1 The Trajectory PHD Filter for Coexisting Point and Extended Target Tracking

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The Trajectory PHD Filter for

Coexisting Point and Extended Target Tracking

Shaoxiu Wei, ´

Angel F. Garc´

ıa-Fern´

andez, and Wei Yi

Abstract—This paper develops a general trajectory probability

hypothesis density (TPHD) ﬁlter, which uses a general density

for target-generated measurements and is able to estimate

trajectories of coexisting point and extended targets. First,

we provide a derivation of this general TPHD ﬁlter based

on ﬁnding the best Poisson posterior approximation by

minimizing the Kullback-Leibler divergence, without using

probability generating functionals. Second, we adopt an efﬁcient

implementation of this ﬁlter, where Gaussian densities correspond

to point targets and Gamma Gaussian Inverse Wishart densities

for extended targets. The L-scan approximation is also proposed

as a simpliﬁed version to mitigate the huge computational cost.

Simulation and experimental results show that the proposed ﬁlter

is able to classify targets correctly and obtain accurate trajectory

estimation.

Index Terms—Multi-target tracking, random ﬁnite set,

Kullback-Leibler divergence.

I. INTRODUCTION

AUTONOMOUS vehicles promise the possibility of

fundamentally changing the transportation industry, with

an increase in both highway capacity and trafﬁc ﬂow [1].

It is required to simultaneously extract the environmental

information that incorporates dynamic as well as static

objects through road infrastructure and other vehicles [2]–[5].

Accordingly, the multi-target tracking (MTT) approaches are

widely used for estimating the states and number of dynamic

targets, which may appear, move and disappear, given noisy

sensor measurements in time sequence [6].

There are two main kinds of approaches to solve MTT

problems. The ﬁrst category is based on random vectors, such

as the joint probabilistic data association (JPDA) ﬁlter [7],

[8] and the multiple hypotheses tracking (MHT) [6], [9]. The

second category is based on random ﬁnite set (RFS) [10]–

[12]. Among them, RFS-based algorithms have been proved to

possess an excellent tracking performance in various scenarios

[13]–[17]. The PHD ﬁlter, known for its low computational

burden among all RFS based ﬁlters, possesses a high efﬁciency

in solving real time tracking problem [18], [19]. It propagates

the ﬁrst-order multi-target moments [10], also called intensity,

through prediction and update step. The PHD ﬁlter can also

be derived by propagating a Poisson multi-target density

S. X. Wei and W. Yi are with the School of Information and Communication

Engineering, University of Electronic Science and Technology of China. (e-

mail: sxiu wei@hotmail.com; kusso@uestc.edu.cn).

Angel F. Garc´

ıa-Fern´

andez is with the Department of Electrical Engineering

and Electronics, University of Liverpool, Liverpool L69 3GJ, U.K., and also

with ARIES Research Centre, Universidad Antonio de Nebrija, 28015 Madrid,

Spain (e-mail: angel.garcia-fernandez@liverpool.ac.uk).

through the ﬁltering recursion, obtained via Kullback-Leibler

divergence (KLD) minimization [10], [20].

In MTT, an important topic is to obtain accurate trajectory

estimates and mitigate trajectory fragmentation [21]. Recently,

calculating the posterior over a set of trajectories [22]–[26] or

a (labeled) multi-target state sequence [21] provide an efﬁcient

approach to the above requirements. Among these approaches,

the trajectory PHD (TPHD) ﬁlter [23] establishes trajectories

from ﬁrst principles using trajectory RFSs. The TPHD ﬁlter

propagates the best Poisson multi-trajectory density under the

standard point target dynamic and measurement models [20].

The Gaussian mixture is proposed to obtain a closed-form

solution of the TPHD ﬁlter. Other trajectory-based ﬁlters for

point targets are the trajectory multi-Bernoulli ﬁlter, trajectory

PMBM ﬁlter and trajectory PMB ﬁlter. [24], [25]. Compared

to ﬁlters based on sets of targets, ﬁlters based on sets of

trajectories contain all information to answer trajectory-related

questions, which are of major importance in autonomous

vehicles and smart trafﬁc systems.

In order to develop Bayesian ﬁlters, we need to model

the distribution of the measurements given the targets as

well as clutter. There are two main types of modeling for

target-generated measurements: the point target model and the

extended target model. In general, a point model is applied

for the target measurement that is smaller than the sensor

resolution, given its size and distance from the sensor [13],

[18], [27].

Conversely, a target may generate more than one

measurement if multiple resolution cells of the sensor are

occupied by a single target, which is referred to as the extended

target [28]. The Poisson point process (PPP) is widely used

in the measurement model for an extended target [29], i.e.,

a Poisson distributed random number of measurements are

generated, distributed around an extended target at each time

step. There are extended target ﬁlters for sets of targets [28],

[30]–[34] and also sets of trajectories [35], [36]. Except for

extracting dynamic target state information, the mean number

of generated measurements from an extended target or the size

of the target are also modeled in these ﬁlters. Besides, there

are also approaches for Bayesian smoothing of target extent

based on random matrix model [36], [37].

There are many applications in which it is important to

develop models and algorithms for simultaneous point and

extended targets [12], [38]. For example, in a self-driving

vehicle application, pedestrians may be modeled as point

targets while some vehicles are taken as extended targets.

Besides, the distinction between point and extended targets

may also depend on the distance.

arXiv:2210.03412v1 [eess.SP] 7 Oct 2022

Prediciton Bayes’rule KLD

minimization

Poisson

Update

PMB

Fig. 1. Diagram of the proposed TPHD ﬁlter, with Poisson birth model

and no spawning. This ﬁlter propagates a Poisson density on the set of

alive trajectories, whose form is kept in the prediction. Bayes’ rule provides

a Poisson multi-Bernoulli density that is projected back to Poisson via

minimizing the KLD.

In this paper, we propose a TPHD ﬁlter for coexisting point

and extended targets. This ﬁlter combines the computational

efﬁciency of PHD ﬁlters, with the ability to estimate

trajectories from ﬁrst principles in situations with point and

extended targets. In particular, the contributions of this paper

are:

1) A derivation of the TPHD ﬁlter update for general

target-generated measurement density and PPP clutter

is proposed based on direct KLD minimization (see

Fig. 1). The corresponding derivation for the general

PHD ﬁlter was derived using probability generating

functionals (PGFLs) [11], [39]. Therefore, the proposed

derivation increases the accessibility of the proof to a

wider audience.

2) We apply the general TPHD ﬁlter for tracking coexisting

point and extended targets. The implementation is

provided for a linear Gaussian model for point targets

[18], [23], [40] and a Gamma Gaussian Inverse Wishart

(GGIW) model for extended targets [28], [32]–[34].

3) Simulation and real-world experimental results

demonstrate the general TPHD ﬁlter can achieve

excellent performance in tracking both point and

extended target.

The structure of this paper is organized as follows.

In Section II, theoretical background and the update and

prediction steps of the general TPHD ﬁlter are provided. The

implementation is given in Section III. In Section IV, we give

the simulation and experimental results of the proposed ﬁlter.

Finally, conclusion are drawn in Section IV,

II. THE GENERAL TPHD FILTER

In this section, the recursion steps of the general TPHD ﬁlter

are provided. The main idea is to use a general likelihood for

target-generated measurements in the update step. We deﬁne

sets of trajectories, the Bayesian ﬁltering recursion for sets of

trajectories, and provide the update and prediction steps of the

general TPHD ﬁlter.

A. Sets of Trajectories

The trajectory state X= (t, x1:i)consists of a ﬁnite

sequence of target states x1:i= (x1, ..., xi)that starts at the

time step twith length i, where xi∈Rnx[23]. For kdenoting

the current time step and a trajectory (t, x1:i)that exists from

time tto t+i−1, the variable (t, i)belongs to the set

Ik={(t, i):1≤t≤kand 1≤i≤k−t+ 1}. Therefore,

a single trajectory up to the time step kis deﬁned in the

space Tk=](t,i)∈Ik{t}×Ri×nx, where ]denotes the disjoint

union, ×denotes a Cartesian product, and nxrepresents the

dimension of target state. Supposing there are Nktrajectories

at time k, the set of trajectories is denoted as

Xk={X1, ..., XNk} ∈ F(Tk),(1)

where F(Tk)represents the set of all ﬁnite subsets of Tk.

B. Bayesian Filtering Recursion

Given the posterior multi-trajectory density πk−1(·)on the

set of trajectories at time k−1and the set of measurements

zkat time k, the posterior density πk(·)is obtained by using

the Bayes’ recursion [26]

πk|k−1(Xk) = Zφ(Xk|Xk−1)πk−1(Xk−1)δXk−1,(2)

πk(Xk) = `k(zk|Xk)πk|k−1(Xk)

R`k(zk|Xk)πk|k−1(Xk)δXk

,(3)

where φ(·|·)denotes the transition density for trajectories,

πk|k−1(·)denotes the predicted density, `k(zk|X)denotes the

density of measurements of trajectories. As the measurements

zkcome from the target states at the current time step k,

`k(zk|Xk)can be also written as

`k(zk|Xk) = `k(zk|τk(Xk)) ,(4)

where τk(X)denotes the corresponding multi-target state at

the time k.

C. Update

The general TPHD ﬁlter propagates a Poisson density on

the set of alive trajectories through the ﬁltering recursion via

KLD minimization [20], see Fig.1. The measurement model

is:

Assumption 1: Each potential target generates an

independent set zkof measurements with density f(zk|·).

Assumption 2: The set zkof measurements at time kis the

union of target generated measurements and clutter. Clutter is

a PPP with intensity λC(·), which is independent of target-

originated measurements.

Let λk|k(X)be the PHD of the alive trajectory Xat time

k. For X= (t, x1:i), we use λk|kxias the PHD of the

current set of targets, which can be obtained by marginalizing

the PHD for trajectories [23].

For the general TPHD ﬁlter, we use a pseudolikelihood

function Lzk(·)for the update step [12], [39], which is given

Lzkxi=f∅|xi+X

P∠zk

wPX

w∈P

fw|xi

κw+τw

(5)

where

τw=Zfw|xiλk|k−1xidxi,(6)

κw=δ1[|w|]"Y

z∈w

λC(z)#,|w|>0,(7)

wP=Qw∈P (κw+τw)

PQ∠zkQw∈Q (κw+τw).(8)

In eq. (5), the notation P∠zkin the sum denotes that the

product goes over all partitions Pof zk. Then sum w∈ P

goes through all sets in this partition. We use wPto denote

the weight of each partition. The notation w∈ P denotes that

the set wis a cell in the partition P[12]. For eqs. (6)–(8),

the notation κwfor PPP clutter is given by [41, eq. (32)] and

δi(·)represents the Kronecker delta.

Proposition 1. Given the prior trajectory PHD with

λk|k−1t, x1:iat the current time k. Then, the TPHD ﬁlter

update is

λk|kt, x1:i=Lzkxiλk|k−1t, x1:i,(9)

if t+i−1 = kand otherwise λk|k−1t, x1:iequals to zero.

The proof of Proposition 1 via direct KLD minimization

is provided in Appendix A. As a general target-generated

measurement model is considered in this ﬁlter, we can recover

the standard point and extended TPHD ﬁlter updates, see

Appendix B.

D. Prediction

The general TPHD ﬁlter considers the standard dynamic

models, and therefore, the prediction step is similar to the

standard TPHD ﬁlter [23]. The multi-target dynamic model

is:

Assumption 3: A target xsurvives to the next time step with

probability pS(x)and moves with a transition density g(·|x).

Assumption 4: New targets are born independently

following a PPP with intensity λγ. The current set of targets

is the union of surviving targets and new born targets.

Proposition 2. Given the posterior trajectory PHD

λk−1|k−1t, x1:i−1at the last time k−1and birth

PHD at the current time k, the prediction of the general

TPHD ﬁlter is [23]

λk|k−1(X) = λγ,k|k(X) + λS

k|k−1(X),(10)

where

λγ,k|k(X) =λγx1δ1[i]δk[t],(11)

λS

k|k−1(X) =pSxi−1gxi|xi−1λk−1|k−1t, x1:i−1.

(12)

Eq. (10) is the sum of the intensities for new born

trajectories (11) and surviving trajectories (12).

III. THE GAMMA GAUSSIAN INVERSE WISHART

IMPLEMENTATION

In this section, we apply the general TPHD ﬁlter recursion

in section II to track coexisting point extended targets. First,

we explain the space and then the implementations. Finally,

some strategies are given to decrease the computational cost.

A. The Coexisting Space Model

A common scenario for coexisting extended and point target

is a trafﬁc monitoring situation, as illustrated in Fig. 2. To track

both kinds of targets, it is important to deﬁne a general space

model. First, for the scenario in Fig. 2, we deﬁne the state of

point target trajectory at time kwith the notation:

Xp= (t, x1:i

p),(13)

where trepresents its birth time and x1:i

p= (x1

p, ..., xi

denotes a sequence including the point target states at each

time step of the trajectory with length i[23]. The state Xp

belongs to the space Tp,k, which is written as

Xp∈Tp,k =](t,i)∈Ik{t} × Ri×nx,(14)

To describe an extended target state X+

eat time k, we

respectively deﬁne the notation Xefor trajectory state, γfor

the expected number of measurements per target and ˜

Xfor

the extent state [28]. More speciﬁcally,

X+

e= (Xe, γ, ˜

X)∈Te,k,(15)

Xe= (t, x1:i

e)∈ ](t,i)∈Ik{t} × Ri×nx,(16)

γ∈R+,(17)

X∈Sd

+,(18)

where x1:i

edenotes the sequence of target states, R+represents

the space of positive real numbers and Sd

+is the space of

positive deﬁnite matrices with size d. The value of dis taken

as the dimension of the target extent. Here, we only consider

target extent ˜

Xand γat the latest time step for simplicity. If

the historical information needs to be stored, we just add the

corresponding sequence to the trajectory space. Therefore, the

coexisting trajectory space for both point and extended targets

is given as Tk=Tp,k ]Te,k.

B. The GGIW Model

The implementation is provided for a Gaussian model for

point target trajectory [18], [23], [40] and a Gamma Gaussian

Inverse Wishart (GGIW) model for extended target trajectory.

[28], [32]–[34].

The density of γis given as the Gamma distribution

G(γ;a, b)with parameters a > 0and b > 0. The

Inverse Wishart density on matrices IW(˜

X;v, V )is used

to describe ˜

X, which is deﬁned in space Sd

+with v > 2d

degrees of freedom and parameter matrix V∈Sd

+. The

Gamma and Inverse Wishart distributions are known as the

conjugate prior to the Poisson distribution and multivariate

Gaussian distribution, respectively. The derivation of these two

distributions in Bayesian recursion can be found in [42], [43].

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1TheTrajectoryPHDFilterforCoexistingPointandExtendedTargetTrackingShaoxiuWei,´AngelF.Garc´a-Fern´andez,andWeiYiAbstractThispaperdevelopsageneraltrajectoryprobabilityhypothesisdensity(TPHD)lter,whichusesageneraldensityfortarget-generatedmeasurementsandisabletoestimatetrajectoriesofcoexistingpointa...

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