}}_{kk1}$, a gating method is applied to extract the associated observations. With the extracted observations, we can finally obtain ${\mathbf{m}}_{k}$ and covariance ${\mathbf{P}}_{k}$.
We denote ${\mathbf{z}}_{kk1}$ as the predicted observation, computed by
where
and
$$\begin{array}{ccc}\hfill {\mathbf{z}}_{kk1}& =& \frac{1}{2n}\sum _{j=1}^{2n}{\mathbf{\chi}}_{kk1,j}^{*}\hfill \end{array}$$
$${\mathbf{\chi}}_{kk1,j}^{*}=\phi \left({\mathbf{\chi}}_{kk1,j}\right)$$
$${\mathbf{\chi}}_{kk1,j}=\sqrt{{\mathbf{P}}_{kk1}}{\mathbf{\alpha}}_{j}+{m}_{kk1}$$
Utilizing the predicted observation ${\mathbf{z}}_{kk1}$ of Equation (20), the error cross covariance matrix of state and observation can be evaluated by
$$\begin{array}{ccc}\hfill {\mathbf{P}}_{kk1}^{mz}& =& \frac{1}{2n}\sum _{j=1}^{2n}({\mathbf{\chi}}_{kk1,j}{m}_{kk1}){({\mathbf{\chi}}_{kk1,j}^{*}{\mathbf{z}}_{kk1})}^{T}\hfill \\ & =& \frac{1}{2n}\sum _{j=1}^{2n}{\mathbf{\chi}}_{kk1,j}{\left({\mathbf{\chi}}_{kk1,j}^{*}\right)}^{T}{m}_{kk1}{\left({\mathbf{z}}_{kk1}\right)}^{T}\hfill \end{array}$$
With the above obtained parameters, we can calculate the state contribution and its corresponding information matrix as
and
where ${\mathbf{\mu}}_{j}$ is the innovation of the jth observation ${\mathbf{z}}_{j}$ (${\mathbf{z}}_{j}\in {\mathbf{Z}}_{k}$), expressed by
$${\mathbf{i}}_{k,j}={\mathbf{Y}}_{\mathbf{k}\mathbf{k}\mathbf{1}}{\mathbf{P}}_{kk1}^{mz}{\mathbf{R}}_{k}^{1}({\mathbf{\mu}}_{j}+{\left({\mathbf{Y}}_{\mathbf{k}\mathbf{k}\mathbf{1}}{\mathbf{P}}_{kk1}^{mz}\right)}^{T}{\mathbf{m}}_{kk1})$$
$${\mathbf{I}}_{k}={\mathbf{Y}}_{\mathbf{k}\mathbf{k}\mathbf{1}}{\mathbf{P}}_{kk1}^{mz}{\mathbf{R}}_{k}^{1}{\left({\mathbf{Y}}_{\mathbf{k}\mathbf{k}\mathbf{1}}{\mathbf{P}}_{kk1}^{mz}\right)}^{T}$$
$${\mathbf{\mu}}_{j}={\mathbf{z}}_{j}{\mathbf{z}}_{kk1}$$
In our scenario, ${\mathbf{z}}_{j}$ is a twodimension vector, and ${\mathbf{\mu}}_{j}$ follows a twodimension Gaussian distribution.
In practice, the observation set may contain large clutters. The existence of these clutters cannot only degenerate the estimation accuracy, but also increase the computational complexity. Recently, several gating technologies have been proposed to remove the clutters from the observation set [29,30]. Inspired by [30], we utilize the gating technology to reduce the influence of clutters.
Intuitively, observations far away from the predicted observation are subject to be generated by clutters. These observations must be removed from the observation set. With the gating technology, the left observations can be represented by
where ${\mathbf{P}}_{kk1}^{zz}$ is the covariance matrix of the predicted observation ${\mathbf{z}}_{kk1}$, and ${(\xb7)}^{1}$ is the matrix inversion. ${T}_{h}$ is the threshold of the gate. According to Equation (27), the innovation ${\mathbf{\mu}}_{j}$ follows the Chisquare distribution. Thus, ${T}_{h}$ can be determined by the dimension of ${\mathbf{\mu}}_{j}$ and association probability. Commonly, the square root of ${T}_{h}$ is known as the number of Sigma. Literature [30] proved that the number of Sigma gates ranging from 3 to 5 (corresponding to ${T}_{h}=925$) can guarantee the true observation lying inside the gate with “enough” probability ($\ge 0.971$), when the dimension of ${\mathbf{\mu}}_{j}$ is less than three. In this paper, we select the number of Sigma gates being to 4 (corresponding to ${T}_{h}=16$). When the dimension of ${\mathbf{\mu}}_{j}$ is less than three, such a selection guarantee that the association probability $\ge 0.998$.
$$\begin{array}{cc}\hfill {\widehat{\mathbf{Z}}}_{k}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\left\{{\mathbf{z}}_{k,j}\right{\mathbf{\mu}}_{j}^{T}{\left({\mathbf{P}}_{kk1}^{zz}\right)}^{1}{\mathbf{\mu}}_{j}<\sqrt{{T}_{h}}\}& {\mathbf{z}}_{k,j}\in {\mathbf{Z}}_{k}\end{array}$$
Then, we concentrate on computing ${\mathbf{P}}_{kk1}^{zz}$ of Equation (27). Let ${\mathbf{P}}_{kk1}^{mz}$ be the cross covariance matrix between observation and state space. According to the linear error propagating of [31], ${\mathbf{P}}_{kk1}^{mz}$ can be rewritten as
where ${\mathbf{H}}_{k}$ is the linearized matrix.
$${\mathbf{P}}_{kk1}^{mz}\simeq {\mathbf{P}}_{kk1}{\mathbf{H}}_{k}^{T}$$
Obviously, ${\mathbf{H}}_{k}$ can be approximated by
$${\mathbf{H}}_{k}\simeq {\left({\mathbf{P}}_{kk1}^{mz}\right)}^{T}{\mathbf{P}}_{kk1}^{1}$$
With the achieved ${\mathbf{H}}_{k}$, according to [32], ${\mathbf{P}}_{kk1}^{zz}$ can be calculated by
$${\mathbf{P}}_{kk1}^{zz}={\mathbf{H}}_{k}{\mathbf{P}}_{kk1}{\mathbf{H}}_{k}^{T}+{\mathbf{R}}_{k}$$
Substituting the achieve ${\mathbf{P}}_{kk1}^{zz}$ into Equation (27), ${\widehat{\mathbf{Z}}}_{k}$ can be extracted from the current observation set ${\mathbf{Z}}_{k}$.
With the extracted observation set ${\widehat{\mathbf{Z}}}_{k}$, the information state vector ${\mathbf{y}}_{k}$ and matrix ${\mathbf{Y}}_{k}$ are represented as:
$$\begin{array}{c}\hfill {\mathbf{y}}_{k}={\mathbf{y}}_{kk1}+\sum _{j=1}^{{\widehat{\mathbf{Z}}}_{k}}{\mathbf{i}}_{k,j}\end{array}$$
$$\begin{array}{c}\hfill {\mathbf{Y}}_{k}={\mathbf{Y}}_{kk1}+\sum _{j=1}^{{\widehat{\mathbf{Z}}}_{k}}{\mathbf{I}}_{k,j}\end{array}$$
Given information state ${\mathbf{y}}_{k}$ and matrix ${\mathbf{Y}}_{k}$, posterior state ${\mathbf{m}}_{k}$ can be reconstructed based on Equation (18) :
$${\mathbf{m}}_{k}={{\mathbf{Y}}_{k}}^{1}{\mathbf{y}}_{k}$$
Moreover, the posterior covariance ${\mathbf{P}}_{k}$ is recovered based on Equation (19):
$${\mathbf{P}}_{k}={\left({\mathbf{Y}}_{k}\right)}^{1}$$
If there is no observation that lies inside the gate (${\widehat{\mathbf{Z}}}_{k}=\varnothing $), we approximate ${\mathbf{m}}_{k}$ and ${\mathbf{P}}_{k}$ in the following,
$$\begin{array}{c}\hfill {\mathbf{m}}_{k}={\mathbf{m}}_{kk1}\end{array}$$
$$\begin{array}{c}\hfill {\mathbf{P}}_{k}={\mathbf{P}}_{kk1}\end{array}$$
Substituting the above obtained ${\mathbf{m}}_{k}$ and ${\mathbf{P}}_{k}$ into Equations (10) and (11), we can approximate the IS functions of survival and birth targets for our CISMCPHD approach.
3.2. The Birth Intensity Initialization Method
According to Equation (2), the birth intensity has large effect on the posterior intensity estimation. Targets may “born at anywhere” of the state space. In other words, birth intensity $\gamma \left(\mathbf{x}\right)$ may cover the whole state space, which is rather exhaustive [25]. To avoid such a disadvantage, observationdriven birth intensity initiation methods were proposed [20,33,34]. Inspired by these methods, an adaptive birth intensity initialization method is proposed for the CISMCPHD approach. Instead of initializing birth intensity across the whole state space, the proposed method of CISMCPHD approach utilizes the current observations and estimated targets to initialize the birth intensity at the next recursion. Compared with the conventional SMCPHD filters, our method can initialize the birth intensity without knowing it as a prior.
The implementation of our method consists of two steps. First, in order to initialize the birth intensity, we remove observations generated by the estimated targets That is because the current survival targets cannot be newborn targets at the next recursion. Second, we use the remaining observations to estimate the birth target components, which can be used to calculate the birth intensity. With these two steps, the birth intensity can be initialized for the next recursion.
Step1. Remove observations generated by the estimated targets.
In the basic PHD filter, it is assumed that each target can yield at most one observation [35]. According to this assumption, each target has one and only one corresponding observation. Influenced by the noises and clutters, the observation generated by the target may appear around the target. In other words, observations around the target has the large probability to be generated by the same target. Therefore, the birth target state set can be estimated by removing states of estimated targets from the multitarget state.
Here, we adopt the bearing and range tracking model [36] to illustrate the birth intensity initialization method of our CISMCPHD approach. Let ${\mathbf{x}}_{k,i}^{e}$ be the state of the ith target in the estimate state set ${\mathbf{X}}_{k}^{e}$. ${\mathbf{x}}_{k,i}^{e}$ consists of position and velocity, while ${\mathbf{z}}_{k,j}$ (${\mathbf{z}}_{k,j}\in {\mathbf{Z}}_{k}$) consists of the bearing angle and range. We define the distance between ${\mathbf{x}}_{k,i}^{e}$ and ${\mathbf{z}}_{k,j}$ (${\mathbf{z}}_{k,j}\in {\mathbf{Z}}_{k}$) as
where ${(\xb7)}_{r}$ denotes the rangedimension element, and $\xb7$ is the absolution value.
$${d}_{i,j}=\left{({\mathbf{z}}_{k,j}\phi \left({\mathbf{x}}_{k,i}^{e}\right))}_{r}\right$$
We follow the way of [30] to select the certain threshold for Equation (37),
where ${\sigma}_{r}$ is the error of the rangedimension (known as a prior). l is the confidence level, commonly selected from $l=3\sim 5$. Here, we use $l=3$, which can guarantee that the associated probability equals to 0.997.
$${d}_{i,j}<l\xb7{\sigma}_{r}$$
With Equations (37) and (38), we can remove the observations associated with the estimated targets. Let ${\tilde{\mathbf{Z}}}_{k}$ be the observations of birth targets, the removing procedure is summarized in Table 1.
Step2. Estimate the birth target components.
Once ${\tilde{\mathbf{Z}}}_{k}$ is obtained, we turn to estimate the birth target components (the mean of the ith target state vector ${\mathbf{m}}_{k,b}^{\left(i\right)}$ and its corresponding covariance ${\mathbf{P}}_{k,b}^{\left(i\right)}$) by the unbiased model of [37].
Let ${\tilde{\mathbf{z}}}_{k,i}\in {\tilde{\mathbf{Z}}}_{k}$, we map ${\tilde{\mathbf{z}}}_{k,i}$ into state space denoted by ${\tilde{\mathbf{z}}}_{k,i}^{c}={[{p}_{k,i}^{x},{p}_{k,i}^{y}]}^{T}$. ${p}_{k,i}^{x}$ and ${p}_{k,i}^{y}$ can be computed by ${p}_{k,i}^{x}={\beta}_{\theta}^{1}{r}_{k,i}cos{\theta}_{k,i}$ and ${p}_{k,i}^{y}={\beta}_{\theta}^{1}{r}_{k,i}sin{\theta}_{k,i}$. ${\beta}_{\theta}={\sigma}_{\theta}$ is a biased comparison factor, where ${\sigma}_{\theta}$, as a prior, is the error of bearing angle ${\theta}_{k,i}$. According to [37], ${\mathbf{m}}_{k}^{\left(i\right)}$, the mean of the ith birth target state, can be estimated as
$${\mathbf{m}}_{k}^{\left(i\right)}={[{p}_{k,i}^{x},0,{p}_{k,i}^{y},0,0]}^{T}$$
The covariance can be approximated by
where ${\sigma}_{v}$, as a prior, is the standard deviation of velocity. In Equation (40), the following exists,
$${\mathbf{P}}_{k}^{\left(i\right)}=\left[\begin{array}{ccccc}{\sigma}_{xx}& 0& {\sigma}_{xy}& 0& 0\\ 0& {\sigma}_{v}& 0& 0& 0\\ {\sigma}_{yy}& 0& {\sigma}_{xy}& 0& 0\\ 0& 0& 0& {\sigma}_{v}^{2}& 0\\ 0& 0& 0& 0& {\sigma}_{\theta}^{2}\end{array}\right]$$
$$\left\{\begin{array}{cc}{\sigma}_{xx}=\hfill & ({\beta}_{\theta}^{2}2){\left({\tilde{r}}_{k,i}\right)}^{2}{cos}^{2}\left({\theta}_{k,i}\right)+0.5({\left({\tilde{r}}_{k,i}\right)}^{2}\hfill \\ & +{\sigma}_{r}^{2})(1+{\beta}_{\theta}^{4}cos\left(2{\theta}_{k,i}\right))\hfill \\ {\sigma}_{xy}=\hfill & ({\beta}_{\theta}^{2}2){\left({\tilde{r}}_{k,i}\right)}^{2}cos\left({\theta}_{k,i}\right)sin\left({\theta}_{k,i}\right)+0.5({\left({\tilde{r}}_{k,i}\right)}^{2}\hfill \\ & +{\sigma}_{r}^{2})(1+{\beta}_{\theta}^{4}cos\left(2{\theta}_{k,i}\right))\hfill \\ {\sigma}_{yy}=\hfill & ({\beta}_{\theta}^{2}2){\left({\tilde{r}}_{k,i}\right)}^{2}{sin}^{2}\left({\theta}_{k,i}\right)+0.5({\left({\tilde{r}}_{k,i}\right)}^{2}\hfill \\ & +{\sigma}_{r}^{2})(1{\beta}_{\theta}^{4}cos\left(2{\theta}_{k,i}\right))\hfill \end{array}\right.$$
Finally, we can construct the newborn targets as ${\{{\mathbf{m}}_{k,i},{\mathbf{P}}_{k,i}\}}_{i=1}^{{N}_{k},b}$, where ${N}_{k,b}=\left{\tilde{\mathbf{Z}}}_{k}^{c}\right$ is considered as the number of birth targets. We use Equation (11) to sample states of birth particles. The weights of these birth particles are initialized with the same values, ${w}_{k,b}^{\left(i\right)}=\frac{\int \gamma \left(\mathbf{x}\right)d\mathbf{x}}{N\xb7{N}_{k,b}}$, where N is the number particles for each target, and $\gamma \left(\mathbf{x}\right)$ is defined in Section 2. On this basis, these birth particles become survival particles at time $k+1$. That is to say, the IS functions of these particles at time $k+1$ can be computed by the method of Section 3.1. Notice that the newborn target in this section may contain clutters, and these clutters can be removed in the resampling step of the CISMCPHD approach.
3.3. State Estimation
In multitarget tracking, it is rather important to estimate the target number and states. As for the state estimation, clustering methods, are commonly used in SMCPHD filters [16,18]. However, they are subject to biased estimation [21]. Ristic et al. [21] proposed an method that clusters the particles into several groups at the update stage. In this paper, we intend to adopt the method of [38] for state estimation, which is an improved method of [21]. There are also several alternative methods, such as Zhao’s method [39] and MEAP method [40], which have better estimation performance.
According to Equation (6), the updated weight ${w}_{k}^{i}$ of the ith particle consists of two parts,
$${w}_{k}^{(i,j)}=\left\{\begin{array}{cc}(1{p}_{d}\left({x}_{k}\right)){w}_{kk1}^{\left(i\right)}\phantom{\rule{0.166667em}{0ex}}\hfill & \phantom{\rule{0.166667em}{0ex}}j=0\hfill \\ \frac{{p}_{d}\left({\mathbf{x}}_{k}^{\left(i\right)}\right){g}_{k}\left({\mathbf{z}}_{k,j}\right{\mathbf{x}}_{k}^{\left(i\right)})}{\kappa \left({\mathbf{z}}_{k,j}\right)+{C}_{{z}_{k,j}}}{w}_{kk1}^{i}\phantom{\rule{0.166667em}{0ex}}\hfill & \phantom{\rule{0.166667em}{0ex}}j=1,\dots ,{\widehat{\mathbf{Z}}}_{k}\hfill \end{array}\right.$$
In Equation (42), $j=0$ denotes that there is no observation, and ${C}_{{z}_{k,j}}$ can be computed by Equation (7). For state estimation, we aggregate ${W}^{j}$ of particle weights corresponding to observation ${\mathbf{z}}_{k,j}$,
$$\begin{array}{cc}\hfill {W}^{j}=\sum _{i=1}^{{L}_{k1}+{J}_{k}}{w}_{k}^{(i,j)}& j=0,\dots ,{\widehat{\mathbf{Z}}}_{k}\end{array}$$
According to Equations (42) and (43), if ${z}_{k,j}$ is generated by the clutter, then the likelihood ${g}_{k}\left({\mathbf{z}}_{k,j}\right{\mathbf{x}}_{k}^{\left(i\right)})$ may be small, leading to low value of ${W}^{j}$. However, if ${z}_{k,j}$ is generated by the target, then ${W}^{j}$ may be large due to the large value of ${g}_{k}\left({\mathbf{z}}_{k,j}\right{\mathbf{x}}_{k}^{\left(i\right)})$. Thus, setting certain threshold ${W}_{th}$ for ${W}^{j}$, we can assign particles $\{{\mathbf{x}}_{k,j}^{\left(i\right)},{w}_{k}^{(i,j)}\}$ that satisfy ${W}^{j}>{W}_{th}$ to the jth target. In this paper, we set ${W}_{th}=0.5$, the same as [21]. Then, ${\mathbf{x}}_{k,j}$ and ${\mathbf{P}}_{k,j}$ can be calculated in the following
$$\begin{array}{ccc}\hfill {\mathbf{x}}_{k,j}& =& \frac{1}{{W}^{j}}\sum _{i=1}^{{L}_{k1}+{J}_{k}}{w}_{k}^{i,j}{\mathbf{x}}_{k,j}^{\left(i\right)}\hfill \end{array}$$
$$\begin{array}{ccc}\hfill {\mathbf{P}}_{k,j}& =& \frac{1}{{W}^{j}}\sum _{i=1}^{{L}_{k1}+{J}_{k}}{w}_{k}^{i,j}({\mathbf{x}}_{k,j}^{\left(i\right)}{\mathbf{x}}_{k,j})({\mathbf{x}}_{k,j}^{\left(i\right)}{\mathbf{x}}_{k,j})\hfill \end{array}$$
Given Equations (44) and (45), the states of multitarget can be finally output. We summarize our approach at time k in Table 2.
4. A Fast Approach For The CISMCPHD Filter
In this section, we focus on reducing the computational complexity of our CISMCPHD approach. Section 4.1 presents a fast implementation of the CISMCPHD approach, namely the FCISMCPHD approach. Then, we analyze the computational complexity of the CISMCPHD and FCISMCPHD approaches in Section 4.2. The framework of the improved approach is illustrated in Figure 1.
4.1. The Fast CISMCPHD Filter
In this section, we introduce the FCISMCPHD approach, which can save the computational time of the CISMCPHD approach. Inspired by [41], we consider the particle groups of targets as the Gaussian mixture components. Recall that ${N}_{k1,b}$ is the number of birth components. ${\mathbf{m}}_{k1,b}^{\left(i\right)}$ and ${\mathbf{P}}_{k1,b}^{\left(i\right)}$ denote the mean and covariance of the ith birth component, respectively. The posterior intensity of Equation (1) can be approximated by
where ${\mathbf{m}}_{k1,s}^{\left(i\right)}$ and ${\mathbf{P}}_{k1,s}^{\left(i\right)}$ are the mean and covariance of the ith survival component, respectively. ${G}_{k1,s}^{\left(i\right)}$ is its corresponding weight, ${N}_{k1,s}$ is the number of survival components, and ${G}_{k1,b}^{\left(i\right)}$ is the weight of the ith birth component. Here, ${G}_{k1,b}^{\left(i\right)}=\frac{\int \gamma \left(\mathbf{x}\right)d\mathbf{x}}{{N}_{k1,b}}$.
$${D}_{k1k1}\left({\mathbf{x}}_{k1}\right{\mathbf{Z}}_{1:k1})\approx \sum _{i=1}^{{N}_{k1,s}}{G}_{k1,s}^{\left(i\right)}N({\mathbf{x}}_{k1};{\mathbf{m}}_{k1,b}^{\left(i\right)},{\mathbf{P}}_{k1,b}^{\left(i\right)})+\sum _{i=1}^{{N}_{k1,b}}{G}_{k1,b}^{\left(i\right)}N({\mathbf{x}}_{k1};{\mathbf{m}}_{k1}^{\left(i\right)},{\mathbf{P}}_{k1}^{\left(i\right)})$$
Commonly, birth components at time $k1$ become survival components at time k. We combine birth and survival components into one set. That is ${\{{G}_{k1}^{\left(i\right)},{\mathbf{m}}_{k1}^{\left(i\right)},{\mathbf{P}}_{k1}^{\left(i\right)}\}}_{i=1}^{{N}_{k1,b}+{N}_{k1,s}}={\{{G}_{k1,b}^{\left(i\right)},{\mathbf{m}}_{k1,b}^{\left(i\right)},{\mathbf{P}}_{k1,b}^{\left(i\right)}\}}_{i=1}^{{N}_{k1,b}}\cup {\{{G}_{k1,s}^{\left(i\right)},{\mathbf{m}}_{k1,s}^{\left(i\right)},{\mathbf{P}}_{k1,s}^{\left(i\right)}\}}_{i=1}^{{N}_{k1,s}}$, where ${G}_{k1}^{\left(i\right)},{\mathbf{m}}_{k1}^{\left(i\right)},{\mathbf{P}}_{k1}^{\left(i\right)}$ denote the weight, mean and covariance of the ith combined component.
With the combined components, we use the CIF method of Section 3.1 to approximate the IS function of each component. Here, we utilize $q\left(\mathbf{x}\right{\mathbf{m}}_{k1}^{\left(i\right)},{\mathbf{Z}}_{k})$ to represent the IS function of the ith component. On this basis, the jth predicted particle, which is generated by the ith component, can be represented by
where $f\left({\mathbf{x}}_{k}^{\left(j\right)}\right{\mathbf{x}}_{k1}^{\left(i\right)})=N({\mathbf{x}}_{k}^{\left(j\right)};{\mathbf{m}}_{kk1}^{\left(i\right)},{\mathbf{P}}_{kk1}^{\left(i\right)})$. In addition, ${\mathbf{m}}_{kk1}^{\left(i\right)}$ and ${\mathbf{P}}_{kk1}^{\left(i\right)}$ are the predicted mean and covariance of ith component, respectively, which can be computed by Equations (15) and (16). $\lfloor \xb7\rfloor $ denotes the nearest floor integer, and $\lfloor {G}_{k1}^{\left(i\right)}\xb7N\rfloor $ is the number of particles generated by the ith component. According to Equations (47) and (48), the number of predicted particles is ${N}_{k}={\sum}_{a=1}^{{N}_{k,s}+{N}_{k,b}}\lfloor {G}_{k1}^{\left(a\right)}\xb7N\rfloor $.
$${\mathbf{x}}_{k}^{\left(j\right)}\sim \begin{array}{cc}q\left({\mathbf{x}}_{k}^{\left(j\right)}\right{\mathbf{m}}_{k1}^{\left(i\right)},{\mathbf{Z}}_{k})\hfill & j=\sum _{a=1}^{i1}\lfloor {G}_{k1}^{\left(a\right)}\xb7N\rfloor +1,\sum _{a=1}^{i1}\lfloor {G}_{k1}^{\left(a\right)}\xb7N\rfloor +2,\dots ,\sum _{a=1}^{i}\lfloor {G}_{k1}^{\left(a\right)}\xb7N\rfloor \hfill \end{array}$$
$${w}_{kk1}^{\left(j\right)}\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\begin{array}{cc}\frac{{p}_{s}\left({\mathbf{x}}_{k1}^{\left(i\right)}\right)\xb7f\left({\mathbf{x}}_{k}^{\left(j\right)}\right{\mathbf{x}}_{k1}^{\left(i\right)})}{q\left({\mathbf{x}}_{k}^{\left(j\right)}\right{\mathbf{m}}_{k1}^{\left(j\right)},{\mathbf{Z}}_{k})}\xb7\frac{{G}_{k1}^{\left(a\right)}}{\lfloor {G}_{k1}^{\left(a\right)}N\rfloor}\phantom{\rule{0.166667em}{0ex}}\hfill & \phantom{\rule{0.166667em}{0ex}}j=\sum _{a=1}^{i1}\lfloor {G}_{k1}^{\left(a\right)}\xb7N\rfloor +1,\sum _{a=1}^{i1}\lfloor {G}_{k1}^{\left(a\right)}\xb7N\rfloor +2,\dots ,\sum _{a=1}^{i}\lfloor {G}_{k1}^{\left(a\right)}\xb7N\rfloor \hfill \end{array}$$
Then, the predicted weight of Equation (48) is substituted into Equation (42) for particle grouping and state estimation. Recall that, weight ${W}^{j}$ of Equation (43) can be used to assign the particles into the jth group, where the mean ${\mathbf{x}}_{k,j}$ and covariance ${\mathbf{P}}_{k,j}$ can be calculated by Equations (44) and (45) in Section 3.3. Given ${W}^{j}$, ${\mathbf{x}}_{k,j}$ and ${\mathbf{P}}_{k,j}$, we model the jth group as a Gaussian mixture component $\{{G}_{k}^{\left(j\right)},{\mathbf{x}}_{k,j},{\mathbf{P}}_{k,j}\}$, where we set ${G}_{k}^{\left(j\right)}=\frac{{W}_{j}}{{max}_{j}{W}_{j}}$. With such a selection, it can guarantee that the groups with large ${W}^{j}$ have enough number of sampling particles. Note that the group with small ${W}^{j}$ has the large probability to be generated by the clutter, and such a group should be neglected to avoid the waste of computational time. We set a certain threshold ${T}_{g}$ for $\left\{{G}_{k}^{\left(j\right)}\right{G}_{k}^{\left(j\right)}>{T}_{g}\}$, subject to ${T}_{g}\xb7N\ll N$. In this paper, we set ${T}_{g}=0.1$. The construction procedure of the target components is summarized in Table 3.
The target components of Table 3 refer to the survival target components. The birth components and target state estimation can be achieved by Section 3.2 and Section 3.3, respectively.
4.2. Computational Complexity
In this section, we analyze the computational complexity of the CISMCPHD, FCISMCPHD and conventional SMCPHD approaches. For justice, we adopt the same state estimation and birth target initialization methods for the three approaches. The computational complexity of the three approaches on state estimating and birth target initializing is the same, when they have same particle numbers and observations. Thus, it can be neglected for the computational complexity comparing. In addition, we select the multinomial resampling method as the resampling methods of the CISMCPHD and conventional SMCPHD approaches. The particle numbers of these approaches are equal to ${N}_{p}$.
We begin with the computational complexity analysis of the CISMCPHD approach. Incorporating the CIF and gating methods into the SMCPHD approach, the CISMCPHD approach can achieve a good estimation accuracy in nonlinear target tracking. As mentioned in Section 3.1, each particle is applied to approximate the IS functions. The computational complexity of the CIF and gating method per particle is nearly $O\left({n}_{d}^{3}\right)$, where ${n}_{d}$ is the dimension of the particle state. The computational complexity of the CIFbased IS functions is $O({N}_{p}\xb7{n}_{d}^{3})$. Besides, the computational complexity of the PHD update step is $O({N}_{p}\xb7M)$, where M is the number of observations. In addition, the resampling step of our CISMCPHD approach consumes $O\left({N}_{p}\right)$ computational complexity. Thus, the computational complexity of our CISMCPHD approach in total is $O({N}_{p}\xb7{n}_{d}^{3}+{N}_{p}\xb7M+{N}_{p})$.
Then, we turn to the analysis of the computational complexity of the FCISMCPHD approach. Since the FCISMCPHD approach adopts the target components to compute the CIFbased IS functions, the computational complexity of the CIFbased IS functions is $O({N}_{G}\xb7{n}_{d}^{3})$, where ${N}_{G}$ is the number of target components, ${N}_{G}\ll {N}_{p}$. Assuming that in the PHD prediction step, ${N}_{G}$ target components generate ${N}_{p}$ particles. The computational complexity of the PHD update step is $O({N}_{p}\xb7M)$. Besides, the Gaussian target component forming consumes $O(M+1)$. Combining the computational complexity of these steps together, the computational complexity of the FCISMCPHD approach is $O({N}_{G}\xb7{n}_{d}^{3}+{N}_{p}\xb7M+M+1)$. In practice, $M\ll {N}_{p}$, the computational complexity of FCISMCPHD filter is much less than the CISMCPHD filter.
The conventional SMCPHD approach uses the transitional density as the IS function. Compared with the CISMCPHD and FCISMCPHD approaches, it does not need to the IS function computing. Hence, the computational complexity of the conventional SMCPHD approach can be approximated as $O({N}_{p}\xb7M+{N}_{p})$, where $O({N}_{p}\xb7M)$ and $O\left({N}_{p}\right)$ are the computational complexity of the update and resampling steps. Obviously, $O({N}_{p}\xb7M+{N}_{p})<O({N}_{G}\xb7{n}_{d}^{3}+{N}_{p}\xb7M+M+1)<O({N}_{p}\xb7{n}_{d}^{3}+{N}_{p}\xb7M+{N}_{p})$. Thus, the computational complexity of the conventional SMCPHD approach is smaller than the CISMCPHD and FCISMCPHD approaches.
With the above discussion, we can conclude that the conventional SMCPHD approach has the lowest computational complexity, and the CISMCPHD approach has the highest computational complexity. However, the estimation accuracy of the conventional SMCPHD approach is the lowest, and the estimation accuracy of the CISMCPHD approach is the highest. The FCISMCPHD approach can make a tradeoff between the computational complexity and estimation accuracy. Such a conclusion can be observed in Section 5.
5. Simulation Results
In this section, we validate the tracking performance of the proposed CISMCPHD and FCISMCPHD approaches. In Section 5.1, a nonlinear simulation scenario composed of five targets is constructed. Then, Section 5.2 compares the estimation results of the IBSMCPHD [20], proposed CISMCPHD and FCISMCPHD approaches, in terms of the optimal subpattern assignment (OSPA) metric [42] and Root Mean Square Error (RMSE). At last, we compare the simulation results of all three approaches with different numbers of clutters and detection probabilities, to validate the effectiveness of our approaches.
5.1. Simulation Scenarios
In our simulations, we use the nonlinear scenario, the same as [36]. Let $\mathbf{x}$ be the target states, represented by $\mathbf{x}={[{p}_{x},{v}_{x},{p}_{y},{v}_{y},\alpha ]}^{T}$. In this paper, $({p}_{x},{p}_{y})$ is the position, $({v}_{x},{v}_{y})$ is the velocity, and α is the turn rate. With the above definitions, we model the nonlinear dynamic equation as
where $T=1$ second (s) is the sampling interval. In addition, ${\mathbf{\epsilon}}_{k1}$ is the noise, defined by ${\mathbf{\epsilon}}_{k1}\sim N({\mathbf{\epsilon}}_{k1};0,\mathbf{Q})$. We denote $\mathbf{Q}=diag({\sigma}_{x,\mathbf{\epsilon}}^{2},{\sigma}_{y,\epsilon}^{2},{\sigma}_{\alpha}^{2})$ as covariance matrices of ${\mathbf{\epsilon}}_{k1}$. In this paper, we set ${\sigma}_{x,\epsilon}={\sigma}_{y,\mathbf{\epsilon}}=1$ meter/second${}^{2}$ (m/s${}^{2}$), ${\sigma}_{\alpha}=\pi /180$ rad. The initial states, appearing times, and disappearing times of targets are listed in Table 4.
$$\begin{array}{cc}\hfill {\mathbf{x}}_{k}=& \\ & \left[\begin{array}{ccccc}1& \frac{sin\left({\alpha}_{k1}T\right)}{{\alpha}_{k1}}& 0& \frac{1cos\left({\alpha}_{k1}T\right)}{{\alpha}_{k1}}& 0\\ 0& cos\left({\alpha}_{k1}T\right)& 0& sin\left({\alpha}_{k1}T\right)& 0\\ 0& \frac{1cos\left({\alpha}_{k1}T\right)}{{\alpha}_{k1}}& 1& \frac{sin\left({\alpha}_{k1}T\right)}{{\alpha}_{k1}}& 0\\ 0& sin\left({\alpha}_{k1}T\right)& 0& cos\left({\alpha}_{k1}T\right)& 0\\ 0& 0& 0& 0& 1\end{array}\right]{\mathbf{x}}_{k1}\\ & +\left[\begin{array}{ccc}\frac{{T}^{2}}{2}& 0& 0\\ T& 0& 0\\ 0& \frac{{T}^{2}}{2}& 0\\ 0& T& 0\\ 0& 0& 1\end{array}\right]{\mathbf{\epsilon}}_{k1}\end{array}$$
Besides, the observation model is given by
where ${\mathbf{\eta}}_{k}$ is the observation noise defined by ${\mathbf{\eta}}_{k}\sim N({\mathbf{\eta}}_{k};0,\mathbf{R})$, and $\mathbf{R}=diag({\sigma}_{\theta}^{2},{\sigma}_{r}^{2})$ is the covariance. Here, we set ${\sigma}_{\theta}=\pi /180$ rad and ${\sigma}_{r}=1$ meter (m). We also assume that clutters are uniformly distributed in the detection region, where the angle range is $(0,\pi /2)$ rad, and distance range is $(0,1000)$ m. Trajectories of the five targets in both scenarios are shown in Figure 2, where the clutter number is set to be 10 for each scenario.
$${\mathbf{z}}_{k}=\left[\begin{array}{c}arctan\left(\frac{{p}_{y}}{{p}_{x}}\right)\\ \sqrt{{p}_{x}^{2}+{p}_{y}^{2}}\end{array}\right]+{\mathbf{\eta}}_{k}$$
For parameters, we set the gating threshold ${T}_{h}=16$ according to [32], the probability of detection and survival are ${p}_{d}\left({\mathbf{x}}_{k}\right)=0.95$, and ${p}_{s}\left({\mathbf{x}}_{k}\right)=0.99$, in accordance with [36]. The particle numbers for each birth target and survival targets are 5 and 100, respectively. All of the simulations are run in a computer with MATLAB 2015a, and i5 3.2 GHz processor with 4GB RAM.
5.2. Comparison of Estimation Accuracy on Certain Number of Clutters
To compare the estimation accuracy, we adopt the first order OSPA and RMSE as the metric. Here, we discuss the first order OSPA distance in the following. Let $\mathbf{X}=\{{\mathbf{x}}_{1},\dots ,{\mathbf{x}}_{n}\}$ and $\mathbf{Y}=\{{\mathbf{y}}_{1},\dots ,{\mathbf{y}}_{n}\}$ be two RFSs, where m and n are numbers of elements in $\mathbf{X}$ and $\mathbf{Y}$, respectively. Supposing that ${\mathrm{\Omega}}_{n}$ represents the set of permutations of $\{1,2,\dots ,n\}$, the first order OSPA metric can be rewritten by
where ${d}^{c}(\mathbf{x},\mathbf{y})=min(c,d(\mathbf{x},\mathbf{y}))$, $c>0$ is a cutoff factor, and $d(\mathbf{x},\mathbf{y})$ is the distance between $\mathbf{x}$ and $\mathbf{y}$. In this paper, we set $c=150$ in accordance with [43], and use the Euclidean distance to compute $d(\mathbf{x},\mathbf{y})$.
$${\overline{d}}_{p}^{c}(\mathbf{X},\mathbf{Y})=\frac{1}{n}\left(\underset{\varsigma \in {\mathrm{\Omega}}_{n}}{min}\sum _{i=1}^{m}{d}^{c}{({\mathbf{x}}_{i},{\mathbf{y}}_{\varsigma \left(i\right)})}^{p}+{c}^{p}(nm)\right)$$
Then, we conducted 500 Monte Carlo runs for multitarget tracking with the IBSMCPHD, our CISMCPHD, and FCISMCPHD approaches. The estimated trajectories of all three approaches are demonstrated in Figure 3. We can observe that most of estimated points are covered with the true trajectories in Figure 3b,c, while most of points are not covered with the groundtruth trajectories.
Furthermore, Figure 4 depicts the OSPA distances of all three approaches. In these figures, the OSPA distances of the proposed CISMCPHD and FCISMCPHD approaches are smaller than the BSMCPHD approach. Note that large OSPA distance denotes large tracking error. Thus, the proposed CISMCPHD and FCISMCPHD approaches have the smaller tracking error than the IBSMCPHD approach. We can also observe that the OSPA distances of the proposed CISMCPHD and FCSMCPHD approaches have four peaks at time 2, 9, 26, and 60. That is to say, targets may appear at these times.
We also plot the estimated numbers and corresponding RMSEs of all three approaches in Figure 5. As seen from Figure 5a, the estimated number of the proposed CISMCPHD approach is closest to the ground truth among three approaches, thus enjoying the lowest RMSE in Figure 5b. In addition, the numerical results of the averaged OSPA distances and RMSEs are listed in Table 5, demonstrating that our CISMCPHD approach achieves the best estimation on numbers and states. According to Table 5, the FCIMSPHD approach can make a compromise between the computational time and estimation accuracy.
5.3. Comparison of Estimation Accuracy on Various Numbers of Clutters
To validate the influence of clutters on multitarget tracking, the IBSMCPHD, CISMCPHD and FCISMCPHD approaches were implemented with 500 Monte Carlo simulations alongside the clutter number changing from 1 to 30. The results are illustrated in Figure 6a,b. From this figure, we can see that the averaged OSPA distances of all three approaches are enhanced, when the clutter number increases from 1 to 30. Among these approaches, the CISMCPHD approach has the smallest averaged OSPA distance and RMSE.
5.4. Comparison of Estimation Accuracy over Different Probabilities of Detection
In this section, we compare the estimation accuracy at different detection probabilities (varying from 0.92 to 0.98). Here, the clutter number is chosen to be 10, and 500 Monte Carlo simulations are run for the comparison. Table 6 reports the OSPA distances and RMSEs of the IBSMCPHD, CISMCPHD and FCISMCPHD approaches.
From Table 6, we can observe that the estimated accuracy of the FCISMCPHD approach get close to the CISMCPHD approach, when the probabilities of detection increase from $0.92$ to $0.98$. Thus, the FCISMCPHD approach is suitable for the high probabilities of detection.
5.5. Comparison of Estimation Accuracy at Challenging Nonlinear Scenarios
In this section, we compare the estimation accuracy of the IBSMCPHD, CISMCPHD and FCISMCPHD approaches at challenging nonlinear scenarios. For the challenging nonlinear scenarios, the standard deviation ${\sigma}_{\theta}$ varies from $\frac{1.5\pi}{180}$ to $\frac{3\pi}{180}$. We implement each approach with 500 Monte Carlo simulations.
The averaged accuracy, evaluated by OSPA and RMSE, is listed in Table 7. Table 7 indicates that the estimation accuracy of all three approaches decreases, when ${\sigma}_{\theta}$ increases from $\frac{1.5\pi}{180}$ to $\frac{3\pi}{180}$. Compared with the IBSMCPHD approach, the OSPA distances and RMSEs of the CISMCPHD and FCISMCPHD approaches are smaller at all four values of ${\sigma}_{\theta}$. It means that the estimation accuracy of the CISMCPHD and FCISMCPHD approaches is more stable than the IBSMCPHD approach in the challenging nonlinear scenarios.
6. Conclusions
In this paper, we have proposed the CISMCPHD and FCISMCPHD approaches, which can estimate the timevarying number and states of multitarget nonlinear tracking. In our CISMCPHD approach, a novel IS function approximation method is presented, which incorporates a gating method into the CIF method. To initiate the birth intensity of the next recursion, we use the current observations and estimated states to estimate the birth target components. In addition, we also present a fast implementation of the CISMCPHD approach, namely FCISMCPHD, to reduce the time complexity of the CISMCPHD approach. By clustering the particles into several groups, the target components can be obtained by representing the groups as Gaussian mixture components. Utilizing these components to approximating the IS functions, the computational time can be reduced magnificently. The simulation results demonstrate that the proposed CISMCPHD and FCISMCPHD approaches outperform the conventional BSMCPHD approach.
This paper concentrates on improving efficiency and accuracy of the conventional SMCPHD filter. It simply utilizes the multinomial resampling method as the resampling method. Other resampling methods may be integrated in the CISMCPHD approach for the future work. Besides, the FCISMCPHD approach is only suitable for the high probability of detection and small number of clutters. Study on improving the tracking performance of the FCISMCPHD approach at low probability of detection and large number of clutters may be seen as another direction of the future work.
Acknowledgments
This work was supported by the National Nature Science Foundation of China projects under Grants 61471022 and 61573037, and the Fok Ying Tung Education Foundation under Grant 151061.
Author Contributions
Zhe Liu provided insights in formulating the ideas, performed the simulations, and analyzed the simulation results. Zulin Wang provided some insights on motivation and basic idea in introduction. Mai Xu offered some insights on complexity analysis and technical derivations. Zhe Liu wrote the paper. Mai Xu and Zulin Wang carefully revised the paper.
Conflicts of Interest
The authors declare no conflict of interest.
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Figure 1.
Framework of the FCISMCPHD approach. In the IS function Approximating step, we utilize the survival and birth components to approximate the IS functions. Then the predicted particles are generated and updated in the PHD prediction and update step to achieve the posterior particles. By clustering the particles into several groups, the Gaussian Mixture components (namely survival components) can be constructed in the survival components construction step. Meanwhile, we also apply the survival components to estimate the birth components in the birth components construction step. These components are used to approximate the IS functions in the next iteration.
Figure 2.
Groundtruth trajectories of five targets with the clutter number setting to be 10. The target trajectories are depicted by circlesolid lines with different colors, while the asterisks denote clutters.
Figure 3.
Estimated trajectories of three approaches with clutter number being 10. The estimated trajectories are represented with the red (light) points, while the true trajectories are with the black (dark) solid line. (a) Trajectories of IBSMCPHD; (b) Trajectories of CISMCPHD; (c) Trajectories of FCISMCPHD.
Figure 4.
OSPA distances of the IBSMCPHD, CISMCPHD and FCISMCPHD approaches with clutter number being 10.
Figure 5.
Estimated numbers and RMSEs of IBSMCPHD, CISMCPHD and FCISMCPHD approaches with clutter number being 10. (a) Estimated numbers of the three approaches; (b) RMSEs of the three approaches.
Figure 6.
Estimated error of the IBSMCPHD, CISMCPHD and FCISMCPHD approaches along with the clutter number changing from 1 to 30. (a) Averaged OSPA distances; (b) Averaged RMSEs of estimated number.



Target  State  Appearing(s)  Disappearing(s) 

1  $[320,5,320,5,0]$  1  40 
2  $[400,5,400,5,0]$  8  50 
3  $[375,5,375,5,0]$  25  70 
4  $[400,5,325,5,0]$  59  70 
5  $[325,5,375,5,0]$  59  70 
Approaches  OSPA (m)  RMSE  Time (s) 

IBSMCPHD  46.20  0.44  0.02 
CISMCPHD  25.48  0.24  0.2 
FCISMCPHD  31.93  0.30  0.06 
${p}_{d}=0.92$  ${p}_{d}=0.94$  

IBSMCPHD  CISMCPHD  FCISMCPHD  IBSMCPHD  CISMCPHD  FCISMCPHD  
OSPA(m)  52.13  34.40  40.21  43.55  27.39  29.82 
RMSE  0.54  0.37  0.47  0.47  0.32  0.37 
${\mathit{p}}_{\mathit{d}}=\mathbf{0}.\mathbf{96}$  ${\mathit{p}}_{\mathit{d}}=\mathbf{0}.\mathbf{98}$  
IBSMCPHD  CISMCPHD  FCISMCPHD  IBSMCPHD  CISMCPHD  FCISMCPHD  
OSPA(m)  43.55  27.39  29.82  39.35  23.20  23.02 
RMSE  0.42  0.27  0.31  0.35  0.22  0.24 
${\sigma}_{\theta}=\frac{1.5\pi}{180}$  ${\sigma}_{\theta}=\frac{2\pi}{180}$  

IBSMCPHD  CISMCPHD  FCISMCPHD  IBSMCPHD  CISMCPHD  FCISMCPHD  
OSPA(m)  49.62  27.32  29.66  50.40  27.64  29.99 
RMSE  0.43  0.33  0.37  0.47  0.34  0.38 
${\mathit{\sigma}}_{\mathit{\theta}}=\frac{\mathbf{2}.\mathbf{5}\mathit{\pi}}{\mathbf{180}}$  ${\mathit{\sigma}}_{\mathit{\theta}}=\frac{\mathbf{3}\mathit{\pi}}{\mathbf{180}}$  
IBSMCPHD  CISMCPHD  FCISMCPHD  IBSMCPHD  CISMCPHD  FCISMCPHD  
OSPA(m)  51.04  27.43  30.74  52.56  28.02  31.3 
RMSE  0.50  0.35  0.39  0.45  0.35  0.39 
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