hybrid algorithms for multitarget tracking using mht and gm-cphd

Hybrid Algorithms for

Multitarget Tracking using

MHT and GM-CPHD

EVANGELINE POLLARD

BENJAMIN PANNETIER

ONERA

MICHELE ROMBAUT Grenoble Image Parole Signal

Automatique Laboratory

The Gaussian mixture cardinalized probability hypothesis

density (GM-CPHD) is a new original algorithm for multitarget

tracking adapted to false alarms, nondetection and closely

spaced objects. It models the target set as a random finite set

(RFS) and estimates the target state as the first-order moment

of a joint probability distribution. In the classical version

no track assignment is implemented; this is a limit to scene

understanding in a multitarget context. A technique for choosing

the peak to track association is therefore proposed. With this

implementation the main strength of the GM-CPHD is shown: it

drastically improves the performances concerning the estimation

of the number of targets and gives acceptable performances

concerning the state of each individual target even if targets

are close together, but it cannot rival an interacting multiple

model estimator with multiple hypothesis tracking (IMM-MHT)

in regards to velocity estimation, which is also the case with

other multitarget tracking algorithms not equiped with IMM.

However, MHT performance decreases due to poor estimation

of the number of targets when targets are close together. It is

worth noting that combining a probability hypothesis density

(PHD) filter with a multiple-model approach should improve the

velocity estimation but is unnecessary because we have developed

a hybrid algorithm, combining the precision of the estimation of

the number of targets given by the GM-CPHD, used in a labeled

version, with the precision of the estimation of each individual

state given by the MHT. These noteworthy performances can

be observed for individual targets as well as for convoys. This

hybrid algorithm is extended by using an IMM-MHT with road

constraints.

Manuscript received November 13, 2008; revised June 12 and

October 30, 2009; released for publication November 4, 2009.

IEEE Log No. T-AES/47/2/940816.

Refereeing of this contribution was handled by B-N. Vo.

Authors’ addresses: E. Pollard and B. Pannetier, Dept. of

Modeling and Information Processing, ONERA, 29 Avenue

de la Division Leclerc, Chatillon, 92322, France, E-mail:

([email protected]); M. Rombaut, Image and Signal

Department (DIS) of Grenoble Image Parole Signal Automatique

Laboratory (GIPSA-lab), Grenoble 38031, France.

0018-9251/11/$26.00 c° 2011 IEEE

I. INTRODUCTION

In the battlefield surveillance domain, ground

target tracking is crucial to evaluating the situation

awareness. Data used for tracking comes from a

ground moving target indicator (GMTI) sensor

which detects moving targets only by measuring

their Doppler frequency. The goal is to have a real

ground picture: how many targets are on the scene,

what their dynamics during the simulation are, how

they are correlated. But the ground environment is

very complex, and its characteristics have to be taken

into account. First the ground traffic density is very

high and generates a large number of measurements.

These measurements are noisy and can contain false

alarms. Also vehicles on the ground are usually quite

manoeuvrable over short periods of time. Sensor

scanning time, denoted by T, is long, and vehicles

are detected by the sensor with the probability PDaccording to the sensor resolution.

Conventional techniques consider the case of

each target individually using a Kalman filter [1],

but that implies the use of data association [2]. This

operation can be very elaborate, particulary when

the number of targets increases or when targets

are close together as in a convoy. One of the most

famous methods is multiple hypothesis tracking [3]

(MHT), a method whereby multiple data association

hypotheses (false alarm, new track, continuity of a

new track) are formed at each scan. But this algorithm

is very time consuming and has weaknesses in its

estimation of the exact number of targets when targets

are close together. Another approach is the joint

probabilistic data association filter [2] (JPDAF),

but this only allows tracking of a known number of

targets, whereas in practice many targets appear and

disappear from the observed area at each iteration.

An alternative to Kalman tracking is particle filtering

which has inspired work in ground target tracking

[4, 5] and particulary convoy tracking [6].

The probability hypothesis density (PHD) filter

was developed by Ronald Mahler using his work on

finite set statistics [7] (FISST) and random sets. This

filter leads to a new class of algorithms [8] based on

the study of joint density probability of the random

finite sets (RFS) describing target dynamics and

measurements. The first-order moment of this RFS,

called the intensity function, is the function whose

integral in any region on state space is the expected

number of targets in that region. Points with the

highest density are then expected targets. To improve

the number of target estimations, Mahler proposes a

generalization of the PHD filter called the cardinalized

PHD (CPHD) filter [9], which jointly propagates

the intensity function and the entire probability

distribution of the number of targets. Under Gaussian

assumptions on target dynamics and birth process,

Vo et al. proposes a CPHD filter recursion called

832 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 2 APRIL 2011

Fig. 1. IMM-MHT/GM-CPHD hybridization.

the Gaussian mixture cardinalized PHD [10, 11]

(GM-CPHD). This approach gives very encouraging

results, in particular for the estimation of the numberof targets. Performance evaluation of MHT versus

GM-CPHD can be seen in [12].

Nevertheless as we show in Table IV, with

manoeuvring targets the GM-CPHD filter without

IMM approach, like other multitarget tracking

algorithms not equiped with IMM, has problems with

velocity estimation. Punithakumar et al. and Pasha

et al. in [13], [14] propose to introduce multiple

motion models in a PHD filter, but our proposed

approach is quite different and is based on the

observation that the GM-CPHD and the IMM-MHTcan be seen as complementary algorithms: the first

for the estimation of the number of targets and for

an approximate position estimation and the latter to

specify state estimation. Consequently combining the

multiple-model approach in both algorithms seems

unnecessary. Our proposed hybridization is described

in Fig. 1. The idea of using the IMM-MHT with the

PHD filter is not new; we see for instance Panta’s

work [15—17], where the particle PHD is used as a

clutter filter or is combined with a track-to-estimate

association technique to produce tracks. Our approachis not unrelated, but we propose to take advantages

of the two methods. We take into account the good

estimation of the number of targets of the CPHD

algorithm. As using particles can be time consuming

in a real multitarget context, we choose the Gaussian

mixture recursion. As PHD techniques do not give

tracks directly, which are necessary to identify targets,

we propose a labeled version of the GM-CPHD.

Further discussion on labelization techniques is

proposed in Section III. Finally we take advantage

of the good state estimation given by the MHTand its different versions as the VS-IMMC-MHT

(variable structure—interacting multiple model with

constraints—multiple hypothesis tracker) [18] by

introducing a hybridization.

The paper is organized as follows. Section II

is a theoretical introduction of the PHD and its

main recursions; Section III describes our approach

concerning GM-CPHD labeling; Section IV describes

precisely how the IMM-MHT and the GM-CPHD

are combined into one algorithm. In Section V, thealgorithm is extended by using the VS-IMMC-MHT.

Finally Section VI describes our simulation and

performances before we conclude in Section VII.

II. BACKGROUND

A. The Random Finite Set

Let M(k) be the the cardinality of the target set Xkat time k and N(k) the cardinality of the measurement

set Zk. These sets are defined as follows:

Xk = fxk,1, : : : ,xk,M(k)g (1)

Zk = fzk,1, : : : ,zk,N(k)g: (2)

An RFS is a finite-set valued random variable

which can be generally characterized by a discrete

probability distribution and a family of joint

probability densities representing the existence

probabilities of the target set. Considering the RFS

of survival targets Skjk¡1 between scans k¡ 1 andk, the RFS of spawned targets Bkjk¡1 and the RFSof spontaneous birth targets ¡k, the global RFS

characterizing the multitarget set can be written as

Xk =

24 [³2Xk¡1

Skjk¡1(³)

35[24 [³2Xk¡1

Bkjk¡1(³)

35[¡k:(3)

In the same manner the multitarget set observation

Zk can be seen as a global RFS composed by the RFS

of measurements originally from the targets Xk and by

the RFS of false alarms Kk:

Zk =

24 [x2Xk

£k(x)

35[Kk: (4)

B. The Probability Hypothesis Density

The notion of PHD was introduced by Ronald

Mahler [7, 9]. From FISST [19] and random set

theory, he extended the optimal recursive Bayesian

filter for single-sensor and single detection to a

theorical multitarget multisensor filter. The PHD filter

traditionally involves two steps: prediction and update

that propagate the multitarget posterior PHD of the

target RFS, also called the intensity function v. Finally

the intensity function of an RFS X in the geometrical

space X can be seen as a nonnegative function which

satisfies, for any closed subset Aμ X representing the

observed area, the following property:

E[jX \Aj] =ZA

v(x)dx (5)

where E denotes the expectation, jX \Aj thecardinality of the target set on the space A, and v(x)

the intensity function at point x.

Now we can define the a priori intensity

function vkjk¡1 of the target random set Xk at time k

considering the a posteriori intensity function vk¡1at the previous time k¡ 1, the probability Ps(³) fora target to survive between times k¡1 and k, the

POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 833

transition function fkjk¡1(: j ³) given the previous state³, and the intensity of target birth °k.

vkjk¡1(x) =μZ

Ps(³):fkjk¡1(x j ³):vk¡1(³)d³¶+ °k(x):

(6)

Knowing the measurement random set Zk, it is

possible to update the intensity function as follows:

vk(x) = (1¡PD(x))vkjk¡1(x)

+Xz2Zk

PD(x):gk(z j x)vkjk¡1(x)·k(z)+

RPD(³):gk(z j ³)vkjk¡1(³)d³

(7)

where gk(z j x) is the likelihood of a measurementz knowing the state of a target x; ·k is the

clutter intensity which is modeled by a Poisson

process.

Because of the presence of integrals, these

equations are not directly tractable, but a solution

proposed and/or used by Zajic et al. [20], Vo et al.

[21, 22], Schubert [23], Sidenbladh [24], Maggio [25],

and Clark [26] is to use a sequential Monte Carlo

method to approximate it. Another approach is to

consider Gaussian mixture assumptions as described

in Section IID.

C. The Cardinalized PHD

To improve the estimation of the number of

targets, Mahler proposed in 2007 [27] to jointly

propagate the intensity function and the entire

probability distribution of the number of targets,

called the cardinality distribution p, as suggested by

Erdinc [28]. In this way we can calculate the predicted

probability pkjk¡1(n) to have n targets as the sum of all

hypothesis probabilities among the n targets as either

survivor targets or birth targets while knowing that

additionnal targets can die between iterations k¡ 1and k. Finally 8n 2 N?,

pkjk¡1(n) =nXj=0

p¡ (n¡ j)

£1Xl=j

CljhPs,vk¡1ijh1¡Ps,vk¡1il¡j

h1,vk¡1ilpk¡1(l)

(8)

with p¡ (n¡ j) the birth probability of (n¡ j) targetand Clj the binomial coefficient with parameters (l,j).

Following Bayes’ Theorem the estimated

cardinality distribution pk can be written as a

likelihood ratio:

pk(n) =¤(Zk j n)¤(Zk)

pkjk¡1(n) (9)

where ¤(Zk j n) is the likelihood of the measurementset Zk knowing that there are n targets and ¤(Zk) is a

normalizing constant.

The intensity propagation in the CPHD filter

becomes more complex. In fact given these

probabilities, the measurement likelihood can

be calculated more precisely by separating the

cases D where the target x is not detected and D

where it is detected. Finally the posterior intensity

becomes

vk(x) =

"(1¡PD)

¤(Zk j D)¤(Zk)

+PD¤(Zk jD)¤(Zk)

#vkjk¡1(x):

(10)

Details concerning the way to calculate these

likelihoods are given in the next subsection.

D. The Gaussian Mixture Cardinalized ProbabilityHypothesis Density

1) The Gaussian Assumptions: As proposed by

Vo in 2006 [29, 30], a Gaussian mixture recursion

is possible by considering some linear Gaussian

assumptions.

1) Each individual target follows a linear Gaussian

model as well as measurements

fkjk¡1(x j ³) =N (x;Fk¡1³,Qk¡1) (11)

gk(z j x) =N (z;Hx,Rk) (12)

where N (:;Hx,Rk) denotes the Gaussian densitywith mean Hx and covariance Rk with Fk¡1 thestate transition matrix, Qk¡1 the process noisecovariance, H the measurement to target space

transformation matrix, and Rk the observation noise

covariance.

2) The detection probability PD and the survival

probability Ps are constant over the entire observed

area A:

Ps(x) = Ps (13)

PD(x) = PD: (14)

3) The birth intensity used in (6) can be

considered as a Gaussian mixture:

°k(x) =

J°

kXi=1

w°k,iN (x;m°k,i,P°k,i) (15)

where w°k,i, m

°k,i, and P

°k,i are the weight, mean, and

covariance of the birth Gaussians, respectively, and

J°k is their number.

4) The posterior target intensity on surface S can

be written as a Gaussian mixture:

vk(x) =

JkXi=1

wk,iN (x;mk,i,Pk,i) (16)

where wk,i, mk,i, and Pk,i are the weight, mean, and

covariance of the current Gaussians, respectively, and

Jk is their number.


Moreover mk is the state vector of the target i at time

k and is written

mk,i = [xk,i, _xk,i,yk,i, _yk,i]T (17)

where (xk,i,yk,i) is the position and ( _xk,i, _yk,i) the

velocity of the target i in the Cartesian model.

2) The Gaussian Mixture CPHD: Combining

Gaussian mixture properties with the CPHD recursion

is a solution proposed by Vo in 2006 [8, 10, 31].

Since then this method was modified to add road

constraints, and this leads to a complete method for

ground moving target tracking [11]. Based on the

CPHD and the Gaussian mixture assumptions, we

can now describe the prediction step for the intensity

function and the cardinality distribution:

vkjk¡1(x) = vS,kjk¡1(x)+ °k(x) (18)

with °k(x) as described in (15) and vS,kjk¡1 thepredicted intensity function of survival targets defined

as follows:

vS,kjk¡1(x)

= Ps

Jk¡1Xi=1

wk¡1,iN (x;Fk¡1mk¡1,i,Fk¡1Pk¡1,iFTk¡1 +Qk¡1)

(19)

and 8n 2N¤,

pkjk¡1(n) =nXj=0

p¡ ,k(n¡ j)1Xl=j

Cljpk¡1(l)Pjs (1¡Ps)(l¡j):

(20)

Given the predicted cardinality pkjk¡1 and thepredicted intensity vkjk¡1, the posterior cardinality canbe calculated as

pk(n) =¨ 0k [vkjk¡1,pkjk¡1](n)pkjk¡1(n)

h¨ 0k [vkjk¡1,pkjk¡1]i

(21)

with the likelihood ¨uk of the measurement set Z,

which can be calculated considering all hypotheses

for a target to be detected or not, knowing that at least

u measurements are associated with a target at time k

and that there are n targets:

¨uk [v,Z](n) =

min(jZj,n)Xj=0

(jZj ¡ j)!p·(jZj ¡ j)Pnj+u

£ (1¡PD)n¡(j+u)

h1,wij+u ej(¥k(w,Z)) (22)

where ej is the elementary symmetric function of

order j (more information on the subject is available

in [32]), Pnj+u is the permutation coefficient with

parameters (j+u,n), p·(jZj ¡ j) is the probability tohave jZj ¡ j false alarms, and ¥k(w,Z) is a measurefor the measurement likelihood knowing the weight

wkjk¡1 of predicted Gaussian components:

¥k(w,Z) = fhv,Ãk,zi : z 2 Zg (23)

with Ãk,z(x) expressed as

Ãk,z(x) =h1,·ki·k(z)

gk(z j x)PD: (24)

Finally the estimation of the intensity function can

be calculated as

vk(x) =h¨ 1

k [vkjk¡1,Zk],pkjk¡1ih¨ 0

k [vkjk¡1,Zk ],pkjk¡1i(1¡PD)vkjk¡1

+Xz2Zk

h¨ 1k [vkjk¡1,Zk n fzg],pkjk¡1ih¨ 0

k [vkjk¡1,Zk],pkjk¡1iÃk,z(x)vkjk¡1(x):

(25)

III. THE LABELED GM-CPHD

In the classical version of the GM-CPHD filter,

the problem of track labeling is not considered.

Some authors study this problem. Clark et al. in

[33] proposed for the GM-PHD filter to assign a tag

to each Gaussian component and to keep as tracks

Gaussians with weights above a certain threshold.

And when a measurement is not received, the weight

falls below the desired threshold but the Gaussian

component is not deleted, and the target trajectory is

specified a posteriori after the weight is again above

the desired theshold. In another publication [26] he

offered two methods to overcome the problem in a

classical particle PHD filter: first the particles are

labeled and a k-means method is used to attribute

a label to a peak. Secondly he proposed to take the

best combination of association between peaks and

predicted tracks. This last method is very close to

Lin’s method [34]. However when the number of

targets is large, the association is limited to a nearest

neighbour standard filter (NNSF) approach. Finally in

some recent works [35, 36], Panta et al. proposes to

manage track labeling by assigning tags to individual

Gaussian components and by using tree structures for

propagating these tags. Then closely spaced targets

are specifically processed by using a track-to-estimate

association based on distance between Gaussians

components and predicted tracks.

Our approach is close to the latter, but take

into account the estimated number of targets. The

goal is to estimate an association matrix between

Gaussian components and tracks and to ensure the

track continuity even if targets are not detected at

each iteration. If we specify the target trajectories

a posteriori like Clark in [33], we would not take

into account the estimation of the number of targets

processed by the CPHD filter. A first track-to-estimate

association is searched by maximizing the global

weight of the association (Section IIID); in other

words, Gaussian components that are strongly

weighted are selected often as possible. A Gaussian

component is strongly weighted depending on the

likelihood of a measurement z knowing the Gaussian

state as shown in (24). However when targets are


close together, some association combinations are not

discriminated, and the statistical distance between

peak and predicted track by using the track score is

then used (Section IIIE). An optimization algorithm

is finally proposed (Section IIIF) to reduce computing

time.

A. Notations

Let G be the Gaussian component set given by theGM-CPHD written:

Gk = fwk,i,mk,i,Pk,igi2f1,:::,NGkg = fGk,1, : : : ,Gk,NG

kg(26)

where NGk is the number of Gaussian components( 6= Nk) at time k. A track can be defined as a sequenceof estimated states describing the dynamics of one

target. The goal of tracking is to offer a list of tracks

corresponding to all of the targets. That is why this

labeling step is necessary in order to provide a track

set chosen amongst the Gaussian components Gk. Atrack Tk,j is defined by a state xk,j , a covariance Pk,j ,and a score sk,j at time k:

Tk,j = fxk,j ,PTk,j ,sk,jgj2f1,:::,Nkg: (27)

The track set is written

Tk = fTk,1, : : : ,Tk,Nkg (28)

with Nk the estimation of the number of targets given

by the GM-CPHD.

B. Score Calculation

Blackman [37] recursively calculates the score

of a track l by using the Napierian natural logarithm

denoted ln of the associated measurement likelihood

gk(z j xk,j):

sk,l = sk¡1,l+ lnμ

PD¯FA+¯NT

gk(z j xk,j)¶

(29)

with ¯FA and ¯NT the false alarm and birth spatial

densities. As previously, gk denotes a Gaussian

density.

If a track l is not associated with any

measurement, the score is calculated as follows:

sk,l = sk¡1,l+ ln(1¡PD): (30)

Regarding the initialization,

s0,l = ln

μPD:¯NT¯FA

¶: (31)

To prevent the score from increasing, this is

calculated only on ns iterations. This means

sk,l =

nsXt=1

sk¡t,l+ lnμ

PD¯FA+¯NT

:gk(z j xk,j)¶: (32)

C. The Association Matrix

We define the set of Na feasible association

matrices Ak. Each association matrix Ak,i of sizeNk £NGk associates the Gaussian set with the tracks,8(m,n)· (Nk,NGk ) knowing that a track is associated atmost with one Gaussian:

Ak,i(m,n) =

8<:1 if Gk,n can be associated

to Tkjk¡1,m0 otherwise:

:

(33)

A Gaussian component n is said to be associable to a

track m if it satisfies a gating test around the predicted

position of the track and if the track is not associated

with another component.

D. The Weight Matrix

We define the weight matrix Wk of size Nk £NGkdefined as follows, 8(m,n)· (Nk,NGk ):

Wk(m,n) =

8<:wk,n if Gk,n satisfies a gating

test around Tkjk¡1,m0 otherwise

:

(34)

If Nk > Nk¡1, one or more new tracks must beinitialized, and each Gaussian component is a potential

new track. In matrix Wk, 8m 2 f1, : : : ,NGk g, 8l 2fNk¡1 +1, : : : ,Nkg,

Wk(m, l) = wk,l: (35)

In the same way if Nk < Nk¡1, some tracksmust be deleted. Weakly weighted tracks cannot

be deleted based on their weights alone because

if a measurement is missing, the weight of the

corresponding Gaussian is low, which is why tracks

with the lowest score are deleted.

Finally we define the global weight set Wgk =

fWgk,ig8i<Na . Each global weight of an association i is

calculated as

Wgk,i =

NkXm=1

NGkX

n=1

Ak,i(m,n):Wk(m,n): (36)

And the association matrices which maximize the

weight are written as

A?k = argmaxAkWgk : (37)

E. The Cost Matrix

If targets are close together, the previous step

produces too many association matrices. In order to

choose one amongst the N?a matrices A?k, the cost isthen used. Similary the cost matrix Ck of size Nk £NGk


is written as 8(m,n)· (Nk,NGk ),

Ck(m,n) =

8<:c(m,n) if Gk,n satisfies a gating

test around Tkjk¡1,m0 otherwise

(38)

with c(m,n) the cost of the association of the predicted

track m with the Gaussian n written as the negative

Napierian logarithm of the likelihood ratio, 8(m,n)·(Nk,N

Gk ),

c(m,n) =¡ lnÃPD:gk(Gk,n j xkjk¡1,j)

¯FA

!(39)

with ¯FA the spatial false alarm density and gk(Gk,n jxkjk¡1,j) the likelihood of the Gaussian component nknowing the predicted position xkjk¡1,j of the track j.Finally the global association cost set Cgk =

fCgk,ig8i<N?a is used, and each global cost is calculatedas

Cgk,i =

NkXm=1

NGkX

n=1

A?k,i(m,n):Ck(m,n) (40)

and the best association Ak is calculated like the

minimal cost matrix

Ak = argminA?k

Cgk : (41)

F. Optimization Algorithm

Under the conditions discussed above, our problem

is very close to the S-D assignment problem [38].

The direct calculation of the association matrix that

maximizes the global weight and minimizes the global

cost of the association is time consuming and becomes

intractable when the number of closely spaced targets

increases. By considering an association matrix of

size Nk £NGk , the number of possible associations isNG!=Nk!. For example, if the estimated number oftargets is 10 and if the maximum number of Gaussian

components is limited to Jmax = 50, the number of

feasible associations is 8:1057. Fortunately this number

is reduced by using the gating step, but this is not

enough to have a tractable algorithm. Therefore we

propose an optimization algorithm. The main idea is

to separate, as much as possible, the association space

search. For example, if the most strongly weighted

Gaussian component can be associated with only one

track, then this association is automatically selected.

More generally the first step of our optimization

algorithm is to sort the weight matrix in ascending

order. Then by considering columns one after the

other while the number of processed columns is

lower than the number of rows, a column which

is a single non-zero element contains the element

which maximizes the weight matrix. Similarly still

considering columns one after the other, if the

number of processed columns is equal to the number

of the corresponding non-zero element rows, it is

easier to find the association that maximizes the

weight and minimizes the cost. The optimization

algorithm principle is illustrated by the pseudo code

in Appendix.

IV. THE HYBRID ALGORITHM

Before describing the hybridization step, we

provide a brief review of the MHT principle and

describe how this algorithm is combined with an IMM

approach.

A. The Multiple Hypothesis Tracker

The MHT can be described with seven successive

steps:

1) The first step of an MHT is track prediction.

Usual motion models such as constant velocity (CV)

or constant acceleration (CA) can be used to predict

the states.

2) In order to avoid the association problem, we

need a probabilistic expression for the evaluation

of the track formation hypotheses that includes

all aspects of the data association problem. It is

convenient to use the log-likelihood ratio (LLR)

or track score which can be expressed at current

time k in the recursive form given in [37]. When

the new measurement set Zk is received, a standard

gating procedure is applied in order to determine

the viable moving target indicator (MTI) reports to

track pairings. The existing tracks are updated, and

extrapolated confirmed tracks are formed.

3) After the track score calculation of track Tk,l,

the Wald’s sequential probability ratio test (SPRT)

is used to set up the track status either as deleted,

tentative, or confirmed. The tracks that fail the test

are deleted, and the surviving tracks are kept for the

next stage.

4) The clustering process consists of listing

the collection of all tracks linked by a common

measurement. The clustering technique is used to limit

the number of hypotheses and therefore to reduce the

complexity. The clustering result is a list of tracks that

are interacting. The next step is to form hypotheses of

compatible tracks.

5) In the fifth step for each cluster, multiple

coherent hypotheses are formed to represent the

different compatible track scenarios. Each hypothesis

is evaluated according to the track score function

associated with the different tracks. Then a technique

is required to find the hypothesis set that represents

the most likely track collection. The unlikely

hypotheses and associated tracks are deleted by a

pruning process, and only the NHypo best hypotheses

are conserved.

6) For each track the a posteriori probability

is computed, and a well-known N-Scan pruning

approach [37] is used to select and delete the


confirmed tracks. With this approach the most likely

track is selected to reduce the number of tracks.

7) A pruning step is used to remove the unlikely

hypotheses among the N-Scan hypotheses. In order

to reduce the number of tracks, a merging technique

(selection of the most probable tracks which have

common measurements) is also implemented.

Finally the MHT is classically combined with an

IMM. The idea is to use, during the prediction step,

a finite number of models, describing the target

behaviour globally. The IMM estimates the state

sytem for each model and the model occurence

probability. The estimated state is a combination of

the different estimations using each model.

B. The Hybridization Step

After the estimation step the system gives a track

set T MHTk originally from the IMM-MHT and a track

set T PHDk from the GM-CPHD in which the estimation

of the number of targets is supposed good. The goal

is now to combine this information to give the best

estimation possible of the number of targets and target

states, along with the best track continuity possible.

Our strategy is to assume that the target positions

given by the GM-CPHD are approximately good.

Then a gating process is applied to find MHT tracks

that are statistically close to PHD tracks, regarding

the Mahalanobis distance. Finally among the selected

MHT tracks, tracks which have the highest score are

selected as final tracks. However if a PHD track is

not associated with any MHT track, the PHD track is

maintened as a final track.

V. THE HYBRID ALGORITHM WITH ROADCONSTRAINT

Ground target tracking with map information can

substantially improve usual target tracking algorithms.

In the literature several approaches are presented

to use this prior information and introduce it in the

tracking process. Recent work proposes to combine

road network location and GM-CPHD to track

individual targets and convoys on the road network

[11]. Other approaches based on familiar association

algorithm like S-D assignment sequentially modify the

IMM motion models according to the road network

configuration [39].

A. The VS-IMMC-MHT Principle

Our algorithm [40], the VS-IMMC-MHT,

ressembles MHT except for the first and the sixth step

described in the following paragraphs.

1) Track Prediction: The first functional part

of the MHT is the track prediction. Each track is

constrained to the road, and consequently the motion

models of the associated IMM are also constrained

to the road. In addition the prediction depends on the

road network topology, and the variation of the road

segment configurations carry out the constraint motion

models modification. The variation of the motion

models set is described in [18].

2) Validation Step: When the new set Zk of

measurements is received, a standard gating procedure

is applied in order to determine the viable MTI reports

of track pairings. The existing tracks are updated

with VS-IMMC, and extrapolated confirmed tracks

are formed. When the track is not updated with MTI

reports, the stop motion model is activated. According

to each constraint motion model, the associated

updated state is projected on the road in the manner

that the track is constrained to the road.

3) Track Confirmation and Maintenance: In order

to reduce the severity of the association problem,

we need a probabilistic expression for the evaluation

of the track formation hypotheses that includes

all aspects of the data association problem. It is

convenient to use the LLR or track score of a track.

After the track score calculation of the track, the

SPRT is used to set up the track status either as

deleted, tentative, or confirmed track. The tracks that

fail the test are deleted, and the surviving tracks are

kept for the next stage.

4) Clustering Step: The process of clustering

is the collection of all tracks that are linked by a

common measurement. The clustering technique is

used to limit the number of hypotheses generated

and therefore to reduce the complexity. The result of

clustering is a list of tracks that are interacting. The

next step is to form hypotheses of compatible tracks.

5) Hypotheses Formation: For each cluster, in the

fourth level multiple coherent hypotheses are formed

to represent the different compatible tracks scenarios.

Each hypothesis is evaluated according to the track

score function associated with the differents tracks.

Then a technique is required in order to find the

hypotheses set that represents the most likely tracks

collection. The unlikely hypotheses and associated

tracks are deleted by a pruning process, and only the

NHypo best hypotheses are conserved.

6) N-Scan Pruning Step: For each track the

a posteriori probability is computed, and a well-known

N-Scan pruning approach is used to select and delete

the confirmed tracks. With this approach the most

likely track is selected to reduce the number of

tracks. But the N-Scan technique combined with the

constraint implies that other tracks hypotheses (i.e.,

constrained on other road segments) are arbitrarily

deleted. That is why we must modify the N-Scan

pruning approach in order to select the Nk best tracks

on each Nk road sections.

7) Wald Test: Wald’s SPRT is used to delete the

unlikely hypotheses among the Nk hypotheses. The

tracks are then updated and projected on the road

network. In order to reduce the number of tracks kept


TABLE I

The MHT Parameters

Name Value

Birth target density 8:92 ¢ 10¡9Threshold for track confirmation 10¡4Threshold for track deletion 10¡1Threshold for hypothesis deletion 10¡2

Number of branches to keep 2

Threshold for global track probability 50

Number of scans before pruning 50

Gating probability 0.95

TABLE II

The GM-CPHD Parameters

Name Value

Survival probability 0.98

Initial Gaussian weight 10¡3

Pruning threshold 10¡2

Merging threshold 20

Maximum number of targets 50

Maximum number of Gaussians 50

Average number of births 0.06

Model noise 2

Maximum velocity 20

TABLE III

The Hybridization Parameters

Name Value

Number of iterations for score calculation 3

Weight threshold for new tracks 0.8

in the memory of the computer, a merging technique

(selection of the most probable tracks which have

common measurements) is also implemented.

Moreover we use a three-model IMM: one “stop”

model and two CV models, one with a very small

plant noise and the other with a large plant noise to

be robust to manoeuvers. During the gating step when

the track is not updated with MTI reports, the stop

motion model is activated.

B. The Constrained Hybrid Version

By adding Geographical Information System

(GIS) data, state estimation is improved, as in the

VS-IMMC-MHT, with the introduction of road

constraints. That is why we propose to use the same

method as previously but to substitute the IMM-MHT

with the VS-IMMC-MHT described in Section VA.

The principle is shown in Fig. 2.

VI. SIMULATION RESULTS

In the following we present some simulation

results that evaluate the performances of our

approaches. These are compared with the

performances of a classical IMM-MHT as well as

Fig. 2. GM-CPHD–VS-IMMC-MHT hybridization.

with a labeled GM-CPHD and a VS-IMMC-MHT.

In the next section, we present the parameters of these

algorithms.

A. The Simulation Parameters

We use a three-model IMM, two CV models (one

with a very small plant noise of 0:05 m ¢ s¡2 and thethe other with a large plant noise of 0:8 m ¢ s¡2) andone stop model [41].

MHT parameters are described in Table I,

GM-CPHD in Table II, and hybridization in Table III.

B. Scenario

The scenario time is limited to 400 s. It contains

9 targets (2 convoys) whose trajectories are shown in

Fig. 3. The accumulated MTI reports are shown in

Fig. 4.

1) Target 1 is moving on the left road (to the East)

with the CV of 12 m ¢ s¡1 from North to South from

time t= 1 s to time t= 391 s.

2) Target 2 is moving on the middle road with

the CV of 15 m ¢ s¡1 from North to South from time

t= 1 s to time t= 341 s.

3) Target 3 is moving on the middle road with

the CV of 15 m ¢ s¡1 from South to North from time

t= 1 s to time t= 351 s.

4) Targets 4—6 form a convoy moving on the

middle road with a CV of 8 m ¢ s¡1 from North to

South. The lead target 4 starts at time t= 1 s, the

middle target 5 at time t= 21 s, and the tail target 6

at time t= 41 s. The convoy stops at time t= 341 s.

5) Targets 7—9 form a convoy moving on the right

road with a CV of 10 m ¢ s¡1 from South to North.

The lead target 7 starts at time t= 111 s, the middle

target 8 at time t= 131 s, and the tail target 9 at time

t= 151 s. The convoy stops at time t= 391 s.

In our scenario GMTI reports are simulated. We

assume that the GMTI sensor has a linear trajectory,

its velocity is 30 m ¢ s¡1 and its altitude is 4000 m.The typical measurement error is 20 m in range

and 0.008 in azimuth. The false alarm density is

8:92 ¢10¡9, and the detection probability PD is equalto 0.9. The scanning time is T = 10 s. The minimum

detection velocity is 2 m ¢ s¡1.C. Results

The performances of tracking algorithms have

been compared for 500 independant Monte Carlo


Fig. 3. Scenario.

Fig. 4. Snapshot of accumulated MTI reports.


Fig. 5. Completeness, redundant track rate, and false alarm rate. Completness is same for GM-CPHD, hybrid MHT, and hybrid

VS-IMMC.

Fig. 6. Cardinality, Wasserstein distance, and OSPA metric. GM-CPHD, hybrid MHT, and hybrid VS-IMMC are very difficult to

distinguish because results are similar for cardinality.


TABLE IV

Average RMS Error in Position and Velocity

IMM-MHT GM-CPHD VS-IMMC-MHT Hybrid MHT Hybrid VS-IMMC

Target 1

Position error (in m) 35.85 42.36 21.33 35.54 21.14

Velocity error (in m/s) 3.57 7.48 2.93 3.01 2.31

Target 2



Target 3



Target 4



Target 5



Target 6



Target 7



Target 8



Target 9



runs. Fig. 5 shows completeness, redundant track

rate, and false track rate. Completeness is the ratio

of the number of detected targets over the number of

real targets over the time, the false track rate is the

ratio of the number of false tracks over the number

of detected targets, and the redundant track rate is

the number of tracks associated with at least one

real target over the number of detected targets. Two

appropriate criteria for comparing PHD performance

are the Wasserstein distance [42] and the optimal

subpattern assignment (OSPA) metric [43] which are

shown at the bottom of Fig. 6, illustrating notably the

cardinality estimation (at the top). The Wasserstein

distance is defined as a metric capturing the difference

between two sets of vectors which do not have

necessarily the same cardinality. This metric does not

have a physically consistent interpretation when the

two sets have different cardinalities. This problem is

corrected with the OSPA metric, noted d(c)p as shown

in [44]. This last metric depends on two parameters:

p and c. The parameter p represents the sensitivity to

outliers (false tracks), and the parameter c represents

the penality assigned to cardinality and localization

errors. Before calculating, the set of permutations

¦n between the two compared sets must be defined.

Finally the OSPA metric between the estimated

target set Xk = fxk,1, : : : , xk,mg and the real target setXk = fxk,1, : : : ,xk,ng at time k with n¸m is calculatedas

d(c)p (X,X)

=

Ã1

n

Ãmin¼2¦n

mXi=1

d(c)(xi, x¼(i))p+ cp(n¡m)

!!1=p(42)

where d(c)(x,y) is defined as

d(c)(x,y) = min(c,kx¡ yk): (43)

If m¸ n, d(c)p (X,X) = d(c)p (X,X).


Fig. 7. Track length ratio.

The track length ratio is the ratio of the estimatedtrack length over the actual track length and is shownin Fig. 7. Finally the average mean square errors inposition and velocity are compared for each algorithmin Table IV.

D. Interpretation

The results are consistent with our expectations.

The IMM-MHT is an efficient algorithm for

multitarget tracking. For independant manoeuvring

targets, the track length ratios are higher than 0.85

and the state error is reasonable (approximately 35 m

in distance and 4 m/s in velocity). However, it has

problems tracking targets that are close together, and

in this case the track length ratios decrease and the

errors increase (up to 50 m in position and 7 m ¢ s¡1in velocity). The Wasserstein distance increases with

the number of targets, reflecting the problem of

estimating the number of targets. Effectively in the

IMM-MHT there is no estimation of this variable;

also we consider the number of targets as the number

of confirmed tracks, and this is overestimated at

each iteration. In consequence, the false track rate is

non-zero and the redundant track rate is not equal to

one.

The same observations apply to the

VS-IMMC-MHT except for the state estimation which

is better (approximately 25 m in distance and 4 m/s

in velocity for independant targets versus 30 m and

5 m/s for convoy targets). In consequence the track

length ratio is sometimes better for a convoy target

(for example target 6: 0.54 for IMM-MHT versus 0.66

for VS-IMMC).

Now by considering GM-CPHD results, we can

enumerate interesting observations. The track length

ratio of independant targets is slightly lower, but is

quite a bit larger for convoy targets (for example for

target 9: it is 0.88 for GM-CPHD versus 0.55 for

IMM-MHT). The estimation of the number of targets

is very close to the reality and in consequence the

Wasserstein distance, is lower. The negative point is

for the estimation state. It is higher for the position

estimation (approximately 42 m) and much higher for

the velocity (more than 7:5 m ¢ s¡1).These are the reasons why we develop the hybrid

version of the GM-CPHD algorithm. With the two

versions, we have nearly the same results for the

estimation of the number of targets and for the track

length ratio, but we improve the state estimation. For

the IMM-MHT hybrid version, errors are similar to an

IMM-MHT for independant targets and significantly

better for the other targets. The same is true for

the VS-IMMC hybrid version: the position error is

23.71 m and 3:24 m ¢ s¡1 for the velocity error.By examining the Wasserstein distance, we

confirm that the cardinality estimation error is strongly

penalized. That is the reason why two behaviors

are detected for this metric. Algorithms that use

the CPHD filter have similar results for cardinality

estimation, and we observe the same shape for

the Wasserstein distance with lower values for the

VS-IMMC hybrid version than for the IMM-MHT

hybrid version and also than for the GM-CPHD filter.

The order corresponds to the algorithm ability for

state estimation. The second behavior corresponds to

MHT algorithms. The Wasserstein distance is higher


than for PHD algorithms and strongly increases with

cardinality estimation error.

Concerning the OSPA metric, conclusions are

different because the OSPA metric does not penalize

cardinality errors, localization errors, and outliers

in the same manner. Consequently due to the low

false track rate and the good state and cardinality

estimation, the lowest OSPA metric is given by the

VS-IMMC hybrid version. Then the VS-IMMC-MHT

algorithm gives the lowest OSPA value except

from the time t= 330 s when the cardinality

estimation increases a lot. Finally the IMM-MHT

hybrid algorithm gives better performances that the

IMM-MHT algorithm probably due to the better

cardinality estimation of the hybrid algorithm. And

the GM-CPHD filter produces the higher value for the

OSPA metric, probably due to the high false track rate

and the state estimation error, which is higher than for

hybrid algorithms.

The computability of our algorithms is not

discussed in this article because data is produced by a

Matlab simulator, but this will be the object of future

work.

APPENDIX. PSEUDO ALGORITHM FOR THELABELING STEP

See page 846.

VII. CONCLUSION

In this paper we first propose a labeled version

of the GM-CPHD and an optimization algorithm for

the labeling. Obtained results on a complex scenario

show that the GM-CPHD can be improved in regard

to the velocity estimation. The MHT approach usually

produces better state estimates but is not easily

adapted to problems with closely spaced objects.

That is why we propose secondly to overcome the

weakness of the algorithm by using a hybrid version

of the labeled GM-CPHD and the IMM-MHT (or the

VS-IMMC if the road network is available). These

hybrid versions give very encouraging results. As the

velocity estimation is very close to the reality, it could

be a first step towards convoy detection.

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APPENDIX. PSEUDO ALGORITHM FOR THELABELING STEP


Evangeline Pollard was born in Roubaix on July 27th, 1984. She got her signaland image processing M.Sc. in 2007 from the University of Lyons. That same

year she graduated from the “CPE-Lyons” engineering school in electronics,

telecommunications, and computing.

During 2005—2006, she worked at the German Aerospace Laboratories

(DLR) in Munich on the TerraSAR-X processor. Since 2007, she has been a

Ph.D. student at the French Aerospace Lab (ONERA). With Professor Rombaut

(University of Grenoble) and Dr. Pannetier, she is working on a new approach for

abnormal behaviour detection.

Benjamin Pannetier was born in Paris on November 30th, 1979. He received his

B.Sc. in math from the University of Marne la Vallee and his Ph.D. in automatic

control and signal processing from the University of Grenoble in 2006.

Since 2005, he has been a research engineer at the French Aerospace Lab

(ONERA). His research interests include target tracking, detection/estimation

theory and data fusion for battlefield surveillance systems for the French army.

With Professor Rombaut (University of Grenoble) he is working on a new

approach for the abnormal behaviour detection.

Michele Rombaut graduated from the Ecole Universitaire d’Ingenieurs de Lille,

France, and received her Ph.D. from the University of Lille in electronic systems.

Since 1985, she has been with UTC Compiegne as an associate professor

and since 1994 as a professor. Since 2002, at the University Joseph Fourier of

Grenoble, she has been conducting her research at GIPSA-Lab (Grenoble Images

Speech Signals and Automatics) especially in data fusion and transferable belief

model.


hybrid algorithms for multitarget tracking using mht and gm-cphd

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