A Comparative Analysis of QADA-KF with JPDAF for Multitarget Tracking in Clutter

2015·0.63 MB·

by  Jean Dezert

A Comparative Analysis of QADA-KF with JPDAF
for Multitarget Tracking in Clutter
Jean Dezert Albena Tchamova Pavlina Konstantinova Erik Blasch
ONERA Inst. of I&C Tech., BAS Eur. Polytech. Univ. AFRL
Palaiseau, France. Sofia, Bulgaria. Pernik, Bulgaria. Rome, U.S.A.
[e-mail] [e-mail] [e-mail] [e-mail]
Abstract—This paper presents a comparative analysis of per- way in which Kalman filter (KF) could be enhanced in order
formances of two types of multi-target tracking algorithms: 1) toreflecttheknowledgeobtainedbasedontheQADAmethod,
the Joint Probabilistic Data Association Filter (JPDAF), and
called QADA-KF method.
2) classical Kalman Filter based algorithms for multi-target
Taking into accountthat QADA assumes the reward matrix
tracking improved with Quality Assessment of Data Association
(QADA) method using optimal data association. The evaluation is known, regardlessof the manner in which it is obtained by
is based on Monte Carlo simulations for difficult maneuvering the user,in this paperwe proposeand test the performanceof
multiple-target tracking (MTT) problems in clutter. two possible versions of QADA-KF. The first one utilizes the
Keywords: Data association, JPDAF, Belief Functions, assignmentmatrix,providedbytheGlobalNearestNeighbour
QADA, PCR6 rule, Multitarget Tracking. (GNN)method,calledQADA-GNNKFapproach.Thesecond
oneutilizestheassignmentmatrix,providedbytheProbabilis-
I. INTRODUCTION
tic Data Association (PDA) method, called QADA-PDA KF
Multiple-target tracking (MTT) is a principle component method.
of surveillance systems. The main objective of MTT is to ThesetwoQADA-KFmethodsarecomparedwiththeJoint
estimatejointly,ateachobservationtimemoment,thenumber Probabilistic Data Association Filter (JPDAF) [7]–[9] which
of targets continuously moving in a given region and their is an extension of the Probabilistic Data Association Filter
trajectoriesfromthenoisysensordata.Inasingle-sensorcase, (PDAF) [1] to a fixed and known number of targets. JPDAF
the multitargettracker receives a randomnumberof measure- usesjointassociationeventsandjointassociationprobabilities
mentsduetotheuncertaintywhichresultsinlowdetectionand inordertoavoidconflictingmeasurement-to-trackassignments
false alarms, arising independently of the targets of interest. by making a soft (probabilistic) assignment of all validated
Because of the fact that detection probability is not perfect, measurements to multiple targets.
some targets may go undetected at some sampling intervals. The main objective of this paper is to compare the perfor-
Additional complications appear, apart from the process and mances of: (i) classical MTT algorithms based on the GNN
measurement noises, associated with a measurement origin approach for data association, utilizing Kinematic only Data
uncertainty,misseddetection,cancelling(death)oftargets,etc. (KDA) and Converted Measurement Kalman Filter (CMKF);
Dataassociation(DA)isaprimarytaskofmodernMTTsys- (ii) QADA-GNN KF based MTT; (iii) QADA-PDA KF based
tems[1]–[3].Itentailsselectingthemosttrustableassociations MTT; (iiii) JPDAF based MTT. The evaluation is based on
between uncertain sensor’s measurements and existing targets a Monte Carlo simulation for particular difficult maneuvering
ata giventime.In thepresenceofa dense MTTenvironment, MTT problems in clutter.
with false alarms and sensor detection probabilities less than Thispaperis organisedasfollows.InSectionII theJPDAF
unity, the problem of DA becomes more complex, because it is described and discussed. Section III is devoted to QADA
should contend with many possibilities of pairings, some of based KF. Data associationmethods, providingan assignment
whichareinpracticeveryimprecise,unreliable,andcouldlead matrix for QADA are discussed in Section IV. Two particular
to criticalassociationmistakesin the overalltrackingprocess. simulation MTT scenarios and results are presented for the
In order to deal with these complex associations the most KDA, QADA-GNN KF, QADA-PDA KF, and JPDAF in
recent method to evaluate the Quality Assessment of Data Section V. Conclusions are made in Section VI.
Association (QADA) encountered in multiple target tracking
applications in a mono-criterion context was proposed by II. JOINT PROBABILISTIC DATAASSOCIATION FILTER
Dezert and Benameur [4], and extended in [5] for the multi- The Joint Probabilistic Data Association Filter (JPDAF)
criteria context. It is based on belief functions (BF) for is an extension of the Probabilistic Data Association Filter
achievingthequalityofpairingsbelongingtotheoptimaldata (PDAF) for tracking multiple targets in clutter [1], [2], [10]–
assignment solution based on its consistency with respect to [12]. This Bayesian tracking filter uses the probabilistic as-
all the secondbest solutions, providedby a chosen algorithm. signmentofallvalidatedmeasurementsbelongingtothetarget
Most recently, in [6] the authors did discuss and propose the gateto updateitsestimate.ThepreliminaryversionofJPDAF
was proposed by Bar-Shalom in 1974 [13], then updated measurements available up to time k, and ωˆ (Θ(k)) are the
it
and finalized in [7]–[9]. The assumptions of JPDAF are the corresponding components of the association matrix charac-
following: terizing the possible joint association Θ(k).
• thenumberN ofestablishedtargetsinclutterisknown;
T
• all the information available from the measurements Zk B. Feasible joint association events
up to time k is summarized by the sufficient statistic
Validation gates are used for finding the feasible joint
xˆt(k|k) (the approximate conditional mean), and covari-
events but not in the evaluation of their probabilities [12] (p.
ance Pt(k|k) for each target t;
388–389). To describe the observation situation, it uses the
• the real state xt(k) of a target t at time k is modeled by
validationmatrixΩ=[ω ],i=1,...,m andt=0,...,N
a Gaussian pdf N(xt(k);xˆt(k|k),Pt(k|k)); it k T
with elements ω ∈ {0,1} to indicate whether or not the
• each target t follows its own dynamic model; it
measurementz liesinthevalidationgateoftargett.Because
• each target generates at most one measurement at each i
each measurement can potentially originate from a FA, all
observationtime andthere arenomergedmeasurements;
elementsofthefirstcolumnofΩcorrespondingtoindext=0
• each target is detected with some known detection prob-
(meaning FA, or none of the targets) are equal to one. From
ability Pt;
d this validation matrix, all possible feasible joint association
• thefalsealarms(FA)areuniformlydistributedinsurveil-
events Ωˆ(Θ(k)) = [ωˆ (Θ(k))] where ω (Θ(k)) = 1 if
lance area and their number follows a Poisson pmf with it it
Θt(k)∈Θ(k), and zero otherwise, are realized satisfying the
FA density λ . i
FA following feasibility conditions:
In JPDAF, the measurement to target association proba-
• a measurementcan have only one origin, that is for all i
bilities are computed jointly across the targets and only for
the latest set of measurements. This appealing theoretical
NT
(0-scan-back) approach however can give rise to very high
ωˆ (Θ(k))=1 (1)
combinatorics complexity if there are several persistent inter- X it
t=0
ferences,typically whenseveraltargetsare crossing or if they
movecloselyduringseveralconsecutivescans.Moreoversome • at most one measurement can originate from a target
track coalescence effects may also appear which degrades
mk
substantially the JPDAF performances as it will shown in
ωˆ (Θ(k))≤1, for t=1,...,N (2)
section V. These limitations of JPDAF have already been X it T
i=1
reported in [14]. Here we briefly recall the basics of JPDAF.
For more details, please refer to [1], [2], [10]–[12],[15]. Thegenerationof allpossiblefeasiblejointassociationevents
is computationallyexpensivefor complicated MTT scenarios,
A. JPDAF principle which is a serious limitation of JPDAF for real-world sce-
Let’s consider a cluster1 of T ≥ 2 targets t = 1,...,T. narios. A simple MatlabTM algorithm for the generation of
matricesΩˆ(Θ(k))isgivenin[15](pp.56–57),whichisbased
The set of m measurements available at scan k is denoted
k
Z(k) = {z (k),i = 1,...,m }. Each measurement z (k) of on DFS (Depth First Search) detailed by Zhou in [16], [17],
i k i
Z(k) either originates from a target or from a FA. Denote previously coded in FORTRAN in [18].
ˆzt(k|k − 1) as the predicted measurement for target t, and
all the possible innovationsthat could be used in the Kalman C. Feasible joint association probabilities
Filter to update the target state estimate are denoted z˜t(k) ,
i Thanksto Bayes formula,the computationof the a posteri-
zi(k)−zˆt(k|k−1),i=1,...,mk.InJPDAF,insteadofusing orijointassociationprobabilitiesP{Θ(k)|Zk}involvedinthe
a particular innovation z˜t(k), it uses the weighted innovation
i derivationofβt(k)canbeexpressedas(see[1],[2],[10]–[12],
˜zt(k)= mk βt(k)˜zt(k), where βt(k) is the probabilitythat i
the measPurei=m1enit z (ki) originates firom target t. βt(k) is the [15] for full derivations)
i 0
probability that none measurements originate from the target 1
P{Θ(k)|Zk}= ·p[Z(k)|Θ(k),m ,Zk−1]P{Θ(k)|m }
t. The core of JPDAF is the computation of the a posteriori c k k
patΘasortissgios(eciktbi2)alettiiosj,on0itnhp≤teroabetsvasoeb≤ncilitiaNtitteihosa.ntβMeimtv(okeersn)eu,tsrpierΘm=ec(eik0ns),et1l=yz,,i.(To.k.nmi),e=mk1ohrkΘaisgtibiitan(oskaetc)ed,osmwofnhproeuarmtleel = 1c×· φm(Θmk(kkf!))(!zµF(k()φ)(Θτi((kΘ)()k))V) −φ(Θ(k))
afonrdiβ0=t(ki1),.=..,1mi−k,PβTmiit=(kk1)βit=(k)P, wΘ(hke)rPe{ZΘk(kis)|Zthke}ωˆseitt(Θof(ka)l)l ×iY=T1(cid:2)(Ptti)δti(Θ(k)(cid:3))(1−Pt)1−δt(Θ(k)) (3)
Y d d
1Aclusterisagroupoftargetswhichhavesomemeasurementsincommon t=1
intheirvalidation gates (i.e.non-emptyintersections).
2Byconvention andnotation convenience, ti=0meansthattheoriginof where c is a normalization constant, V is the volume of the
measurementzi isaFA. surveillance region,and the indicators δt(Θ(k)) (targetdetec-
tion indicator),τ (Θ(k)) (measurementassociation indicator), Using (11) and (12) in (10), then
i
φ(Θ(k)) (FA indicator) are defined by
mk
mk xˆt(k|k)=xˆt(k|k−1)+Kt(k)Xβit(k)˜zti(k) (13)
δt(Θ(k)),Xωˆit(Θ(k))≤1 t=1,...,NT (4) i=1
i=1 Pt(k|k)=βt(k)Pt(k|k−1)+ 1−βt(k) Pt(k)+P˜t(k)
0 ! 0 (cid:1) c
T (14)
τ (Θ(k)), ωˆ (Θ(k)) (5)
i X it with
t=1
mk Ptc(k)=[I−Kt(k)H(k)]Pt(k|k−1) (15)
φ(Θ(k)), [1−τ (Θ(k))] (6)
X i mk
i=1 P˜t(k)=Kt(k) βt(k)˜zt(k)˜zt′(k)−˜z(k)˜z′(k) K′(k)
hX i i i i
µF(φ(Θ(k))) is the prior pmf of the number of false mea- i=1
(16)
surements (the clutter model) and
and
fti(zi(k)),N[zi(k);ˆzti(k|k−1),Sti(k)] (7) Kt(k),Pt(k|k−1)H′(k)St(k)−1 (17)
where Sti(k) is the predictedcovariancematrix of innovation ˜zt(k),z (k)−ˆzt(k|k−1) (18)
zi(k)−ˆzti(k|k−1). i mik
Two versions of JPDAF have been proposed [1], [7]–[9]: ˜zt(k), βt(k)˜zt(k) (19)
X i i
• Parametric JPDAF: Knowing the spatial density λFA i=1
of the false measurements, and using a Poisson pmf It has been proved in [1] that P˜(k) is always a semi-positive
µF(φ(Θ(k)))= (λFφAV(k))φ!(k)e−λFAV, results in matrix.Thetargetstatepredictionxˆt(k+1|k)andPt(k+1|k)
are obtained by the classical Kalman Filter (KF) equations
P{Θ(k)|Zk}= 1 · mk[λ−1·f (z (k))]τi(Θ(k)) [1] (assuming linear kinematic models), or by Extended KF
c Y FA ti i equations. They will not be repeated here [2], [10].
1
i=1
In summary, JPDAF is well theoretically founded and it
×NT [Pt]δt(Θ(k))[1−Pt]1−δt(Θ(k)) (8) doesnotrequirehighmemory(0-scan-back).Itprovidespretty
Y d d goodresultsonsimpleMTTscenarios(withnonpersistingin-
t=1
terferences)with moderateFA densities. Howeverthe number
where c is a normalization constant.
1 of feasible joint association matrices increases exponentially
• Non parametric JPDAF: Using a diffuse prior pmf of
with problem dimensions (m and N ) which makes the
number of FA µ (φ(k))=ǫ, ∀φ(k), results in k T
F JPDAF intractable for complex dense MTT scenarios.
P{Θ(k)|Zk}= φ(k)! · mk[Vf (z (k))]τi(Θ) III. QADABASED KALMAN FILTER
c Y ti i
2 The aim of this paperis to comparethe performanceof the
i=1
×NT [Pt]δt(Θ(k))[1−Pt]1−δt(Θ(k)) (9) rJiPthDmA,FubsiansgedthMeTCTMaKlgForbitahsmedwointhktihneemclaatsicsiscaml4eaMsuTreTmaelngtos-,
Y d d
t=1 but improved by the QADA method.
where c is a new normalization constant. ThemainideabehindtheQADAmethod,proposedrecently
2
by Dezert and Benameur[4] is to comparethe valuesa (i,j)
D. JPDAF state estimation 1
in the first optimal DA solution A with the corresponding
1
Once all feasible joint association events Θ(k) have been values a (i,j) in second assignment solution A , and to
2 2
generated and their a posteriori probabilities P{Θ(k)|Zk} identify if there is a change of the optimal pairing (i,j). In
determined,all the marginalassociation probabilitiesβit(k)= theMTTcontext(i,j)meansanassociationbetweenmeasure-
PΘ(k)P{Θ(k)|Zk}ωˆit(Θ(k)) and β0t(k) = 1−Pmi=k1βit(k) ment zj and target Ti. QADA establishes a quality indicator
are computed. The state update and prediction are done with associated with this pairing, depending on the stability of
PDAF equations3 given by the pairing and also, on its relative impact in the global
mk reward.Theproposedmethodworksalsowhenthe1stand2nd
xˆt(k|k)= βt(k)xˆt(k|k) (10) optimalassignmentsA andA are notunique,i.e., thereare
X i i 1 2
i=0 multiplicities available. In such a situation, the establishment
with xˆt(k|k) given by of quality indicators could help in selecting one particular
i
optimalassignmentsolutionamongmultiple possiblechoices.
xˆt (k|k)=xˆt(k|k−1)+Kt(k)˜zt(k) (11)
i>0 i The construction of the quality indicator is based on belief
xˆt (k|k)=xˆt(k|k−1) (12) functions (BF) and the Proportional Conflict Redistribution
i=0
3forthe decoupled version of JPDAF.Forthe coupled version ofJPDAF, 4ClassicalMTTalgorithmsarethosebasedonhardassignmentofachosen
see[10],[12]. measurementtoagiventarget.
fusion rule no.6 (PCR6), defined within Dezert-Smarandache obtain the assignment matrix, utilized in QADA. In this case
Theory (DSmT) [19]. It depends on the type of the pairing the elements of assignment matrix ω(i,j),i = 1,..,m;j =
matching, and it is described in detail in [4]. 1,...,n representthe normalizeddistancesd(i,j),[(z (k)−
j
In [6], the authors discuss and propose the way in which ˆz (k|k − 1))′S−1(k)(z (k) − ˆz (k|k − 1))]1/2 between the
i j i
Kalmanfiltercouldbe improvedinorderto reflecttheknowl- validated measurement z and target T satisfying the con-
j i
edge obtained based on the QADA method. dition d2(i,j) ≤ γ. The distance d(i,j) is computed from
Let’s briefly recall what kind of information is obtained, the measurementz (k) and its predictionˆz (k|k−1) (see [1]
j i
having in hand the quality matrix, derived by QADA, in the for details), and the inverse of the covariance matrix S(k) of
MTT context. It gives knowledgeaboutthe confidenceq(i,j) the innovation, computed by the tracking filter. The threshold
in all pairings (T ,z ),i = 1,..,m;j = 1,..,n, chosen in the γ, for which the probability of given observation to fall in
i j
firstbestassignmentsolution.Thesmallerquality(confidence) the gate is 0.99, could be defined from the table of the Chi-
of hypothesis “z belongstoT ” means, that the particular square distribution with M degreesof freedom and allowable
j i
measurement error covariance R was increased and the filter probability of a valid observation falling outside the gate.
shouldnottrustfullyintheactual(true)measurementz(k+1). In this case the DA problem consists in finding the best
Havingthisconclusioninmind,theauthorspropose,sucha assignment, that minimizes the overall cost.
behaviourofthemeasurementerrorcovariancetobemodelled
by R = R , for every pairing, chosen in the first best B. Assignment matrix based on PDA method
assignmenqt(Tain,zdj)based on the corresponding quality value TheProbabilisticDataAssociation(PDA)method[1]calcu-
obtained. Then, when the Kalman filter gain decreases the lates the association probabilities for validated measurements
truemeasurementz (k+1)istrustedlessintheupdatedstate at a current time moment to the target of interest. PDA
j
estimate xˆ(k+1|k+1). assumesthe followinghypothesesaccordingto each validated
measurement:
IV. BUILDING ASSIGNMENT MATRIX FORQADA • H (k):z (k)isameasurement,originatedfromthetarget
i j
Data Association (DA) is a central problem in the modern of interest, i=1,...,m
MTTsystems[1],[2].Itconsistsinfindingtheglobaloptimal • H (k): no one of the validated measurement originated
0
assignments of targets T ,i=1,...,m to some measurements from the target of interest
i
zj,j = 1,...,n at a given time k by maximizing the overall If N observations fall within the gate of track i, N +1
gaininsuchaway,thatnomorethanonetargetisassignedtoa hypotheses will be formed.
measurement,andreciprocally.Them×nreward(gain/payoff) The probability of H is proportional to p = λN(1 −
0 i0 FA
matrix Ω = [ω(i,j)] is defined by its elements ω(i,j) > 0, P P ), and the probability of H (j = 1,2,..,N) is propor-
g d j
representing the gain of the association of target T with the tional to
i
meTahsuesreemveanltuezsj.are usually homogeneous to the likelihood pij = λ(NF2A−π1)PMg/P2.d·|eS−d22i|j (20)
ratios and could be established in different ways, described p ij
below.TheyprovidetheassignmentmatrixutilizedbyQADA where Pg is the a priori probability that the correct measure-
in order to obtain the quality of pairings (interpreted as a ment is in the validation gate5 [1]; Pd is the target detection
confidence score) belonging to the optimal data assignment probability;λFA is the spatial density of FA. The probabilities
solution based on its consistency (stability) with respectto all pij can be rewritten as [1]
the second best solutions, provided for a chosen algorithm.
b for j =0 (no valid observ.)
QADA assumes the reward matrix is known, regardless b+PNl=1αil
of the manner in which it is obtained by the user. In this pij = (21)
paper we propose two versions of QADA-KF. The first one αij for 1≤j ≤N
utilizestheassignmentmatrixbuiltfromthesinglenormalized b+PNl=1αil

distances, providedby the GlobalNearest Neighbourmethod, where
called QADA-GNN KF method. The second one utilizes b,(1−PgPd)λFA(2π)M/2q|Sij| (22)
the assignment matrix, built from the posterior association
and
p(ProDbAab)imliteiethso,dp,rocvalildeeddQbAyDthAe-PPDroAbaKbFilismtiecthDoadt.a Association αij ,Pd·e−d22ij (23)
TheassignmentmatrixusedinQADAmethodisestablished
A. Assignment matrix based on GNN method
fromallp givenby(21)relatedwithallassociationhypothe-
ij
The GNN method finds and propagates the single most
ses. Thismatrixwill havem rows(wherem is the numberof
likely hypothesis during each scan to update KF. It is a
all targets of interest), and N+1 columnsfor the hypotheses
hard (i.e., binary) decision approach, as compared to the
generated. The (N +1)th column will include the values p
i0
JPDAF which is a soft (i.e., probabilistic) decision approach
associated with H (k).
0
using all validated measurements with their probabilities of
association. GNN method was applied in [6] and [20] to 5Inoursimulations, weusePg =0.99andPd=0.99.
V. SIMULATION SCENARIOS AND RESULTS
The Converted Measurement KF is used in our MTT
algorithm. We assume constant velocity target model. The
processnoisecovariancematrixis:Q=σ2Q ,whereT isthe
υ T
sampling period, σ is the standard deviation of the process
υ
noise and Q is as given in [3]. Here are the results of KDA
T
KF, QADA-GNN KF, QADA-PDA KF, and JPDAF for two
interesting MTT scenarios.
A. Groups of targets simulation scenario
The noise-freegroupsof targetssimulationscenario(Fig.1) Figure1. Noise-free groupsoftargets Scenario.
consists of five air targets movingfrom North-West to South-
East. For the clear explanation of the results, targets are x 104
numbered starting at the beginning with 1st target that has
1
the greater y-coordinate and continuing to 5th target with the 5
smallesty-coordinate.Thethreetargets2nd,3rd,and4thmove 0.5
togetherbetweenthem6.Thestationarysensorislocatedatthe m]
originwith range20000m.ThesamplingperiodisT =5 Y[ 0
scan
sec and the measurement standard deviations are 0.2 deg and −0.5
35 m for azimuth and range respectively. The targets move
−1 FF
with constant velocity V = 100m/sec. The group of three
−1 −0.5 0 0.5 1
targetsinthemiddlei.e.2ndto4thmovewithoutmaneuvering X[m] x 104
keeping azimuth 135 deg from North. It is the main direction
of the group’s movement. The first target starts with azimuth Figure2. Noisedgroupsoftargets Scenario withλFA=16·10−10m−2.
165 deg and moves towards the middle group of rectilinearly
moving targets. When it approachesthe group, it starts a turn
used for track initiation. The evaluation of MTT performance
to the left with −30 deg. Its initial azimuth of 165 deg is
is based on the criteria of Track Purity (TP),Track Life (TL),
decreased by the angle of turn and becomes 135 deg, the
and percentage of miscorrelation (pMC):
maindirection.Thefifthtargetmakessimilarmaneuverbutin
1) TP criteria examines the ratio between the number of
opposite direction- to the right. Its initial azimuth of 105 deg
particular performed (jth observation - ith track) associations
is increased by the turn of 30 deg and becomes 135 deg, and
(incaseofdetectedtarget)overthetotalnumberofallpossible
alsocoincideswiththemaindirection.From21thscanto48th
associations during the tracking scenario, but TP cannot be
scanallthetargetsmoverectilinearlyinparallel.Thedistance
usedwithJPDAFbecauseJPDAFisasoftassignmentmethod.
betweenthemis150m.From48thscan,thefirsttargetmakes
Instead of TP, we define the Probabilistic Purity Index (PPI).
aleftturntoazimuthof105deg,thatmeans−30degreeswith
It considers the measurement that has the highest association
respect to the main direction and starts to go away from the
probabilitycomputedbythe JPDAF and check,(and count)if
middle group. The fifth target makes right turn to azimuth of
this measurement originated from the target or not. PPI mea-
165degthatmeans+30degfromthemaindirectionandalso
sures the ability of JPDAF to commit the highest probability
starts to go away. All maneuvers are with one and the same
tothe correcttargetmeasurementin thesoftassignmentofall
value of the angle (angle= 30 deg by absolute value), the
validated measurements.
same time duration and linear velocity. The absolute value of
2)TLisevaluatedasanaveragenumberofscansbeforetrack’s
thecorrespondingtransversalaccelerationforallmaneuversis
deletion. In our simulations, a track is cancelled and deleted
1.163m/s2. The total number of scans for the simulations is
from the list of tracked tracks, when during 3 consecutive
65.Fig.2showsthenoisedscenarioforλ =16·10−10m−2
FA scans it cannot be updated with some measurement because
yielding to 0.2 FA per gate on average.
thereisnovalidatedmeasurementinthevalidationgate.When
OurresultsarebasedonMonteCarlo(MC)simulationswith
using JPDAF, the track is cancelled and deleted from the
200 independent runs in applying KDA based KF, QADA-
list of tracked tracks, when during 3 consecutive scans its
GNN KF, QADA-PDA KF, and JPDAF7. We compare the
own measurement does not fall in its gate. We call this,
performance of these methods with different criteria, and we
the “cancelling/deletion condition”. The status of the tracked
use an idealized track initiation in order to prevent uncon-
tracks is denoted “alive”.
trolled impact of this stage on the statistical parameters of
3)pMCexaminestherelativenumberofincorrectobservation-
the tracking process during MC simulations. The true targets
to-track associations during the scans.
positions(knowninoursimulations)forthefirsttwoscansare
The MTT performance results for KDA only KF, QADA-
6Notethatthreetargets movetogether inthecenter. GNN KF, QADA-PDA KF, and JPDAF for a low-noise case
7Wehaveusedthenonparametric versionofJPDAFinoursimulations. (0.2 FA per gate on average) are given in Table 1.
(in%) KDA JPDAF QADA-GNN QADA-PDA
Group of Targets Scenario − rmse X−filtered for track 1
AverageTL 50.27 66.46 81.94 90.85 180
sensor error along X
AveragepMC 3.35 2.98 2.10 1.75 160 rmse X−filtered KDA
AverageTP 45.61 PPI=29.14 79.32 87.61 140 rmse X−filtered QADA−GNN
TableI 120 rrmmssee XX−−ffiilltteerreedd QJPADDAAF−PDA
GROUPSOFTARGETSSCENARIO:COMPARISONBETWEENMTT 100 FA in gate=0.2
PERFORMANCERESULTSFOR0.2FAPERGATE. m] 80
[
60
40
20
Accordingtoallcriteria,theQADA-PDAKFmethodshows 0
the best performance, followed by QADA-GNN KF, and −200 10 20 30 40 50 60 70
scans
JPDAF. The KDA based KF approach, as one could expect,
shows the worst performance.Performanceresults for a more Figure3. RMSEonXfortrack1withthefourtracking methods.
noisy scenario with 0.4 FA per gate on average are given in
Table 2. Group of Targets Scenario − rmse Y−filtered for track 1
140
sensor error along Y
(in%) KDA QADA-GNN JPDAF QADA-PDA 120 rrmmssee YY−−ffiilltteerreedd KQDAADA−GNN
AverageTL 43.54 70.51 70.94 84.17 100 rrmmssee YY−−ffiilltteerreedd QJPADDAAF−PDA
AveragepMC 3.90 3.33 2.71 3.11 80 FA in gate=0.2
AverageTP 38.22 66.43 PPI=25.65 78.51 m] 60
TableII [
GROUPSOFTARGETSSCENARIO:COMPARISONBETWEENMTT 40
PERFORMANCERESULTSFOR0.4FAPERGATE. 20
0
−200 10 20 30 40 50 60 70
As we see, the results for 0.4 FA per gate scenario are scans
degraded in comparison to the low-noise case. The average
Figure4. RMSEonYfortrack1withthefourtracking methods.
miscorrelation for QADA-PDA is slightly higher than for
JPDAF, probably because QADA method is based on the
1st and 2nd best solutions only, and more information (i.e. errors obtained by KDA KF, QADA-GNN KF, QADA-PDA
the 3rd best assignment solution) should be used in such KF, which are less than the sensor’s error. The RMSE on X
case to improveQADA performance,which is left for further filteredobtainedwithJPDAFisunderthesensor’serror,beside
research.AccordingtoTLandTP,stillQADA-PDAKFbased KDA KF, QADA-GNN KF, QADA-PDA KF methods.
MTT shows stably better performance than JPDAF.
JPDAFbasedMTToutperformsQADA-GNNKFandKDA Group of Targets Scenario − rmse X−filtered for track 3
120
KF based MTT approachesaccording to the considered crite- sensor error along X
100 rmse X−filtered KDA
ria. In order to make a fair comparison between QADA KF rmse X−filtered QADA−GNN
rmse X−filtered QADA−PDA
and JPDAF, we will discuss also the root mean square errors 80 rmse X−filtered JPDAF
FA in gate=0.2
(RMSE),associatedwiththefilteredXandYvalues,presented m] 60
in Figs. 3–7. Figs. 3 and 4 show the mean square X and Y [ 40
errorfiltered,associatedwithtarget1,andcomparedforKDA 20
KF, QADA-GNN KF, QADA-PDA KF, and JPDAF. Figs. 5 0
and 6 consider the same errors for the middle of track 3. All −200 10 20 30 40 50 60 70
scans
the results are compared to the sensor’s errors along X and
Y axis. We see that the RMSE on X filtered associated with
Figure5. RMSEonXfortrack3withthefourtracking methods.
KDA KF, QADA-GNN KF, and QADA-PDA KF are a little
bit above from the sensor’s error in the region where target
1 makes maneuvers. For scans [20,50], target 1 is moving Groups of Targets Scenario − rmse Y−filtered for track 3
200
in parallel to the group of other targets running rectilinearly FA in gate=0.2
and then these errors are less than respective sensors’s ones. 150
The RMSE on X filtered associated with JPDAF performance
100 sensor error along Y
is three times bigger in the region between scans 20th and m] rrmmssee YY−−ffiilltteerreedd KQDAADA−GNN
[ rmse Y−filtered QADA−PDA
30th where target 1 starts moving in parallel to the rest of 50 rmse Y−filtered JPDAF
rectilinearly moving targets. The RMSE on Y filtered is high
0
duringthewholeregion,wheretarget1movesinparallelway.
TheRMSEonYerrorfilteredbyJPDAFareespeciallycrit- −500 10 20 30 40 50 60 70
scans
ical for the middle track 3 which shows its poor performance
in state estimation on Y direction. The RMSEs are more than
Figure6. RMSEonYfortrack3withthefourtracking methods.
5 times bigger (in the region between scans 20th and 50th,
where all five targets move in parallel) than the respective The large value of RMSE on Y using the JPDAF can be
explained by the specificity of the scenario because it has
1500
five targets moving closely during more than 30 consecutive
1000
scans with sensor’s measurement errors, and false alarms
density, which yields to spatial persisting interferences and 500
trackcoalescenceeffectsinJPDAF, asshowninFig. 7,where m]0
the red and green plots are the tracks estimates. These effects Y[
−500
degrades significantly the quality of JPDAF performance as
already reported in [14]. −1000
−1500
−1500 −1000 −500 X[0m] 500 1000 1500
Figure8. Noise-freeCrossingtargets Scenario.
1500
1000
500
1 F
Y[m] 0 2 F
−500
Figure7. JPDAFtrackcoalescence (onerun)withλFA=16·10−10m−2. −1000
−1500
−1500 −1000 −500 0 500 1000 1500
X[m]
B. Crossing targets simulation scenario
The second considered (crossing targets) scenario (Fig. 8) Figure9. NoisedCrossingtargets ScenariowithλFA=4·10−7m−2.
consists of two maneuvering targets moving with constant
velocity 38m/sec. At the beginning, both targets move from Table3,andtheperformanceresultsforamorenoisyscenario
West to East. The stationary sensor is located at the origin
with 0.4 FA per gate on average are given in Table 4.
with range 1200 m. The sampling period is Tscan = 1sec
and the measurement standard deviations are 0.2 deg and 25 (in%) KDA QADA-GNN JPDAF QADA-PDA
m for azimuth and range respectively. AverageTL 77.06 88.93 91.25 93.47
AveragepMC 2.40 2.24 2.08 2.11
Thefirsttarget,havingatthebeginninggreatery-coordinate, AverageTP 72.78 85.64 PPI=86.29 87.96
movesstraightforwardfromWesttoEast.Betweenthe8thand TableIII
CROSSINGTARGETSSCENARIO:COMPARISONBETWEENMTT
12th scans it makes a 50 deg right turn, and then it moves PERFORMANCERESULTSFORFAINGATE=0.2.
straightforward during 8 scans. From the 20th scan to the
24th scan it makes a 50 deg left turn, and then it moves in
East direction till the 41th scan. It makes a second 50 deg (in%) KDA QADA-GNN JPDAF QADA-PDA
AverageTL 58.80 77.20 82.87 83.18
left turn between 41th and 45th scans, and then it moves
AveragepMC 3.61 3.63 2.94 3.40
straightforward during 8 scans. From 53th scan it makes a AverageTP 52.90 72.01 PPI=76.94 77.15
TableIV
second 50 deg right turn till the 57th scan and then it moves
CROSSINGTARGETSSCENARIO:COMPARISONBETWEENMTT
in East direction.The trajectoryof target1 correspondsto the PERFORMANCERESULTSFORFAINGATE=0.4.
red plot of Fig. 8.
The second target makes a mirrored trajectory correspond-
According to all criteria, the QADA-PDA KF shows again
ingtothegreenplotofFig.8.Fromscan1to8itmovesfrom
the best performance, but now JPDAF based MTT shows
West to East. During 8th to 12th scans it makes a 50 deg left
closed to QADA-PDA KF performance in comparison to the
turn. Then it moves straightforward during 8 scans. During
previous scenario, and exceeds the performance of QADA-
20th to 24th scans it makes a 50 deg right turn and then it
GNN KF. Nevertheless the performances of all methods are
moves in East direction till the 41th scan. It makes a second
deterioratedinmorenoisedcase,whenonehas0.4FAingate
50 deg right turn between the 41th and 45th scans, and then
on average,this tendencyis still kept. JPDAF hasbetter (than
it moves straightforward during 8 scans. From the 53th scan
in the previous scenario) performance, but still QADA-PDA
it makes a second 50 deg left turn till the 57th scan and then
KF exceeds its performance.
it moves in East direction. The total number of scans for the
simulations is 65. Fig. 9 shows the respective noised scenario Figures 10-13 show that the RMS errors associated with
for λ =4·10−7m−2. X and Y filtered are below the sensor’s error. They confirm
FA
TheMTTperformanceresultsobtainedonthebaseofKDA the better performance of JPDAF in this particular scenario
only KF, QADA-GNN KF, QADA-PDA KF, and JPDAF for with only two maneuveringtargets, which is simpler than the
lessnoisedcasecorrespondingto0.2FApergatearegivenin groups of targets scenario.
Crossing Targets Scenario − rmse X−filtered for track 1 based Kalman Filter; (ii) QADA-GNN KF; (iii) QADA-PDA
25
KF; and (iiii) JPDAF. The first scenario (groups of targets)
20 shows the advantages of applying QADA-KF. According to
all performance criteria, the QADA-PDA KF gives the best
15
m] performance, followed by QADA-GNN KF, and JPDAF. The
[10 KDAKFapproachshowstheworstperformance(asexpected).
sensor error along X This scenario is particularly difficult for JPDAF because of
rmse X−filtered KDA
5 FA in gate=0.2 rmse X−filtered QADA−GNN several closely spaced and rectilinearly moving targets in
rmse X−filtered QADA−PDA
00 10 20 30 40 rmse X50−filtered JP60DAF 70 clutter during many consecutive scans, and it leads to track
scans
coalescenceeffectsduetopersistinginterferences.Asaresult,
the tracking performance of JPDAF is degraded. Because the
Figure10. RMSEonXfortrack1withthefourtracking methods.
complexityofthecalculationforjointassociationprobabilities
grows exponentially with the number of targets, JPDAF re-
Crossing Targets Scenario − rmse Y−filtered for track 1
18 sensor error along X quiresalmost3 timesmorecomputationaltime in comparison
16 FA in gate=0.2 rrmmssee XX−−ffiilltteerreedd KQDAADA−GNN toothermethodsinthefirst(complex)scenario.Inthesecond
14 rmse X−filtered QADA−PDA
12 rmse X−filtered JPDAF (onlytwocrossingtargets)MTTscenario,JPDAFshowsbetter
10 trackingperformancesincomparisontoQADA-GNNKF.Itis
[m] 8 abletotrackmorepreciselytheseonlytwotargets,becauseof
6
non persisting interferences. Overall, our analysis shows that
4
2 QADA-PDA KF method is the best of the four approachesto
0 track multiple targets in clutter with a tractable complexity.
−20 10 20 30 40 50 60 70
scans
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VViieeww ppuubblliiccaattiioonn ssttaattss