Multiple‐model Gaussian mixture probability hypothesis density filter based on jump Markov system with state‐dependent probabilities

IET Radar, Sonar & NavigationVolume 16, Issue 11 p. 1881-1894 ORIGINAL RESEARCHOpen Access Multiple-model Gaussian mixture probability hypothesis density filter based on jump Markov system with state-dependent probabilities Yi-Chieh Sun, Corresponding Author Yi-Chieh Sun sun896@purdue.edu orcid.org/0000-0001-5854-0956 Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, Indiana, USA Correspondence Yi-Chieh Sun, Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, IN 47906, USA. Email: sun896@purdue.eduSearch for more papers by this authorDohyeung Kim, Dohyeung Kim Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, Indiana, USASearch for more papers by this authorInseok Hwang, Inseok Hwang Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, Indiana, USASearch for more papers by this author Yi-Chieh Sun, Corresponding Author Yi-Chieh Sun sun896@purdue.edu orcid.org/0000-0001-5854-0956 Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, Indiana, USA Correspondence Yi-Chieh Sun, Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, IN 47906, USA. Email: sun896@purdue.eduSearch for more papers by this authorDohyeung Kim, Dohyeung Kim Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, Indiana, USASearch for more papers by this authorInseok Hwang, Inseok Hwang Aeronautics and Astronautics Engineering, Purdue University, West Lafayette, Indiana, USASearch for more papers by this author First published: 16 August 2022 https://doi.org/10.1049/rsn2.12304AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinkedInRedditWechat Abstract The Gaussian mixture probability density (GM-PHD) filter has become a popular approach to solve the multiple-target tracking (MTT) problem because it can effectively and efficiently estimate the number of targets and target states that change over time from noisy measurements. In the GM-PHD filter, the target detection and survival probabilities, and the birth rate are assumed to be constant, irrespective of the target state. However, in some applications, for example, when the target follows the planned trajectory, the detection, survival and birth of the target depend on its state. Besides, when the target reaches a waypoint along the planned trajectory, it will take a manoeuvre to go to the next waypoint, but the GM-PHD filter does not accommodate manoeuvring targets that switch between several motion models. To address this, we propose a multiple-model GM-PHD filter with state-dependent probabilities, which can explicitly consider the mode transition probabilities, the probabilities of detection and survival, and the birth rate as a function of the target state. The performance of the proposed algorithm is demonstrated by illustrative MTT scenarios which include target manoeuvres during occlusion. 1 INTRODUCTION The ability to jointly estimate time-varying target numbers and target states from noisy measurements makes multiple target tracking (MTT) methods have wide application in land, sea, and air traffic controls [1-3]. The joint probabilistic data association filter (JPDAF) [4, 5] and multiple hypothesis tracking (MHT) filter [6] are the traditional MTT approaches. These methods mainly focus on the issue of measurement origin uncertainty and solve the issue by using measurement-to-track association. However, when the target number is large, the traditional MTT approaches may lead to high computational complexity due to associations between measurements and tracks. Mahler proposed the random finite set (RFS) framework as an alternative approach that can avoid explicit data association between targets and measurements [7]. By using RFS to model the states and measurements of multiple targets, the probability hypothesis density (PHD) filter recursively propagates the first-order moment (intensity) of the Bayesian multiple target posterior in the filtering recursion [8]. The Gaussian Mixture Probability Hypothesis Density (GM-PHD) filter provides a closed-form solution for the PHD filter, which became a popular method for MTT [9]. In principle, the GM-PHD filter does not provide the trajectory of a target, so several methods have been proposed to achieve data association and track management [10, 11]. Since the performance of the GM-PHD filter could be degraded (possibly significantly) in a highly cluttered environment, some studies have focussed on tracking targets in a highly cluttered environment [12, 13]. In addition, the occurrence of miss detection can also degrade the tracking performance. The robustness of tracking under a limited sensor field of view and closely spaced targets is also a research topic of interest [14, 15]. In order to improve the state estimation accuracy, combining deep learning with the GM-PHD filter has also become a popular research topic [16-18]. However, in the GM-PHD filter, the target manoeuvrings are not well accounted and the target birth rate, and survival and detection probabilities are assumed to be constant. To the best of our knowledge, few recent publications have focussed on addressing this issue. This is not applicable in the land, sea, and air traffic applications where the target follows its planned trajectory (e.g., roads, sea lanes, and air routes, respectively). In this paper, we propose a new GM-PHD filter called multiple-model GM-PHD (MMGM-PHD) filter with state-dependent probabilities (SDP). Hybrid system models are used to model the target dynamics and motion mode transitions. A single model is good enough to describe the motion of a non-manoeuvring target since its motion is fixed. However, for a manoeuvring target, a single model cannot accurately model its multi-model behaviour. Thus, a combination of motion models that characterise different manoeuvres can better handle the manoeuver uncertainty, and thus the multiple-model (MM) approaches have been proposed [19]. Most of the MM approaches are based on the jump Markov system (JMS) [20]. Pasha et al. proposed the GM-PHD filter using JMS models [21, 22] and non-linear JMS models [23]. These methods have been implemented in many applications for manoeuvring target tracking [24-29]. In Refs. [21-23], the assumption of constant model (or mode) transition probabilities is not valid in the application when the target follows its planned trajectory. The variable structure multiple-model GM-PHD (VS-MMGM-PHD) filter considers the set of all possible modes and selects a subset of modes according to the target state [30, 31]. Li and Jia used the best-fitting Gaussian (BFG) approximation to revert the MM estimation to the single model estimation to improve tracking accuracy with less computational cost [32, 33]. However, same as in Ref. [21], these methods use the constant mode transition probability in each mode set. In a heavily cluttered and/or low signal-to-noise ratio (SNR) environment, imprecise mode transition probabilities can cause the existing GM-PHD filters with JMS models to lose the estimates of manoeuvring targets frequently. In land, sea, and air traffic applications, targets typically move along given routes (i.e., roads, sea lanes, or air routes) and can only be on these routes. Under the assumption of constant probability, the filter cannot accurately account for the appearance of a new target, nor can it distinguish whether the disappearance of the target is due to leaving the surveillance area or entering the blind spot. Therefore, in traffic applications, the birth rate, survival and detection probabilities of a target vary depending on its state because its motion is restricted by the route and/or traffic regulations. In the refined GM-PHD (RGM-PHD) tracker, the survival probability of the target is calculated adaptively according to the survival model based on its state [34]. Wang et al. modelled the targets entering and exiting the surveillance area by using state-dependent survival probabilities, and revised the posterior weights when miss detection occurs [35]. Hendeby and Karlsson considered a GM-PHD filter with variable probability of detection which is calculated by the distance between the target and the centre of the low detection area [36]. Sanaga and Frueh focussed on designing the model for state-dependent probability of target survival in the space environment [37]. The multi-frame GM-PHD filter manages the weight of the Gaussian component corresponding to the undetected target to achieve more reliable trajectory estimation in the imperfect detection probability scenario [38]. In order to track hidden but possibly existing targets, Törő et al. distinguished between detected, undetected, and hidden targets by calculating state-dependent detection probabilities based on the object occlusion model [39]. However, all of the aforementioned methods are based on ad hoc tuning and/or do not consider the state-dependent target survival probability, detection probability, and birth rate, all at the same time. In order to improve the performance of the GM-PHD filter for traffic surveillance applications where the target follows its planned trajectory, we propose the MMGM-PHD filter with SDP in this paper. The target birth rate and the probabilities of target motion mode transition, detection, and survival are functions of the target state in the proposed method. In the case of a planned trajectory, the trajectory provides information with which the state-dependent probabilities can be computed. The target (e.g. a ground vehicle, ship or aircraft) follows a series of waypoints and it takes a manoeuvre to go to the next waypoint when the current waypoint is reached. Thus, when the target state meets a condition (called as the guard condition), a mode transition (i.e., target manoeuvre) occurs. The highest birth rate happens near the location where the planned trajectory enters the surveillance area. The probability of detection should decrease when a target is close to a blind spot and be almost zero when it locates inside the blind spot (e.g., occlusion). The probability of survival becomes zero if and only if the target leaves the surveillance area. The target is modelled by the hybrid system model and the mode transitions are determined state-dependently by the guard conditions. We also propose a more sophisticated pruning step since the proposed filter can have Gaussian components of a target in multiple modes. We demonstrate the performance of the proposed filter compared to other methods with illustrative target tracking examples, where multiple targets move along the planned trajectories and the measurements are lost for a period of time due to the blind spot. The rest of the paper is organised as follows. Section 2 introduces the system model, explains the multiple-target state, and discusses the PHD filter. In Section 3, we present the proposed MMGM-PHD filter with SDP in detail. The performance of the proposed MMGM-PHD filter with SDP and the comparison with other methods are discussed in Section 4. Conclusions are given in Section 5. 2 PROBLEM FORMULATION In this section, we show the discrete-time stochastic linear hybrid system model of the target whose mode transitions (i.e., manoeuvres) are state-dependent via a stochastic guard condition. Then, we explain the components of the multiple-target state. Finally, the PHD filter is introduced. 2.1 Hybrid system model The state of the stochastic linear hybrid system (SLHS) at time k consists of a pair of continuous state x k ∈ R n ${x}_{k}\in {\mathbb{R}}^{n}$ and discrete mode m k ∈ M ${m}_{k}\in \mathbb{M}$ . A total of r number of modes can be switched from one to another within a finite set M ≔ 1 , 2 , … , r $\mathbb{M}{:=}\left\{1,2,\text{\ldots },r\right\}$ . The discrete-time stochastic linear hybrid system is described by x k = F k − 1 m k x k − 1 + w k − 1 m k ${x}_{k}={F}_{k-1}\left({m}_{k}\right){x}_{k-1}+{w}_{k-1}\left({m}_{k}\right)$ (1) z k = H k m k x k + v k m k ${z}_{k}={H}_{k}\left({m}_{k}\right){x}_{k}+{v}_{k}\left({m}_{k}\right)$ (2)where z k ∈ R p ${z}_{k}\in {\mathbb{R}}^{p}$ denotes the measurement at time k. Fk−1(mk) and Hk(mk) represent the state transition matrix and the measurement matrix corresponding to mode mk, respectively. The process noise wk−1(mk) and measurement noise vk(mk) corresponding to mode mk are assumed to be white Gaussian noise. The augmented state vector is denoted as ξ k = x k , m k T ∈ X = R n × M ${\xi }_{k}={\left[{x}_{k},{m}_{k}\right]}^{T}\in \mathbb{X}={\mathbb{R}}^{n}\times \mathbb{M}$ . Let N ( ⋅ ; μ , P ) $\mathcal{N}(\cdot ;\mu ,P)$ denote the Gaussian density function with mean μ and covariance P. The JMS with linear Gaussian models is given by p x k | x k − 1 , m k = N x k ; F k − 1 m k x k − 1 , Q k − 1 m k $p\left({x}_{k}\vert {x}_{k-1},{m}_{k}\right)=\mathcal{N}\left({x}_{k};{F}_{k-1}\left({m}_{k}\right){x}_{k-1},{Q}_{k-1}\left({m}_{k}\right)\right)$ (3) p z k | x k , m k = N z k ; H k m k x k , R k m k $p\left({z}_{k}\vert {x}_{k},{m}_{k}\right)=\mathcal{N}\left({z}_{k};{H}_{k}\left({m}_{k}\right){x}_{k},{R}_{k}\left({m}_{k}\right)\right)$ (4)where the conditional probability density function (pdf) is denoted by p(⋅|⋅). p(xk|xk−1, mk) and p(zk|xk, mk) are the state transition density and measurement likelihood conditioned on mode mk, respectively. Corresponding to the mode mk at time k, the covariance matrices of process noise and measurement noise are denoted as Qk(mk) and Rk(mk), respectively. For brevity, the Gaussian distribution is written as N ( μ , P ) = N ( Θ ) $\mathcal{N}(\mu ,P)=\mathcal{N}({\Theta})$ (5)where Θ represents the ordered pair of mean and covariance (μ, P). In addition, the discrete mode evolution follows a finite set Markov chain with the state-dependent mode transition matrix Π x k − 1 = π i | j , x k − 1 i , j = 1 , 2 , … , r ${\Pi}\left({x}_{k-1}\right)={\left\{\pi \left(i\vert j,{x}_{k-1}\right)\right\}}_{i,j=1,2,\text{\ldots },r}$ (6)where π(i|j, xk−1) is the conditional mode transition probability, which represents the probability of switching from mode j to mode i with state xk−1 as the condition, π i | j , x k − 1 ≔ p m k = i | m k − 1 = j , x k − 1 . $\pi \left(i\vert j,{x}_{k-1}\right){:=}p\left({m}_{k}=i\vert {m}_{k-1}=j,{x}_{k-1}\right).$ (7)The transition of the augmented state ξk at time k is governed by p ξ k | ξ k − 1 = p x k | x k − 1 , m k π i | j , x k − 1 . $p\left({\xi }_{k}\vert {\xi }_{k-1}\right)=p\left({x}_{k}\vert {x}_{k-1},{m}_{k}\right)\pi \left(i\vert j,{x}_{k-1}\right).$ (8) A manoeuvring target which follows its planned trajectory is considered for the mode transition model. For example, an aircraft flies along a planned air route, or a ship moves along a sea lane. The dynamics of the target can be illustrated by multiple modes and modelled as a stochastic linear hybrid system to track the target in the route. Since a target is likely to manoeuvre as it approaches the waypoints along the planned trajectory, the mode transition probability in this case is state-dependent. Thus, in this paper, the mode transition probability is modelled as a multivariate Gaussian pdf π i | j , x k − 1 = f L i j x k − 1 − μ i j = a i j + b i j N L i j x k − 1 ; μ i j , Σ i j $\begin{array}{rl}\hfill \pi \left(i\vert j,{x}_{k-1}\right)& =f\left({\boldsymbol{L}}_{ij}{x}_{k-1}-{\mu }_{ij}\right)\hfill \\ \hfill & ={a}_{ij}+{b}_{ij}\mathcal{N}\left({\boldsymbol{L}}_{ij}{x}_{k-1};{\mu }_{ij},{{\Sigma}}_{ij}\right)\hfill \end{array}$ (9)where Lij is a constant q × n matrix and q ≤ n. The scalar constants aij and bij satisfy that π(i|j, xk−1) ≥ 0 for all i, j = 1, …, r and ∑ j = 1 r π i | j , x k − 1 = 1 ${\sum }_{j=1}^{r}\pi \left(i\vert j,{x}_{k-1}\right)=1$ . 2.2 Multiple-target state In a multiple-target scenario, the number of targets can vary at each time step since targets can appear and disappear randomly in the surveillance area. At the sensor, the origins of the measurements are not known. Some of the measurements are due to the targets while the rest are clutter, thus the number of measurements also can vary at each time step. Let Nk and Mk be the number of targets and measurements at time k, respectively. The multiple-target states and measurements at time k are denoted by the random finite sets (RFSs) Xk and Zk, respectively. X k = x k 1 , … , x k N k ${X}_{k}=\left\{{x}_{k}^{1},\text{\ldots },{x}_{k}^{{N}_{k}}\right\}$ (10) Z k = z k 1 , … , z k M k ${Z}_{k}=\left\{{z}_{k}^{1},\text{\ldots },{z}_{k}^{{M}_{k}}\right\}$ (11)where x k i ${x}_{k}^{i}$ and z k j ${z}_{k}^{j}$ denote the state of the ith target and the jth measurement at time k, respectively. The RFS of the augmented states at time k is given by Ξ k = ξ k 1 , … , ξ k N k ${{\Xi}}_{k}=\left\{{\xi }_{k}^{1},\text{\ldots },{\xi }_{k}^{{N}_{k}}\right\}$ (12)where ξ k i ${\xi }_{k}^{i}$ denotes the augmented state of the ith target at time k. Note that there is no ordering on the respective collections of target states and measurements at each time step. The time evolution of the multiple-target state involves target survival, birth, and death. Given a multiple-target state Xk−1 at time k − 1, each xk−1 ∈ Xk−1 either has a probability Psv(xk−1) to survive at time k or a probability 1 − Psv(xk−1) to die. The behaviour of xk−1 at time k is modelled by the RFS Sk|k−1(xk−1) and can take on either x k $\left\{{x}_{k}\right\}$ when the target survives or ∅ when the target dies. In addition to the survival and death of existing targets, a new target can arise at time k either by spontaneous birth or by spawning from a target at time k − 1. The state RFS Γk models the states of spontaneous births at time k that are independent of the previous target states. The states of targets spawned from previous targets with state xk−1 ∈ Xk−1 at time k are modelled by RFS Bk|k−1(xk−1). Then, the multiple-target state Xk at time k is given by the union of the surviving targets, spawned targets, and new born targets: X k = ⋃ x ∈ X k − 1 S k | k − 1 ( x ) ∪ ⋃ x ∈ X k − 1 B k | k − 1 ( x ) ∪ Γ k . ${X}_{k}=\left[\bigcup\limits _{x\in {X}_{k-1}}{S}_{k\vert k-1}(x)\right]\cup \left[\bigcup\limits _{x\in {X}_{k-1}}{B}_{k\vert k-1}(x)\right]\cup {{\Gamma}}_{k}.$ (13)The measurements received by the sensor at time k can either be generated by the targets or clutter. Given multiple-target state xk ∈ Xk at time k, each xk either has a probability PD(xk) to be detected or a probability 1 − PD(xk) to be missed. The measurement generated by xk can take on either z k $\left\{{z}_{k}\right\}$ when the target is detected or ∅ when the target is undetected, and is modelled by the RFS Φ(xk). In addition to the target originated measurements, the sensor also receives a set of false measurements and is modelled by the RFS Kk. Thus, at time k, the union of target-generated measurements and clutter gives the multiple-target measurement Zk: Z k = ⋃ x ∈ X k Φ ( x ) ∪ K k . ${Z}_{k}=\left[\bigcup\limits _{x\in {X}_{k}}{\Phi}(x)\right]\cup {K}_{k}.$ (14) 2.3 Probability hypothesis density filter Based on RFS theory and point process theory, the probability hypothesis density (PHD) filter has been proposed [5-7]. The prior and posterior intensity functions for the PHD filter are given as follows: I x k | Z ( k − 1 ) = b x k + ∫ P s v x k − 1 f x k | x k − 1 + s p x k | x k − 1 I x k − 1 | Z ( k − 1 ) d x k − 1 $\begin{array}{r}\hfill I\left({x}_{k}\vert {Z}^{(k-1)}\right)=b\left({x}_{k}\right)+\int \left(\!\right.{P}_{sv}\left({x}_{k-1}\right)f\left({x}_{k}\vert {x}_{k-1}\right)\\ +sp\left({x}_{k}\vert {x}_{k-1}\right)\left.\!\right)I\left({x}_{k-1}\vert {Z}^{(k-1)}\right)d{x}_{k-1}\hfill \end{array}$ (15) I x k | Z ( k ) ≃ 1 − P D x k I x k | Z ( k − 1 ) + ∑ z k ∈ Z k P D x k p z k | x k I x k | Z ( k − 1 ) κ k z k + ∫ P D x k p z k | η k I η k | Z ( k − 1 ) d η k $\begin{array}{r}\hfill I\left({x}_{k}\vert {Z}^{(k)}\right)\simeq \left[1-{P}_{D}\left({x}_{k}\right)\right]I\left({x}_{k}\vert {Z}^{(k-1)}\right)\\ +\sum\limits _{{z}_{k}\in {Z}_{k}}\frac{{P}_{D}\left({x}_{k}\right)p\left({z}_{k}\vert {x}_{k}\right)I\left({x}_{k}\vert {Z}^{(k-1)}\right)}{{\kappa }_{k}\left({z}_{k}\right)+\int {P}_{D}\left({x}_{k}\right)p\left({z}_{k}\vert {\eta }_{k}\right)I\left({\eta }_{k}\vert {Z}^{(k-1)}\right)d{\eta }_{k}}\hfill \end{array}$ (16)where f(xk|xk−1) is the state transition probability density; b(xk) and sp(xk|xk−1) are the intensities of birth and spawn RFSs, respectively. κk(zk) is the intensity of the clutter RFS. Z ( k ) = Z 1 , … , Z k ${Z}^{(k)}=\left\{{Z}_{1},\text{\ldots },{Z}_{k}\right\}$ is the measurement RFS sequence. 3 STATE-DEPENDENT GM-PHD FILTER WITH STATE-DEPENDENT JUMP MARKOV SYSTEM MODELS In this section we present a closed-form PHD solution with state-dependent probabilities for linear Gaussian JMS models. The proposed system uses state-dependent mode transition probabilities, target survival and detection probabilities, and target birth rate. The following subsections show how the Gaussian components of the posterior intensity are analytically propagated to the next time step. 3.1 Planned trajectory scenario In the planned trajectory scenario, the targets follow the planned trajectory, which indicates that the targets enter and exit the surveillance area of the sensor along the planned trajectory. For illustration, We consider a rectangular sensor field of view (FoV) whose boundaries are denoted as B s v = b s v 1 , b s v 2 , b s v 3 , b s v 4 ${B}_{sv}=\left\{{b}_{sv}^{1},{b}_{sv}^{2},{b}_{sv}^{3},{b}_{sv}^{4}\right\}$ and is shown in Figure 1. The intersections of the FoV boundaries and the planned trajectory are the entrances (and exits) of the targets entering (and exiting) the sensor FoV. The red crosses in Figure 1 are the entrances of the planned trajectory marked with a green dashed line. The entry points are denoted as E b = e b 1 , … , e b G b ${E}_{b}=\left\{{e}_{b}^{1},\text{\ldots },{e}_{b}^{{G}_{b}}\right\}$ , where Gb is the number of entrances of the surveillance area. Each entrance element consists of its position in 2D: e b = e b , x , e b , y T ${e}_{b}={\left[{e}_{b,x},{e}_{b,y}\right]}^{T}$ . A target cannot be observed by the sensor when the target is located in the blind spot. A set of blind spot is denoted by B D = b D 1 , … , b D G D ${B}_{D}=\left\{{b}_{D}^{1},\text{\ldots },{b}_{D}^{{G}_{D}}\right\}$ , where GD is the total number of blind spots. In this paper, each blind spot is approximated by a rectangular shape. Each bD consists of the x and y position of its boundaries: b D = b D , x 1 , b D , x 2 , b D , y 1 , b D , y 2 T ${b}_{D}={\left[{b}_{D,x}^{1},{b}_{D,x}^{2},{b}_{D,y}^{1},{b}_{D,y}^{2}\right]}^{T}$ . Figure 1 shows an example of the planned trajectory scenario and the grey region represents a blind spot. FIGURE 1Open in figure viewerPowerPoint An example of planned trajectory scenario 3.2 Prediction The posterior intensity at time k − 1 is a Gaussian mixture of the form I ξ k − 1 | Z ( k − 1 ) = ∑ i = 1 J k − 1 | k − 1 m k − 1 w k − 1 | k − 1 i m k − 1 ⋅ N x k − 1 ; Θ ˆ k − 1 | k − 1 i m k − 1 $\begin{array}{r}\hfill I\left({\xi }_{k-1}\vert {Z}^{(k-1)}\right)=\sum\limits _{i=1}^{{J}_{k-1\vert k-1}^{{m}_{k-1}}}{w}_{k-1\vert k-1}^{i}\left({m}_{k-1}\right)\\ \cdot \mathcal{N}\left({x}_{k-1};{\widehat{{\Theta}}}_{k-1\vert k-1}^{i}\left({m}_{k-1}\right)\right)\hfill \end{array}$ (17)where w k − 1 | k − 1 i m k − 1 ${w}_{k-1\vert k-1}^{i}\left({m}_{k-1}\right)$ is the weight of the ith posterior Gaussian component for mode mk−1 at time k − 1. J k − 1 | k − 1 m k − 1 ${J}_{k-1\vert k-1}^{{m}_{k-1}}$ represents the number of posterior Gaussian components for mode mk−1 at time k − 1. The ordered pair of mean and covariance x ˆ k − 1 | k − 1 i m k − 1 , P ˆ k − 1 | k − 1 i m k − 1 $\left({\widehat{x}}_{k-1\vert k-1}^{i}\left({m}_{k-1}\right),{\widehat{P}}_{k-1\vert k-1}^{i}\left({m}_{k-1}\right)\right)$ of the ith posterior Gaussian component for mode mk−1 at time k − 1 is represented as Θ ˆ k − 1 | k − 1 i m k − 1 ${\widehat{{\Theta}}}_{k-1\vert k-1}^{i}\left({m}_{k-1}\right)$ . Then, with the union given in Equation (13), the prior intensity at time k is written as I ξ k | Z ( k − 1 ) = b ξ k + I s p ξ k | Z ( k − 1 ) + I s v ξ k | Z ( k − 1 ) $I\left({\xi }_{k}\vert {Z}^{(k-1)}\right)=b\left({\xi }_{k}\right)+{I}_{sp}\left({\xi }_{k}\vert {Z}^{(k-1)}\right)+{I}_{sv}\left({\xi }_{k}\vert {Z}^{(k-1)}\right)$ (18)where b(ξk), Isp(ξk|Z(k−1)), and Isv(ξk|Z(k−1)) represent the intensities of the birth RFS, spawned Gaussian components, and surviving Gaussian components, respectively, for mode mk at time k. By using the transition of the augmented state vector (8), each intensity can be derived. 3.2.1 Intensity of born targets First, the intensity of birth RFS for mode mk at time k is a Gaussian mixture of the form b ξ k = ∑ j = 1 J b , k m k π b m k w b , k j ξ k N x k ; Θ ˆ b , k j m k $b\left({\xi }_{k}\right)=\sum\limits _{j=1}^{{J}_{b,k}^{{m}_{k}}}{\pi }_{b}\left({m}_{k}\right){w}_{b,k}^{j}\left({\xi }_{k}\right)\mathcal{N}\left({x}_{k};{\widehat{{\Theta}}}_{b,k}^{j}\left({m}_{k}\right)\right)$ (19)where J b , k m k ${J}_{b,k}^{{m}_{k}}$ is the number of the newborn Gaussian components for mode mk at time k and πb(mk) is the mode probability that the target is operating in mode mk at time k. The weight of a newborn Gaussian component for mode mk at time k is a state-dependent variable w b , k j ξ k ${w}_{b,k}^{j}\left({\xi }_{k}\right)$ and is calculated independently for each Gaussian component. The ordered pair of mean x ˆ b , k j m k ${\widehat{x}}_{b,k}^{j}\left({m}_{k}\right)$ and covariance P ˆ b , k j m k ${\widehat{P}}_{b,k}^{j}\left({m}_{k}\right)$ of the born Gaussian component is denoted as Θ ˆ b , k j m k ${\widehat{{\Theta}}}_{b,k}^{j}\left({m}_{k}\right)$ . According to the distance between the position of a Gaussian component and Eb, the state-dependent birth rate is derived by w b , k j ξ k = max l ∈ G b Ω b , k x k , e b l N 0 ; 0 , σ b 2 × T e x ${w}_{b,k}^{j}\left({\xi }_{k}\right)=\frac{{\mathrm{max}}_{l\in {G}_{b}}\left\{{{\Omega}}_{b,k}\left({x}_{k},{e}_{b}^{l}\right)\right\}}{\mathcal{N}\left(0;0,{\left({\sigma }_{b}\right)}^{2}\right)\left.\!\right)}\times {T}_{ex}$ (20)where Ω b , k x k , e b l ${{\Omega}}_{b,k}\left({x}_{k},{e}_{b}^{l}\right)$ is the birth rate of the newborn target with continuous state xk with respect to the lth entrance and is given by Ω b , k x k , e b l = N d i s t x k , e b l ; 0 , σ b 2 ${{\Omega}}_{b,k}\left({x}_{k},{e}_{b}^{l}\right)=\mathcal{N}\left(dist\left({x}_{k},{e}_{b}^{l}\right);0,{\left({\sigma }_{b}\right)}^{2}\right)$ (21) d i s t x k , e b l = p x , k − e b , x l 2 + p y , k − e b , y l 2 . $dist\left({x}_{k},{e}_{b}^{l}\right)=\sqrt{{\left({p}_{x,k}-{e}_{b,x}^{l}\right)}^{2}+{\left({p}_{y,k}-{e}_{b,y}^{l}\right)}^{2}}.$ (22)px,k and py,k are the x and y position of the target with state xk. d i s t x k , e b l $dist\left({x}_{k},{e}_{b}^{l}\right)$ calculates the distance between the position of the target with state xk and the lth entrance. σb is the standard deviation and Tex equals to the value of extraction threshold. 3.2.2 Intensity of spawned targets Next, the intensity with the spawned Gaussian component is a Gaussian mixture of the form I s p ξ k | Z ( k − 1 ) = ∑ m k − 1 ∑ j = 1 J k − 1 | k − 1 m k − 1 ∑ t = 1 J s p , k m k , m k − 1 w s p , k | k − 1 j , t m k , m k − 1 N x k ; Θ ˆ s p , k | k − 1 j , t m k , m k − 1 $\begin{array}{r}\hfill {I}_{sp}\left({\xi }_{k}\vert {Z}^{(k-1)}\right)=\sum\limits _{{m}_{k-1}}\sum\limits _{j=1}^{{