hybrid algorithms for multitarget tracking using mht and gm-cphd
TRANSCRIPT
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.
834 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 2 APRIL 2011
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
POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 835
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
836 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 2 APRIL 2011
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
POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 837
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
838 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 2 APRIL 2011
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
POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 839
Fig. 3. Scenario.
Fig. 4. Snapshot of accumulated MTI reports.
840 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 2 APRIL 2011
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.
POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 841
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
Position error (in m) 35.61 42.70 25.63 34.03 20.34
Velocity error (in m/s) 3.78 7.55 4.76 2.30 2.30
Target 3
Position error (in m) 35.52 42.86 21.50 34.64 20.79
Velocity error (in m/s) 4.39 7.93 4.26 2.49 2.87
Target 4
Position error (in m) 39.66 41.10 29.34 35.32 24.84
Velocity error (in m/s) 5.23 7.62 4.57 3.15 3.00
Target 5
Position error (in m) 51.47 41.52 40.84 39.07 28.76
Velocity error (in m/s) 7.46 8.00 6.74 4.14 4.27
Target 6
Position error (in m) 41.33 43.60 35.50 35.11 26.24
Velocity error (in m/s) 4.94 10.95 4.45 3.47 3.58
Target 7
Position error (in m) 38.43 39.82 27.38 33.09 21.37
Velocity error (in m/s) 4.80 7.46 4.75 2.58 2.65
Target 8
Position error (in m) 48.78 47.38 34.08 36.67 25.52
Velocity error (in m/s) 7.32 9.13 6.99 4.15 4.28
Target 9
Position error (in m) 39.91 43.49 32.88 34.61 24.42
Velocity error (in m/s) 5.08 11.20 5.76 3.52 3.95
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).
842 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 2 APRIL 2011
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
POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 843
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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POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 845
APPENDIX. PSEUDO ALGORITHM FOR THELABELING STEP
846 IEEE TRANSACTIONS ON AEROSPACE AND ELECTRONIC SYSTEMS VOL. 47, NO. 2 APRIL 2011
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.
POLLARD ET AL.: HYBRID ALGORITHMS FOR MULTITARGET TRACKING USING MHT AND GM-CPHD 847