Marginal multi-Bernoulli filters (extended version)

1.5 MB·

by  Jason L. Williams

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Marginal multi-Bernoulli filters (extended version)
Jason L. Williams
Intelligence, Surveillance and Reconnaissance Division, Defence Science and Technology Organisation, Australia
and School of Electrical and Electronic Engineering, University of Adelaide, Australia
email [email protected]
Abstract—Random finite sets (RFSs) has been a fruitful area Outside of the RFS field, two major approaches are joint
of research in recent years, yielding new approximate methods probabilistic data association (JPDA) [6] and multiple hy-
for multiple target tracking such as the probability hypothesis
pothesis tracking (MHT) [7,8]; both explicitly formulate and
density (PHD), cardinalised PHD (CPHD), and multiple target
reason over association hypotheses, i.e., hypotheses for the
multi-Bernoulli (MeMBer) filters. These new approaches have
3 largelybeenbasedonapproximationsthatside-steptheneedfor correspondence of measurements and targets. JPDA seeks to
1
measurement-to-trackassociationinordertomaintaintractabil- marginaliseovertherandomvariablesrepresentingassociation
0
ity. In this paper, we derive a conjugate prior form for the in order to calculate the marginal distribution of each track;
2
full Bayes RFS filter under commonly invoked assumptions.
jointintegratedPDA(JIPDA)[9,10]andjointintegratedtrack
n We observe that data association is implicitly present in the
splitting (JITS) [10,11] extend this by incorporating target
a form, and obtain alternative algorithms by approximating the
J discretedistributionofjointassociationeventsutilisingarecently existence as an additional random variable to be estimated.

5 developed, high quality approximation of marginal association MHTseekstocalculatethemostlikelyassociationhypothesis,
1 probabilities. The two resulting algorithms maintain a hybrid returning a tracking estimate conditioned on the hypothesis.
MeMBer/Poisson distribution of objects, but form Bernoulli Track oriented MHT (TOMHT) [12] provides a tractable

] components differently to the conventional MeMBer filter. The
Y track-oriented marginal MeMBer/Poisson (TOM-MeMBer/P) is approachforimplementingMHTbymaintainingahypothesis
S shown to be closely related to a variant of the popular joint treeforeachtrack,andfindingthemostlikelyjointhypothesis
integratedprobabilisticdataassociation(JIPDA)algorithm.The indirectly (e.g., using Lagrangian relaxation [13]).
.
s measurement oriented MeMBer/P (MOM-MeMer/P) adopts the While the RFS framework avoids the need for explicitly
c
structureoftheMeMBerfilter(collectingallhypothesesutilising
[ modellingdataassociation,itimplicitlyarisesinthederivation
a given measurement into a single multi-Bernoulli component),
2 butusesmarginalassociationprobabilitiestoprovideasignificant of the full Bayes RFS filter. This paper seeks to apply
v improvement to performance in low probability of detection the high quality, tractable approximation of the marginal
5 cases. We demonstrate both the performance and tractability association probabilities [14,15] to tracking within the RFS
9 of the filters in a challenging scenario involving multiple targets framework, as an alternative to the approximations of PHD,
9 in close proximity. CPHD or MeMBer. Section III presents a derivation of the
2 full Bayes RFS filter under commonly invoked assumptions,
.
3 I. INTRODUCTION demonstrating a conjugate prior form for the filter (i.e., a
0 form that is preserved by prediction and update); preliminary
2 Recently, much work has been devoted to random finite set
versions were available in the pre-prints [16,17]. The form
1 (RFS)-based approximations such as the probability hypoth-
incorporates a PPP component, and a second component that
: esis density (PHD) [1], cardinalised PHD (CPHD) [2] and
v is structurally similar to TOMHT. In Section IV, we observe
i multiple target multi-Bernoulli (MeMBer) filters [3]. A key
X that data association implicitly arises in the derivation, and
characteristic of these methods has been their tractability,
subsequently approximate the discrete distribution of data
r which is the consequence of clever approximations that avoid
a association hypotheses. The first approximation yields the
data association, i.e., reasoning over the correspondence of
track oriented marginal MeMBer/Poisson (TOM-MeMBer/P)
measurements and targets. For example, the PHD approxi-
filter,whichisverysimilartoJITSandJIPDA;thedifferences
mates the posterior distribution as a Poisson point process
relate to the inclusion of a Bayesian model of target birth in
(PPP), for which updates can be calculated without data
our RFS derivation. Thus, with minor changes, the popular
association. The CPHD improves upon this by calculating the
JITS/JIPDAalgorithms(andextensionsthereof)canbederived
cardinalitydistribution,andusinganindependently,identically
within the RFS framework. The second approximation yields
distributed (iid) cluster process approximation for which up-
the measurement oriented marginal MeMBer/Poisson (MOM-
dates can be calculated efficiently; CPHD has been shown to
MeMBer/P) filter, which collects all hypotheses using a given
improve performance significantly over PHD in challenging
measurement into a single Bernoulli component in a manner
environments (e.g., [4]). The MeMBer makes approximations
similartotheMeMBerfilterof[3],butusingthemarginalas-
in its derivation that avoid association and are reasoned to
sociationweights.Inbothcases,themarginalmeasurement-to-
be of little consequence when the false alarm rate is low;
trackassociationprobabilitiesareapproximatedusingthehigh
these approximations introduced a significant cardinality bias
quality, tractable procedure described in [14,15]. Section VI
that was corrected in the cardinality balanced MeMBer (CB-
demonstrates the performance of the proposed methods in a
MeMBer) [5].
challenging scenario, showing that improved performance is
ThispaperisUNCLASSIFIEDandisapprovedforpublicrelease. obtained in comparison to CPHD and MeMBer (particularly
2
in cases with a lower probability of detection) for a similar an alternative representation of a RFS distribution. If X and
computational load. Y are independent RFSs and Z =X∪Y, then [3, p372,386]
(cid:88)
f (Z)= f (X)f (Z−X)
II. BACKGROUND Z X Y
X⊆Z (6)
RFS-based methods [3] have been developed in order to
G [h]=G [h]·G [h]
Z X Y
conductstatisticalinferenceinproblemsinwhichthevariables
of interest and/or the observations form finite sets. The meth- This convolution identity is analogous to the result that the
ods are particularly applicable to tracking since they address pgf of the sum of two independent variables z = x+y is
two of the major challenges involved: G (s)=G (s)·G (s) [18, p153].
z x y
• The number of targets present in the scene is unknown This work utilises two basic forms of RFS distributions:
• Measurements are unordered, and measurement-to-target PPPs and Bernoulli processes. A PPP with intensity function
correspondence is unknown λ(x) has density and p.g.fl: [3, p373]
Problems such as this can be conveniently formulated by (cid:26) (cid:90) (cid:27)
(cid:89)
modelling the system state as a set of target states1 Xt = f(X)=exp − λ(x)dx · λ(x)
{x1t,...,xntt} (at time t), and incorporating set-valued mea- x∈X (7)
surements Zt = {zt1,...,ztmt}. RFS densities encode both G[h]=exp{λ[h−1]}∝exp{λ[h]}
uncertainty in the number of targets, and their states. A RFS
density can be constructed from a cardinality distribution where λ[h] denotes the inner product of λ(·) and h(·):
p(n), n ≥ 0, and a series of cardinality-conditioned joint
(cid:90)
distributions fn(x1,...,xn), yielding λ[h](cid:44) h(x)λ(x)dx (8)
(cid:88)
f({x ,...,x })=p(n) f (x ,...,x ) (1)
1 n n π(1) π(n) (cid:82) (cid:82)
and thus λ[h−1]= h(x)λ(x)dx− λ(x)dx.
π
A Bernoulli process with probability of existence r and
where the sum over π spans the n! permutation functions, en-
existence-conditionedprobabilitydensityfunction(PDF)f(x)
suringthatf(X)ispermutationinvariant.Bayesianestimation
has density and p.g.fl:
canbeperformedinthisframework(conceptually,atleast)by
interleaving prediction and update steps: [3, p484] 
1−r, X =∅

(cid:90)
f(X)= r·f(x), X ={x}
ft|t−1(X)= ft|t−1(X|X(cid:48))ft−1|t−1(X(cid:48))δX(cid:48) (2) 0, otherwise (9)
f(Z |X)f (X)
f (X)= t t|t−1 ∝f(Z |X)f (X) (3) G[h]=1−r+r·f[h]
t|t f (Z ) t t|t−1
t|t−1 t
By (9) and (6), we can write the p.g.fl of a multi-Bernoulli
where ft|t(cid:48)(X) is the RFS density at time t given measure- process, i.e., the process resulting from the union Y =(cid:83) X
ments up to and including time t(cid:48),2 f (X|X(cid:48)) is the RFS i i
t|t−1 of independent Bernoulli processes X :
transitiondensity,f(Z|X)istheRFSmeasurementlikelihood, i
and the set integral is defined as: [3, p361] (cid:89)
G [h]= (1−r +r f [h]) (10)
Y i i i
(cid:90) (cid:88)∞ 1 (cid:90) i
g(X)δX (cid:44)g(∅)+ g({x ,...,x })dx ···dx
n! 1 n 1 n wherer andf (x)aretheexistenceprobabilityandexistence-
n=1 i i
(4) conditioned PDF of the i-th Bernoulli process X .
i
Probabilitygeneratingfunctions(pgfs)arewidelyusedinanal- Assumption 1 describes the dynamics model that we utilise
ysis of queueing systems, etc, as they greatly simplify anal- in this work.
ysis of independent random variables (e.g., [18]). Likewise,
probability generating functionals3 (p.g.fls) simplify analysis Assumption 1. The multiple target state evolves according to
the following time dynamics process:
ofRFSsthatpossessindependencerelationships.Thep.g.flof
a RFS density f is defined as: [3, p371] • Targets arrive at each time according to a non-
(cid:90) homogeneous PPP with birth intensity λb(x), indepen-
G[h]= hXf(X)δX (5) dent of existing targets
• TargetsdepartaccordingtoiidMarkovianprocesses;the
where hX (cid:44) (cid:81) h(x). Just as the Fourier, Laplace and z survival probability in state x is Ps(x)
x∈X
transforms are alternative representations of signals, a p.g.fl is • Target motion follows iid Markovian processes; the
single-target transition PDF is f (x|x(cid:48))
t|t−1
1Weusetheconventionofcapitalletters(e.g.,X)representingsets,and
lower-caselettersrepresentingsinglemembers(e.g.,x). Assumption 1 describes a multiple target dynamics process
2We leave off conditioning on the measurement set history Zt(cid:48) = whichistheunionofaPPP(describingarrivalofnewtargets),
(Z1,...,Zt(cid:48))throughout,asthisisimplicitinthesecondsubscriptft|t(cid:48). and an independent Bernoulli process for each existing target
3Note that, whereas the argument of a pgf G(s) is the scalar s, the
argumentofap.g.flG[h]isthefunctionh(x). conditioning on the prior existence and state, G[h|x(cid:48)] = 1−
3
Ps(x(cid:48))+Ps(x(cid:48))p (x(cid:48)),wherep (x(cid:48))(cid:44)(cid:82) h(x)f (x|x(cid:48))dx. In analogy with (12), the p.g.fl of the measurement likeli-
h h t|t−1
Thus, the multiple target transition p.g.fl is: [3, p474] hood f(Z|X) is [3, p422]
(cid:90) (cid:90)
G[h|X(cid:48)]= hXf (X|X(cid:48))δX (11) G[g|X]= gZf(Z|X)δZ (20)
t|t−1
=exp{λb[h−1]} (cid:89) {1−Ps(x(cid:48))+Ps(x(cid:48))ph(x(cid:48))} (12) =exp{λfa[g−1]} (cid:89){1−Pd(x)+Pd(x)pg(x)} (21)
x(cid:48)∈X(cid:48) x∈X
Taking the p.g.fl of (2), changing the order of integration and where pg(x)(cid:44)(cid:82) g(z)f(z|x)dz. Thus: [3, p531]
substituting (12), we obtain the p.g.fl form of the prediction (cid:90)
equation: [3, p529] F[g,h]= hXG[g|X]f (X)δX (22)
t|t−1
(cid:90)(cid:90) (cid:90)
Gt|t−1[h]= hXft|t−1(X|X(cid:48))δXft−1|t−1(X(cid:48))δX(cid:48) (13) =exp{λfa[g−1]} {h(1−Pd+Pdpg)}Xft|t−1(X)δX
(cid:90) (23)
=exp{λb[h−1]}· {1−Ps+Psp }X(cid:48)f (X(cid:48))δX(cid:48)
h t−1|t−1 =exp{λfa[g−1]}G [h(1−Pd+Pdp )] (24)
t|t−1 g
(14)
Thus, using (15), (19) and (24), a Bayes filter may be derived
=exp{λb[h−1]}·G [1−Ps+Psp ] (15)
t−1|t−1 h directly in p.g.fl form. A key component of the derivation is
the product rule: [3, p395]
In analogy to the similar result for pgfs [18, p150], the RFS
distribution can be recovered from the p.g.fl via the iterated δ (F [h]···F [h])= (cid:88) δF0 [h]··· δFn [h]
functional derivative: [3, p375-376,384] δZ 0 n δW δW
0 n
W0(cid:93)···(cid:93)Wn=Z
δ (cid:12)(cid:12) δ|X| (cid:12)(cid:12) (25)
f(X)= G[h](cid:12) = (cid:81) G[h](cid:12) (16) where the notation (cid:93) denotes that the sum is over all disjoint
δX (cid:12) δx (cid:12)
h=0 x∈X h=0 sets W0,...,Wn such that W0∪···∪Wn =Z. This permits
calculationofthederivativeofaproductofseveralcomponent
where δδxG[h] (cid:44) lim(cid:15)↓0 G[h+(cid:15)δ(cid:15)x]−G[h], and δx(cid:48)(x) is a Dirac p.g.fls (e.g., a multi-Bernoulli p.g.fl).
delta function concentrated at x = x(cid:48).4 Thus we can recover
the p.g.fl of the updated distribution (3) via [3, p530]
III. RANDOMSETFILTERDERIVATION
(cid:90)
G [h]∝ hXf(Z |X)f (X)δX (17) The model of interest was detailed in Assumptions 1 and
t|t t t|t−1
2. The derivation does not change if the birth density, sur-
= δZδt (cid:90)(cid:90) hXgZtf(Zt|X)ft|t−1(X)δXδZt(cid:12)(cid:12)(cid:12)(cid:12)g=0 (18) vmiveaasluprreombaebnitliltiyk,edliehtoecotdioanreprtoimbaebivlaitryy,ifnagls(eea.gla.,rmdedpeennsdiitnyganodn
δ (cid:12)(cid:12) a known, time-varying sensor state); we omit the time index
= δZ F[g,h](cid:12)(cid:12) (19) from these parameters for notational simplicity. The multiple
t g=0
sensor case may be addressed by performing update steps for
where the derivative with respect to Z operates on the each sensor sequentially in between prediction steps.
t
corresponding functional g, and F[g,h] is the joint p.g.fl of Our derivation is based on an induction process, assuming
measurements Z (functional g) and targets X (functional h). a particular form for the full multiple target probability distri-
t
Assumption2describesthemeasurementmodelthatweutilise bution,andshowingthatthisformismaintainedbyprediction
in this work. andupdatesteps.Theformofthefullmulti-targetdistribution
at time t, conditioned on measurements up to and including
Assumption 2. The multiple target measurement process is time t(cid:48) ≤t, consists of two independent components:
as follows:
(cid:88)
• Each target may give rise to at most one measurement; ft|t(cid:48)(X)= ftp|tp(cid:48)p(Y)ftm|tb(cid:48)m(X−Y)
probability of detection in state x is Pd(x) Y⊆X (26)
• Each measurement is the result of at most one target Gt|t(cid:48)[h]=Gpt|pt(cid:48)p[h]·Gmt|tb(cid:48)m[h]
• False alarm measurements arrive according to a non-
homogeneous PPP with intensity λfa(z), independent of where fppp(X) (and the corresponding p.g.fl Gppp[h]) is a
t|t(cid:48) t|t(cid:48)
targets and target-related measurements PPP representing unknown targets,5 i.e., those that have never
• Each target-derived measurement is independent of all been detected, with intensity λut|t(cid:48)(x):
othertargetsandmeasurementsconditionedonitscorre-
(cid:89)
spondingtarget;thesingletargetmeasurementlikelihood fppp(X)∝ λu (x)
t|t(cid:48) t|t(cid:48)
is f(z|x) x∈X (27)
Gppp[h]∝exp{λu [h]}
t|t(cid:48) t|t(cid:48)
4A measure theoretic exploration of this can be found in [19, Section
2.2.5]. 5Adoptingterminologyfrom[7].
4
and fmbm(X) (and the corresponding p.g.fl Gmbm[h]) is a hypothesis ri,ai, and the hypothesis- and existence-
t|t(cid:48) t|t(cid:48) t|t(cid:48)
multi-Bernoulli mixture (MBM), i.e., a linear combination of conditioned PDF fi,ai(x)
multi-Bernoulli distributions: t|t(cid:48)
The global hypothesis weights wa are related to the single-
fmbm({x ,...,x })=(cid:88)wa (cid:89) fi,ai(X ) target hypothesis weights via the expression
t|t(cid:48) 1 n t|t(cid:48) α(i)
α,a i∈Tt|t(cid:48) (28) wa ∝ (cid:89) wi,ai (29)
Gmbm[h]=(cid:88)wa (cid:89) Gi,ai[h] t|t(cid:48)
t|t(cid:48) t|t(cid:48) i∈Tt|t(cid:48)
a i∈Tt|t(cid:48) where a = (a1,...,ant|t(cid:48)). As opposed to the TOMHT,
where the sum over α spans all functions α : Tt|t(cid:48) → each global hypothesis incorporates a single-target hypothesis
{0,...,n} such that {1,...,n} ⊆ α(Tt|t(cid:48)) and if α(i) > for every track. Since the hypothesis-conditioned distribution
0, i (cid:54)= j then α(i) (cid:54)= α(j) (i.e., permutation-like functions fi,ai(X) is Bernoulli, the probability of existence under
t|t(cid:48)
that map exactly one track index onto each k ∈ {1,...,n}, different single-target hypotheses will vary, thus through the
and the remaining track indices onto 0), and multi-Bernoulli distribution, the fixed set of n Bernoulli
t|t(cid:48)
(cid:40) tracks can represent different numbers of targets.
∅, α(i)=0
X (cid:44) In (28), the sum over a spans all valid global association
α(i)
{xα(i)}, α(i)>0 hypotheses, i.e., all hypotheses that associate each measure-
ment in the history Mt(cid:48) with exactly one track. Ultimately,
We refer to the multi-Bernoulli components in (28) as tracks
wewillseektoavoidexplicitlyenumeratingglobalassociation
since they will tend to collect measurements corresponding to
hypotheses. To permit this possibility, we replace this with
a single target (or Bernoulli tracks to emphasise the inclusion
a sum that spans the Cartesian product of the single-target
ofaprobabilityofexistence).Thesetoftracksattimetusing
hypothesis sets for each track, i.e.,
information up to t(cid:48) ≤t is denoted by T ={1,...,n }.
t|t(cid:48) t|t(cid:48)
(cid:89)
While measurement-to-target association is not explicitly a∈ Hi ={(a1,...,ant|t(cid:48))|ai ∈Hi }
t|t(cid:48) t|t(cid:48)
introduced in the derivation, we will find that it arises im-
i∈Tt|t(cid:48)
plicitly, and each term a in the sum in (28) turns out to
The sum over this set can be made equivalent to (28) by
correspond to a global association history hypothesis, i.e.,
modifying (29) to
one possible grouping of all past and present measurements
into subsets, where each subset is hypothesised to correspond wa ∝(cid:15) (a) (cid:89) wi,ai (30)
to a particular target. The source of the association events t(cid:48) t|t(cid:48)
in the derivation is the sum over all decompositions of the i∈Tt|t(cid:48)
measurements Zt into the disjoint subsets Wi assigned to where (cid:15)t(cid:48)(a)=1 for any valid global association hypothesis,
prior tracks, which appears in the product rule of (25). As and (cid:15)t(cid:48)(a)=0 otherwise, e.g.,
with TOMHT [12], we decompose these global association 0, ∃i(cid:54)=i(cid:48) s.t. Mt(cid:48)(i,ai)∩Mt(cid:48)(i(cid:48),ai(cid:48))(cid:54)=∅,
hypotheses into single-target hypotheses, a=(a1,...,ant|t(cid:48)); 
each single-target hypothesis is re-used across many global (cid:15)t(cid:48)(a)= or (cid:83)iMt(cid:48)(i,ai)(cid:54)=Mt(cid:48)
hypotheses. As illustrated in Fig. 1, the structure incorporates 1, otherwise
atreeforeachtrack,wheretheleavesofeachtreecorrespond (31)
tothesingle-targethypothesesforthattrack.Thesetofsingle- Note also that many of our expressions (e.g., (27), (30))
target hypotheses for a given track i ∈ T is denoted claim proportionality but not equality. This unambiguously
t|t(cid:48)
Hi ={1,...,hi }.Denotingthesetofallpastandpresent characterises the posterior distribution through the knowledge
t|t(cid:48) t|t(cid:48)
measurements indices by6 Mt(cid:48), each single-target hypothesis thatthedensity(andeachofitsindependentcomponents)must
ai ∈Hi comprises: integrate to unity, or, equivalently, that Gt|t(cid:48)[1] = 1. This is
t|t(cid:48)
a simple matter for the PPP component, but it does not need
• the history of measurement indices7 that are hypoth- to be performed since we do not extract estimates from this
esised to correspond to the target under the hypothe-
component (since outputting an estimate on a target that has
sis, Mt(cid:48)(i,ai) ⊆ Mt(cid:48) (e.g., the measurement indices
never been detected would be nonsensical). Normalisation of
collected tracing from the root of the tree to the leaf theMBMcomponentisachievedbyensuringthat(cid:80) wa =1,
corresponding to the hypothesis in Fig. 1) a
wherewafollows(30).Inpractice,thisisaccountedforwithin
• the hypothesis weight wti|,ta(cid:48)i the approximate inference method outlined in Section IV-C.
• the hypothesis-conditioned Bernoulli multi-target distri-
bution fi,ai(X) (with p.g.fl Gi,ai[h]) of the form (9),
t|t(cid:48) t|t(cid:48) A. Prediction step
parameterised by the probability of existence under the
Theorem 1 derives the prediction step by substituting (26)
6The elements of Mt(cid:48) are of the form (t,j), where j ∈ {1,...,mt} into (15). The theorem confirms what one would intuitively
isanindexofameasurementinscant≤t(cid:48). expect: that the prediction of the PPP component (32) follows
7e.g., if Mt(cid:48)(i,ai) = {(1,3),(2,5),(4,7)} at time t(cid:48) = 4, then thestandardPHDpredictionstepof[1,20],andthatthemulti-
ai hypothesises that the target corresponding to track i is associated with
Bernoulli tracks are predicted independently in an equivalent
measurement 3 at time 1, measurement 5 at time 2 and measurement 7 at
time4(implyingthatamisseddetectionoccurredattime3). mannertotheMeMBer[3,21](excludingbirthofnewtargets,
5
since we utilise a PPP birth model as opposed to a multi- Time 1 (two measurements)
Bernoulli birth model). Track 1 (new) Track 2 (new) n.e. = non-existent
m = missed detection
Theorem 1. Assume that the distribution from the previous (t, j) = time t measurement j
n.e. (1,1) n.e. (1,2)
time step G [h] is of the form given in (26), (27), (28),
t−1|t−1
Time 2 (two measurements)
(30).ThenthepredicteddistributionforthenextstepG [h]
t|t−1 Track 1 Track 2
is of the same form, with:
(1,1) (1,2)
(cid:90) Track 3 (new) Track 4 (new)
λu (x)=λb(x)+ f (x|x(cid:48))Ps(x(cid:48))λu (x(cid:48))dx(cid:48)
t|t−1 t|t−1 t−1|t−1
n.e. m (2,1) (2,2) n.e. m (2,1) (2,2) n.e. (2,1) n.e. (2,2)
(32) example of a valid global hypothesis
n =n ; hi =hi ∀i (33)
t|t−1 t−1|t−1 t|t−1 t−1|t−1 Fig.1. MBMstructuremaintainedbyfilter.Structureaftertime1isshownat
wi,ai =wi,ai ∀i,ai (34) thetop(assumingthatthereweretwomeasurements).Anewtrackiscreated
t|t−1 t−1|t−1 foreachmeasurement;thisresultsfromupdatingthePPPofunknowntargets
(cid:90) with each measurement. The new tracks each contain two hypotheses; one
ri,ai =ri,ai Ps(x(cid:48))fi,ai (x(cid:48))dx(cid:48) ∀i,ai (35) hypothesisingthatthemeasurementgoeswithapreviouslyexistingtrackand
t|t−1 t−1|t−1 t−1|t−1
hence that the new track is not required (following eq (48)), and the other
(cid:82) f (x|x(cid:48))Ps(x(cid:48))fi,ai (x(cid:48))dx(cid:48) hypothesisingthatthemeasurementgoeswiththenewtrack,capturingboth
fi,ai (x)= t|t−1 t−1|t−1 ∀i,ai the possibility that it is the result of a target detected for the first time, or
t|t−1 (cid:82) Ps(x(cid:48))fi,ai (x(cid:48))dx(cid:48) a false alarm (following eq (49)-(52)). The structure after time 2 is shown
t−1|t−1 at the bottom (assuming that there were also two measurements in the new
(36) scan). Each track from the prior distribution is continued, incorporating a
hypothesis for each prior hypothesis (marked with “m”, corresponding to a
The proof of the theorem is in Appendix A. misseddetection,followingeq(39)-(42)),andforeachcombinationofaprior
hypothesis and a new measurement (marked with “(t,j)”, corresponding to
B. Measurement update step thepriorhypothesisbeingupdatedwiththemeasurement,followingeq(43)-
(46)).Non-existencehypothesesarecontinuedwithoutbranching.Abranching
Theorem 2 derives the measurement update step by sub- structure similar to TOMHT results. New tracks are created for each new
stituting (24) and (26) into (19). In Lemma 1, we prove the measurement(markedwith“new”),followingthestructuredescribedforthe
firsttimestep;again,thesecorrespondtoupdatingthePPPofunknowntargets
preliminary result that the posterior remains the union of an
with each new measurement. The gray line shows an example of a global
independent PPP and a second distribution which turns out to hypothesis–i.e., a choice of one single-target hypothesis from the tree for
be a MBM. The PPP intensity update equation is (37); this is eachtrack,inwhicheverymeasurementineveryscanisusedexactlyonce.
Eachterminthesumin(28)correspondstoaglobalhypothesis.
identical to a PHD update with no measurements [1,20]. The
MBM update is illustrated and described in Fig. 1.
the previous hypothesis a˜i, updated with measurement zj):8
t
Theorem 2. Assume that the predicted distribution G [h]
t|t−1
isoftheformgivenin(26),(27),(28),(30).Thentheupdated Mt(i,ai)=Mt−1(i,a˜i)∪{(t,j)} (43)
distribution Gt|t[h] (updated with the measurement set Zt = wi,ai =wi,a˜i ri,a˜i fi,a˜i [f(zj|·)Pd] (44)
{z1,...,zmt}) is of the same form, with n =n +m , t|t t|t−1 t|t−1 t|t−1 t
t t t|t t|t−1 t
ri,ai =1 (45)
λu (x)={1−Pd(x)}λu (x) (37) t|t
t|t t|t−1
Mt =Mt−1∪(cid:8)(t,j)|j ∈{1,...,mt}(cid:9) (38) fi,ai(x)= f(ztj|x)Pd(x)fti|,ta˜−i1(x) (46)
t|t fi,a˜i [f(zj|·)Pd]
For tracks continuing from previous time steps (i ∈ t|t−1 t
{1,...,n }), a hypothesis is included for each combi-
t|t−1 Finally, for new tracks, i ∈ {n +j}, j ∈ {1,...,m }
nation of a hypothesis from a previous time, and either a t|t−1 t
(i.e., the new track commencing on measurement zj),
missed detection, or an update using one of the m new t
t
measurements, such that the number of hypotheses becomes hi =2 (47)
hi = hi (1 + m ). For missed detection hypotheses t|t
t|t t|t−1 t Mt(i,1)=∅, wi,1 =1, ri,1 =0 (48)
(i∈{1,...,n }, ai ∈{1,...,h }): t|t t|t
t|t−1 t|t−1
Mt(i,2)={(t,j)} (49)
Mt(i,ai)=Mt−1(i,ai) (39)
wi,2 =λfa(zj)+λu [f(zj|·)Pd] (50)
wi,ai =wi,ai (1−ri,ai +ri,ai fi,ai [1−Pd]) (40) t|t t t|t−1 t
t|t t|t−1 t|t−1 t|t−1 t|t−1 λu [f(zj|·)Pd]
ri,ai fi,ai [1−Pd] ri,2 = t|t−1 t (51)
ri,ai = t|t−1 t|t−1 (41) t|t λfa(ztj)+λut|t−1[f(ztj|·)Pd]
t|t 1−ri,ai +ri,ai fi,ai [1−Pd]
t|t−1 t|t−1 t|t−1 f(zj|x)Pd(x)λu (x)
fi,ai(x)= {1−Pd(x)}fti|,ta−i1(x) (42) fti|,t2(x)= λtut|t−1[f(ztj|·t)|Pt−d1] (52)
t|t fi,ai [1−Pd]
t|t−1
8Ahypothesisattheprevioustimewithri,ai =0neednotbeupdated
For hypotheses updating existing tracks (i ∈ {1,...,n }, t|t−1
t|t−1 sincetheposteriorweightin(44)wouldbezero.Forsimplicity,thehypothesis
ai =a˜i+hit|t−1j, a˜i ∈{1,...,hit|t−1}, j ∈{1,...,mt}, i.e., numberingdoesnotaccountforthisexclusion.
6
Note that we have not specified the distribution fi,1(x) IV. MARGINALASSOCIATIONFILTERS
t|t
for new target tracks. Since ri,1 = 0 (i.e., the probability of The previous section derived a conjugate prior form for the
t|t
existence is zero), the value of this distribution has no effect. full Bayes RFS filter. Not surprisingly, the filter is intractable,
In Lemma 1, we prove the first piece of Theorem 2, i.e., the andapproximationsmustbemade.Notingitssimilaritytothe
structuralresultthatG [h]remainsaunionofanindependent PHD,thePPPcomponentoftheposterioristractable(e.g.,see
t|t
PPP and some other distribution (which will subsequently be [23]).Thedifficultyisthenumberofglobalassociationhistory
shown to be a MBM). hypotheses, each of which gives rise to a term in the sum in
the MBM (28). To motivate our approximation, consider the
Lemma 1. The updated distribution G [h] has the form
t|t definition of a probability of joint association history:
G [h]=Gppp[h]·Gmbm[h] (53) P (a)=wa ∝(cid:15) (a) (cid:89) wi,ai (56)
t|t t|t t|t t|t t t|t
i∈Tt|t
where
following which (28) can be equivalently written as a total
Gppp[h]∝exp{λu [h]} (54) probability expansion
Gmtb|tm[h]∝ δ (cid:8)et|xtp{λfa[g]+λu [hPdp ]}· ftm|tbm(X)=(cid:88)Pt|t(a) (cid:89) fti|,tai(Xα(i)) (57)
t|t δZt t|t−1 g α,a i∈Tt|t
(cid:12)
·Gmt|tb−m1[h(1−Pd+Pdpg)](cid:9)(cid:12)(cid:12)(cid:12) (55) Twhheichfoirnmtu(r2n8s)ugwgaesstschmoestehnodtsothealutcmidaakteeatphpisroxeiqmuaivtiaolennscteo,
g=0
the discrete distribution P (a). The methods in this section
t|t
and λu (x) is given in (37). approximate the posterior distribution fmbm(X) by directly
t|t t|t
approximating the discrete probability distribution of global
The proof of Lemma 1 is in Appendix B. Lemma 1 shows
associationhistoryhypothesesP (a).Hereinwewillassume
that, in the updated distribution, the portion of the component t|t
thatthepriordistributionfmbm (X)ismulti-Bernoulli,such
Gppp [h] that is not detected is independent (in the posterior) t−1|t−1
t|t−1 that the predicted distribution fmbm(X) is multi-Bernoulli,
of the remainder of the distribution (which comprises the t|t−1
portion of Gppp [h] that is detected, and both detected and and our task is to gain an approximation of ftm|tbm(X) that is
t|t−1
multi-Bernoulli (since the true distribution is not). A distri-
undetected portions of the component Gmbm[h]). Note that
t|t−1 bution in the form (28) will be multi-Bernoulli if there is a
this is not claiming that the entire undetected portion is
single association hypothesis, e.g.,
independentofthedetectedportion.TheproofofTheorem2is
inAppendixB.Theupdatesin(42),(46)and(52)arestandard Mt−1=∅, hi =1∀i,
t|t−1
(58)
single target measurement updates, and thus can be calcu- Mt−1(i,ai)=∅∀i,ai, wi,ai=1∀i,ai
t|t
lated with well-known methods (e.g., [22]). The derivation
In this case, we refer to the weight, existence probability and
incorporates hypotheses updating every prior hypothesis with
PDF for track i as wi =1, ri and fi (x) respectively.
every measurement; however, in practical implementations, t|t t|t t|t
gating can be used to reduce the computational burden by
A. Track-oriented marginal MeMBer-Poisson filter
excluding hypotheses with negligible weights. Other standard
Suppose we seek an approximate representation of the dis-
approximations such as clustering, mixture reduction, etc are
tribution P (a) in which the hypotheses for each track ai
also required in practice. t|t
comprising the global hypothesis a = (a1,...,ant|t) are
forced to be independent, i.e., a distribution of the form
C. Initialisation P (a)≈ (cid:89) qi(ai) (59)
t|t
One significant benefit of the inclusion of a Poisson com- i∈Tt|t
ponent is in initialisation of the tracker. Generally, when Then with no further approximation, the joint multi-target
a surveillance system is first activated, the fact that prior distribution in (57) becomes:
measurements are not available does not imply that no targets (cid:88) (cid:89)
fmbm(X)≈ fi (X )
are present. The Poisson distribution provides a convenient t|t t|t α(i)
mechanism for specifying a prior distribution on the number α i∈Tt|t
and position of targets when little information is available. fi (X)= (cid:88) qi(ai)fi,ai(X) (60)
t|t t|t
By initialising n0|0 = 0 (i.e., no MBM components), and ai∈Hi
t|t
setting λu (x) = λuf (x) where λu is the expected number
0|0 0 0 0 which is multi-Bernoulli. The problem of finding the choice
of targets present, and f (x) is the (presumably diffuse)
0 of qi(ai) which best fits the assumed form of the distribu-
prior state distribution, the Bayes prediction and update steps
tion can be solved by minimising the Kullback-Leibler (KL)
incorporatethisknowledge,adjustingtheaccrualofconfidence
divergence:
dsinpeemtehodenesoxtfriastttreeadnccktehaipntritothibaiastiboainpliptiirenosalcrohtiw|,taciasnaigccncooanlr-dstiiodn-egnrloayib.sleIynri[am2ti3po]r,o(viSteNwtRhaes) (cid:2)qi∗(ai)(cid:3)i∈Tt|t =qia(arig),mi∈iTnt|tD(cid:18)Pt|t(a)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)i∈(cid:89)Tt|tqi(ai)(cid:19)
environments. (61)
7
where D(p||q) = (cid:80) p(a)logp(a). It is shown in [24, p277] (48), (51) and (60), the posterior probability of existence of
a q(a)
that qi∗(ai) = Pti|t(ai) = (cid:80)a˜|a˜i=aiPt|t(a˜), i.e., the marginal a track started on a measurement ztj is (i = nt|t−1 + j,
association distribution for track i. j ∈{1,...,mt})
Accordingly, the data carried forward to the next time
Pi (2)Pdλu [f(zj|·)]
for the MBM component are simply the single hypothesis ri = t|t t|t−1 t (64)
Bernoullidistributionsfi (X)(parameterisedbyanexistence t|t λfa(zj)+Pdλu [f(zj|·)]
t|t t t|t−1 t
probability,andapositiondistributionconsistingofaweighted
This is exactly [10, p325, (9.13)], with ρ (z) =
mixture of the old hypothesis-conditioned distributions). We target
Pdλu [f(z|·)]. Thus, track initiation in the variant of
refer to this as the track oriented marginal MeMBer/Poisson t|t−1
filter(TOM-MeMBer/P),asitformstracksoutofthemarginal JITS/JIPDA modelling a birth intensity ρtarget(z) can be seen
track-to-measurement association distributions. The approxi- to be equivalent to TOM-MeMBer/P assuming a stationary
mation made in the TOM-MeMBer/P is exactly that made in intensity of unknown targets.10 The benefit of dynamically
JPDA: approximating the joint association distribution by the estimating this time-varying quantity was demonstrated in
product of its marginals.9 Whereas JPDA assumes target ex- [23]. Accordingly, we have shown that after two minor mod-
istence, applying the marginal approximation within the RFS ifications and the approximation of (60), JITS/JIPDA may be
framework yields an algorithm that naturally captures target derived (and extended) using the RFS formalism. Similar al-
existence.Inwhatfollows,weshowthattheTOM-MeMBer/P gorithmshavealsobeenproposedwithintheRFScommunity;
is equivalent to a particular variant of JITS/JIPDA9 with one for example, [25] updates existing tracks in a similar manner,
minor modification. The specific variant is JITS/JIPDA with but uses a two-point differencing approach to calculate initial
a parametric clutter model, and using the one-point track states, weighting the new target hypotheses using a constant
initialisationmethodincorporatinganewobjectspatialdensity density of new targets (motivated as a truncated Poisson); and
[10,p324].WeassumethatPd(x)=PdandPs(x)=Ps(i.e., [26] considers a bank of joint target-detection and tracking
that both of these parameters are spatially constant). From (JoTT) filters interacting through modified weights.
(56), (40), (44), (50) and (58), the probability of a global
association hypothesis a is:
B. Measurement oriented marginal MeMBer-Poisson filter
P (a)∝(cid:15) (a)n(cid:89)t|twi,ai The most striking difference between conventional filters
t|t t t|t and the MeMBer filter [3,5] is that the MeMBer collects
i=1
(cid:89) (cid:110) (cid:111) all hypotheses corresponding to a particular measurement in
=(cid:15)t(a)· 1−rti|t−1Pd · the most recent scan into one track (i.e., multi-Bernoulli
i∈{1,...,nt|t−1}|ai=0 component), splitting hypotheses from prior tracks among
· (cid:89) (cid:110)ri Pdfi [f(zai|·)](cid:111)· the new tracks. This is the opposite of the TOM-MeMBer/P
t|t−1 t|t−1 t (and most conventional trackers), which favours continuity
i∈{1,...,nt|t−1}|ai>0 of historical components over the constraints relating to the
(cid:89) (cid:110) (cid:111)
· λfa(zj)+Pdλu [f(zj|·)] most recent measurement scan. An alternative MeMBer fil-
t t|t−1 t
j∈{1,...,mt}|ant|t−1+j=2 ter can be obtained by using a different marginalisation of
(62) Pt|t(a) that follows this philosophy. We refer to the result as
the measurement oriented marginal MeMBer/Poisson (MOM-
Since, under any hypothesis, a measurement will either be
MeMBer/P)filter.Forconsistency,wecontinuetousetheterm
explainedasbelongingtoanexistingtrack(∃i s.t. ai =j),or
track to refer to multi-Bernoulli components although, unlike
anewtrack(ant|t−1+j =2)(whichincorporatesthepossibility
the TOM-MeMBer/P, there is no correspondence of tracks
of it being a false alarm), we can divide (62) by the constant
(cid:110) (cid:111) between time steps.
(cid:81)mt λfa(zj)+Pdλu [f(zj|·)] to obtain:
j=1 t t|t−1 t The MOM-MeMBer/P is based on an alternative param-
eterisation of association hypotheses.11 Rather than the pa-
P (a)∝(cid:15) (a)·
t|t t rameterisation a=(a1,...,ant|t) where ai indexes the mea-
(cid:89) (cid:110) (cid:111)
· 1−ri Pd · surement(s) associated with track i, we formulate hypotheses
t|t−1
i∈{1,...,nt|t−1}|ai=0 as b = (b1,...,bmt) where bj indexes the track associated
(cid:40) ri Pdfi [f(zai|·)] (cid:41) with measurement j, e.g., bj = i indicates that measurement
· (cid:89) t|t−1 t|t−1 t j is associated with track i. Under the assumption in (58),
i∈{1,...,nt|t−1}|ai>0 λfa(ztai)+Pdλut|t−1[f(ztai|·)] b is an equivalent parameterisation of the global association
(63) hypotheses. However, it gives rise to an alternative approx-
imation of the distribution. Specifically, rather than seeking
which differs from [10, p179] only in the inclusion of the
the best-fitting distribution that enforces independence in the
term Pdλu [f(zai|·)] in the denominator. Thus it is seen
t|t−1 t association variables for different tracks (i.e., (61)), we seek
that JITS/JIPDA neglects the influence of unknown targets
in the calculation of probabilities of association events. From 10Wealsoincorporatethekinematicdistributionoftargetsintotheinitial
updatein(52);thisdistributionisassumedtobeuniformin[10].
9JPDA/JIPDAadditionallyapproximatestheposteriordistributionofeach 11For clarity of explanation, we initially omit missed detections, new
targetasbeingGaussian. targetsandfalsealarms.
8
(cid:82) f(zj|x)fi(x)dx. The exact posterior for two targets is
t
(a) fi (xi), i∈{1,2} (b) fi (X) shownin(d);thisisthesymmetrisedversionofthedistribution
t|t−1 t t|t−1 t
0.25 10
0.2 w1,1w2,2f1(x1t|zt1)f2(x2t|zt2)+w1,2w2,1f1(x1t|zt2)f2(x2t|zt1)
5
0.15
The posterior approximation used by the TOM-MeMBer/P
2xt
0.1 0 (andJIPDA,etc)isshownin(e);thisisasymmetrisedversion
0.05 of the distribution
0 −5
−5 0 xti; ztj 5 10 −5 0 xt1 5 10 (cid:2)p1,1f1(x1t|zt1)+p1,2f1(x1t|zt2)(cid:3)·
(c) fit|t−1(xti|ztj), i∈{1,2}, j∈{1,2} (d) ft|t(Xt) (cid:2)p2,1f2(x2|z1)+p2,2f2(x2|z2)(cid:3)
0.1 10 t t t t
0.08 where pi,j = Pi (ai = (t,j)) = Pj (bj = i) is the marginal
5 t|t t|t
0.06 probability that measurement j is associated with track i. The
2xt
0.04 previous section showed this to be the best approximation in
0
0.02 which the association variables for different tracks are forced
0 −5 to be independent. In (67), we will see that the approximation
−5 0 5 10 −5 0 5 10
xi x1 obtained by the measurement-oriented marginalisation in (65)
t t
(e) TOM approx of f (X) (f) MOM approx of f (X) is a symmetrisation of
t|t t t|t t
10 10
(cid:2)p1,1f1(x1|z1)+p2,1f2(x1|z1)(cid:3)·
t t t t
5 5 (cid:2)p1,2f1(x2|z2)+p2,2f2(x2|z2)(cid:3)
2xt 2xt t t t t
0 0
as illustrated in (f). In this instance, the MOM-MeMBer/P
approximation clearly provides a better representation of the
−5 −5
−5 0 5 10 −5 0 5 10 true distribution.
x1 x1
t t Before proceeding, we expand our notation to admit new
targets, false alarms and missed detections. Since we have
assumed that the prior distribution was multi-Bernoulli (58),
Fig.2. Scenarioinvolvingtwotargetsinonedimension.Priormulti-Bernoulli posterior hypotheses will be either Mt(i,ai) = ∅ (the target
components are shown in (a) (solid is i = 1, dashed is i = 2), along with
measurements(“×”;leftisj=1,rightisj=2),Thetwo-targetjointprior was missed) or Mt(i,ai) = {(t,j)} (measurement ztj corre-
distribution is shown in (b). Result of updating each component with each spondstotargeti).Thissetofhypothesescanbecharacterised
measurement is shown in (c) (blue is j = 1, green is j = 2). Joint two- equivalently via a set of hypotheses b = (b1,...,bmt+nt|t−1)
targetposteriorisshownin(d).MeMBerapproximationscarriedforwardby
TOM-MeMBer/PandMOM-MeMBer/Pareshownin(e)and(f)respectively. where for j ∈ {1,...,mt}, bj = 0 if measurement j is
hypothesisedtonotcorrespondtoapreviouslydetectedtarget,
andbj =iifmeasurementj correspondstopre-existingtarget
the best-fitting distribution that enforces independence in the i, and for j = mt +i, i ∈ {1,...,nt|t−1}, bmt+i ∈ {0,i},
association variables for different measurements: where bmt+i = i if track i is missed (and 0 otherwise). The
(cid:89) global hypotheses a and b are alternative parameterisations of
P (b)≈ qj∗(bj)
t|t the same event. Thus (57) can be written equivalently as
j
(cid:2)qj∗(bj)(cid:3)j =arg minD(cid:18)Pt|t(b)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:89)qj(bj)(cid:19) (65) ftm|tbm(X)=(cid:88)Pt|t(b)mt+(cid:89)nt|t−1ftβ|t(bj,j)(Xα(j))
qj(bj) j α,b j=1
(cid:80)where, agPain(˜bf)olilsowthiengma[2rg4i,napl2p7r7o]b,aqbjil∗i(tybjt)ha=t mPeatjs|tu(rbejm)e=nt Pt|t(b)(cid:44)(cid:15)¯nt|t−1(b)mt+(cid:89)nt|t−1wtβ|(tbj,j)
˜b|˜bj=bj t|t j=1
j is associated with track i.
where, in analogy with (31),
As an example, consider the case in Fig. 2, involving two
targets and two measurements, neglecting the possibilities of 0, ∃j (cid:54)=j(cid:48) s.t. bj =bj(cid:48) >0

new targets or missed detections. The prior distribution for (cid:15)¯ (b)= or ∃i∈{1,...,n} s.t. bj (cid:54)=i∀j
n
eachBernoullicomponentisillustratedin(a).Thecorrespond- 1, otherwise
ing two-target joint distribution is shown in (b); this is the
symmetrisation of the distribution f1(x1)f2(x2) (leaving off and β(bj,j) maps to the hypothesis and track corresponding
t t
conditioning on prior measurements for simplicity), i.e., to bj:
f1(x1)f2(x2)+f2(x1)f1(x2) • if bj = i ∈ {1,...,nt|t−1} for j ∈ {1,...,mt} then
t t t t β(bj,j) maps to the hypothesis in (43)-(46) updating
The hypothesis-conditioned updated distributions for each track i with measurement j
prior marginal, updated with each measurement are shown • if bj = 0 for j ∈ {1,...,mt} then β(bj,j) maps to the
in (c), weighted by the association likelihood wi,j = new target hypothesis in (49)-(52)
9
• ifbj =iforj =mt+ithenβ(bj,j)mapstothemissed Continuing
detection hypothesis in (39)-(42) Marginal nt! tracks! New tracks!
• if bj =0 for j =mt+i then β(bj,j) maps to the “non- pmreoabsaubrielmitye notf/ xiste
e
existence” hypothesis, i.e., (48) track corres- on- 1! 2! 3! 4! 5!
pondence! N
Subsequently,weobtainanalternativeapproximationbasedon
the representation of P (b) via the product of its marginals:
t|t
Non-existent! 0! 0! 0! 0.8! 0.6! 0.1!
Pt|t(b)≈mt+(cid:89)nt−1Ptj|t(bj), Ptj|t(bj)= (cid:88) Pt|t(˜b) (66) ents! 1! 0! 0.6! 0.2! 0.2! 0! 0!
m
j=1 ˜b|˜bj=bj re 2! 0! 0.2! 0.4! 0! 0.4! 0!
u
s
Again, with no further approximation, the joint distribution is ea 3! 0! 0! 0.1! 0! 0! 0.9!
M
multi-Bernoulli: d on! 4! 0.8! 0.2! 0! 0! 0! 0!
ftm|tbm(X)≈(cid:88)mt+(cid:89)nt−1ftj|t(Xα(j)) Missedetecti 5! 0.7! 0! 0.3! 0! 0! 0!
α j=1 (67)
fj (X)=(cid:88)Pj (bj)fβ(j,bj)(X) Fanigd.B3.ernEouxlalmicpolemopfonmeanrtgsinfoarlmdeisdtrbibyutTioOnMo-fMtreaMckB-mere/Pas(ucroelmleecntitnagsscoocliuamtionns),
t|t t|t t|t
bj and MOM-MeMBer/P (collecting rows). Each cell of the table contains
the marginal probability that a given measurement is associated with a
We have thus obtained an alternative form of update for a given track (which sums over the same global association hypotheses as
filterthatisstructurallysimilartotheMeMBer[3,5].Theonly the probability that the track is associated with the measurement). In the
example, there are two continuing tracks and three new measurements; this
approximation made in the derivation is the approximation of
resultsinfiveposteriortracks(i.e.,Bernoullicomponents).UsingtheTOM-
the joint association distribution Pt|t(b) as the product of its MeMBer/P, these correspond to the two existing tracks, and a new track
marginals.12 for each measurement. Using the MOM-MeMBer/P, a posterior track exists
for each measurement, collecting events from all prior tracks that use the
The difference between the tracks formed by the TOM-
measurement.Theremainingposteriortrackshypothesisemisseddetectionof
MeMBer/P and MOM-MeMBer/P is illustrated in Fig. 3. thecorrespondingpriortrack(andnon-existenceotherwise).
In the TOM-MeMBer/P, there is a track in the posterior
corresponding to each prior track collecting all hypotheses
updating that track, and a track in the posterior for each While there are few cases in which the performance of LBP
measurement considering the possibility of the measurement is guaranteed, its practical performance in many problems is
being a new target. In the MOM-MeMBer/P, there is a track remarkable; for example, turbo decoding has been shown to
for each measurement collecting all hypotheses updated by be an instance of LBP [28]. While LBP is not guaranteed to
themeasurement(collectinghypothesesfromallpriortracks), converge in general, its convergence in the model of interest
and a track for each prior track containing only the “missed is proven in [14], and the underlying optimisation problem is
detection” hypothesis. showntobeconvexin[15].Detailsofthealgorithm,theaccu-
Detailed pseudocode for both TOM-MeMBer/P and MoM- racyofthemarginalestimatesthatityieldsandthenumberof
MeMBer/P can be found in Appendix C. iterationsrequiredforconvergencearestudiedindetailin[14];
detailed pseudocode for the algorithm and its employment in
the TOM-MeMBer/P and MOM-MeMBer/P can be found in
C. Approximating marginal association distributions
AppendixC.Notethatthefilterderivationaboveisnottiedto
The greatest barrier to practical application of the TOM- the LBP approximation; any other method such as Maskell’s
MeMBer and MOM-MeMBer algorithms is the calculation of exact EHM [29], MCMCDA [30], or LMIPDA [10,31] may
the marginal probabilities of association Pi (ai) and Pj (bj), also be used to calculate (approximately or otherwise) the
t|t t|t
which is closely related to the #P-complete calculation of a marginal association probabilities.
matrix permanent. The present work was motivated by the
emergence of a new approximation to this calculation that is
V. CONTRIBUTIONSANDRELATIONSHIPTOOTHERWORK
both accurate and tractable [14]. The approximation is based
on loopy belief propagation (LBP), and the particular model The relationships of the proposed algorithms to the
used has been studied recently in machine learning, physics JITS/JIPDA and MeMBer filters were discussed in Sections
and information theory.13 While LBP was discovered as a IV-A and IV-B. As discussed in Section III, the structure of
heuristic extension of the standard forwards-backwards algo- the MBM component of the distribution is similar to that of
rithm for Markov chains to probabilistic graphs with loops, it the TOMHT (in its use of a hypothesis tree for each target,
has since been understood as a variational (i.e., optimisation- implicitlyrepresentingglobalassociationhypotheses).Ateach
based) inference method with roots in statistical physics [27]. time,MHTmethodsseekthemostprobableglobalassociation
hypothesis, which corresponds to the term in the sum (28)
12Clearly,practicalimplementationswillrequireadditionalapproximations with the largest weight. The derivation in Section III retains
in pruning unlikely tracks, reducing the number of components in the
separability of the weights (30) such that efficient methods
mixture-conditioneddistributions,andincalculatingthemarginaldistributions
Pj (bj)themselves. such as Lagrangian relaxation could be applied to obtain a
t|t
13See[14,15]forreferences. RFS-basedTOMHT;thesewereappliedin[16,32].Ithasbeen
10
suggested that the explicit modelling of associations in MHT shown in Fig. 4. It is well-known that JPDA and related
may induce a bias into the solution (e.g., [3, p340]). While algorithms suffer from coalescence, i.e., after targets come in
the derivation in Section III showed that association naturally close proximity, mean estimates remain on the mid-point of
arisesinRFSfilters,itappearsasatotalprobabilityexpansion thetargetsforasignificantduration(e.g.,[37]).Incase1,tar-
(28).Thusthederivationdoesnotvalidateapproachessuchas gets are completely indistinguishable at the mid-point, hence
MHTthatreplacethesumoverain(28)withamaximisation. coalescence-like effects are at their worst. Coalescence-like
The derivation is related to [33,34], which was developed effects also occur in case 2, but to a lesser extent (mainly due
simultaneously and independently. On a theoretical level, the to the discernible difference in velocity). In case 1, targets all
primary difference is that the latter papers study the labelled exist throughout the simulation (tracks are not pre-initialised).
case (rather than the unlabelled case studied herein). The In case 2, the targets are born at times {0,10,...,10(n−1)}
labelled case has obvious benefits in its inherent ability to (any targets not existing prior to time t=100 are born at that
maintain continuity between time steps. On a practical level, time; consequently, for case 2 with n = 20, ten targets are
both methods effectively involve a total probability expansion bornattimet=100).Targetsfollowalinear-Gaussianmodel
over an intractable number of terms (i.e., global association with nominally constant velocity, x = Fx +w , where
t t−1 t
hypotheses). The labelling in [33] increases the number of w ∼N{0,Q},
t
terms, since unknown targets from the Poisson birth process
must be approximated via additional components in the sum.
The filter in [33] operates by carrying a fixed number (N) (cid:20) 1 T (cid:21) (cid:20) T3/3 T2/2 (cid:21)
F= ⊗I , Q=q ⊗I
of terms from one time step to the next. The update step then 0 1 2×2 T2/2 T 2×2
appliesMurty’salgorithmN timestofindtheN termswiththe
highestweight,truncatingthesumtocontainonlytheseterms.
This is algorithmically similar to the MHT implementation in
andq =0.01,T =1.Target-originatedmeasurementsprovide
[35].Whilethisapproachavoidseffectssuchascoalescence,it
position corrupted by Gaussian noise, i.e., z = Hx + v ,
limitsthenumberofglobalassociationhypothesesthatcanbe t t t
where H = [ 1 0 ] ⊗ I , and v ∼ N{0,I }. The
represented.Forexample,ifourmarginalfilterapproachmain- 2×2 t 2×2
initial unknown target intensity for the proposed methods
tains twenty tracks, each with ten single-target hypotheses,
is assumed to be λu (x) = 10N{x;0,P}, where P =
collectively 1020 global hypotheses are implicitly represented 0|0
diag[1002, 1, 1002, 1] covers the position and velocity
(withtheerrorinweightsinducedbytheapproximation(59));
regionofinterest.Thebirthintensityusesthesamecovariance,
explicit representation of this number of terms is obviously
λb(x) = 0.05N{x;0,P}. Cases are considered with the
intractable. This can be expected to limit the performance of
average number of false alarms per scan is λfa ∈ {10,80},
the approach in [33] in low SNR environments (i.e., low Pd,
and their distribution is uniform on [−100,100]2. All fil-
high λfa).
ters assume Ps = 0.999. We consider cases with Pd ∈
The concept of maintaining a representation of targets that
{0.3,0.5,0.7,0.98}, representing a range of SNR values.
have never been detected is uncommon but not new; e.g.,
see [8]. While this aspect of the filter has not been a focus We compare the proposed methods to CPHD [3,4], CB-
in this paper, it yields significant benefits including in the MeMBer [5], JITS and linear multi-target ITS (LMITS)
speed ofinitiationduring theinitial transientwhen thetracker [10,11] (a variant of JITS which replaces the marginal asso-
commences and in environments with non-stationary sensor ciation probabilities with a heuristic approximation). The CB-
coverage [23]. Furthermore, it is shown in [23] that multi- MeMBer approximates the initial distribution by 30 identical
Bernoulli tracks with a low probability of existence can be Bernoulli components with r = 1/3, and uses a single birth
efficiently represented via the Poisson component, greatly component with r = 0.05 at each time. JITS and LMITS
reducing the complexity associated with initiating tracks in truncate the unknown target intensity to use ρ (z) =
target
highfalsealarmenvironments.Theadaptiveclutterestimation Pdλb[f(z|·)]. The implementation of JITS initiates a new
approach of [36] may also be incorporated tractably via the track on every measurement (via (64)), but only updates new
Poisson component. tracks with measurements that are not in the gate of any
hypothesis of an existing track. Exact marginal association
VI. RESULTS probabilities were calculated using the junction tree algorithm
We demonstrate the performance of the proposed ap- describedin[14],whichinturnusesthelibraryfordiscreteap-
proaches in a challenging scenarios involving n∈{6,10,20} proximateinference[38].Theexactcalculationisnotpossible
targets which are in close proximity at the mid-point of forthesimulationswithn>6targets,orwithλfa >10.Since
the simulation, achieved by initialising at the mid-point and the weights provided by LMITS do not provide a probability
running forward and backward dynamics. We consider two thatameasurementisnotusedbyanyothertrack,theLMITS
cases for the mid-point initialisation (i.e., t=100): implementation initiates tracks only on measurements that are
Case 1: x ∼N{0,10−6×I } not within the gate of any hypothesis. Where applicable, all
100 4×4 tracks with ri ≥10−4 are maintained (the threshold needed
Case 2: x ∼N{0,0.25×I } t|t
100 4×4 tobeincreasedto10−3 forJITStobeabletosuccessfullyex-
where the target state is position and velocity in two di- ecute).AllmethodsutiliseGaussianmixtureimplementations,
mensions. Snapshots of one dimension of both cases are withhypothesismanagementperformedusingn-scanmerging