Measurement-to-Track Association and Finite-Set Statistics

0.2 MB·

by  Ronald Mahler

1
Measurement-to-Track Association and Finite-Set
Statistics
Ronald Mahler, Random Sets LLC, Eagan, MN, USA, January 1, 2017
Abstract—Multi-hypothesistrackers(MHT’s),whicharebased This paper expands upon an argument originally advanced
on the measurement-to-track association (MTA) concept, have in 2007 in Section 10.7.2 of [8]. Its purpose is fourfold:
7 long been asserted to be “Bayes-optimal.” Recently, rather
1 bolder claims have come to the fore: “The right model of 1) Systematicallyassess the“Bayes-optimal”and“theoret-
0 themultitarget state is that used in themulti-hypothesistracker ically rigorous” claims made for MHT.
2 (MHT) paradigm, not the RFS [random finite set] paradigm.” 2) In particular, assess the mathematical and phenomeno-
n Or,theRFSapproachisessentiallyamathematicallyobfuscated logical underpinnings of the concept that underlies
reinvention of MHT. In this paper it is shown that: (a)
a MHT’s and related algorithms: the measurement-to-
J although MTA’s can be given a Bayesian formulation, this
track association (MTA).
formulation is not fully consistent with Bayesian statistics; (b)
5 phenomenologically, an MTA is a heuristic extrapolation of an 3) Clarify the relationship between classical MTA-based

intuitivespecial case togeneral multitarget scenarios; (c) MTA’s approaches, and the random finite set (RFS) approach
]
E are,therefore,notphysicallyrealentitiesandthuscannot(aswith of finite-set statistics (FISST) [9].
MHT’s) be employed as state representations of a multitarget
M 4) RefutecertainerroneouscriticismsoftheRFSapproach.
system;(d)MHT’sare,consequently,heuristicapproximationsof
For example, that “The right model of the multitarget
. the actual Bayes-optimal approach, the multitarget Bayes filter;
at (e) the theoretically correct measurement modeling approach is state is that used in the multi-hypothesistracker (MHT)
t the RFS multitarget likelihood function LZ(X) = f(Z|X); (f) paradigm,nottheRFSparadigm.” Ormoreexpansively,
s althoughMTA’sdooccurin f(Z|X),theyaretheconsequenceof that the RFS approach is essentially a mathematically
[
amerechangeofnotationduringtheRFSderivationof f(Z|X); obfuscated reinvention of MHT theory.
1 and (g) the generalized labeled multi-Bernoulli (GLMB) filter
v of Vo and Vo is currently the only provably Bayes-optimal and See [15] foran overviewofmultitargettrackingthatcovers
8 computationallytractableapproachfortruemultitargettracking the MHT, RFS, and other approaches.
7 involving MTA’s. The paper is organized as follows. Following brief sum-
0 Index Terms—Multitarget tracking, finite-set statistics, maries of Bayesian statistics and MTA theory in Sections II
7
measurement-to-track association. and III, we examine the concept of an association likelihood
0
in Section IV. Then, in Section V, we turn to a Bayesian
.
1
assessment of MTA’s. This will lead us to argue that,
0 I. INTRODUCTION even though MTA’s can be given a Bayesian formulation,
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1 Bytheearly1990’satleast,Bayesianstatisticshadbecome this formulation is more consistent with classical than with
: the overwhelmingly dominant foundation for target tracking. Bayesian statistics.
v
i Such was its cache´, in fact, that it was not uncommon A phenomenological assessment of MTA’s in Section VI
X for authors to claim that a proposed approach was “Bayes- will lead us to further conclude that:
ar optimal” merely because Bayes’ rule had been utilized in • MTA’sare notphysicallyrealentities andthuscannotbe
somefashion. Inparticular,itwas—andstillis—claimedthat employedasstaterepresentationsofamultitargetsystem.
multi-hypothesis trackers (MHT’s) are not only theoretically
Which is to say, it is not phenomenologically reasonable
rigorous,buttheoreticallyrigorouswithintheBayesianframe-
to claim that certain measurements originated with certain
work (“Bayes-optimal”). For example: “...MHT algorithms
tracks. Rather,themostthatcanlegitimatelybeassertedisthe
themselvescan be, and indeedwere, derivedthroughrigorous
following: If targetswith state-set X are present,then there
mathematics [“in the theory of Bayesian filtering”].”
is a probability (density) f(Z|X) that they will generate a
Such claims have tended to be made while overlookingthe measurement-set Z.
following points:
This will lead us to consider, in Sections VII and VIII, the
1) Theterm“Bayesoptimal”referstoonethingonly: state RFS notion of a multitarget likelihood function L (X) =
Z
estimation. f(Z|X). There we will note that, even though the formula
2) In target tracking, the term “state variable” also has a for f(Z|X) involves MTA’s, they do not—as in MTA
specific meaning: it must be a faithful mathematical theory—arise from questionable heuristic intuition. Rather,
modelof some unknownbut physically real propertyof they arise from a mathematically rigorous RFS derivation
whatever targets are of interest. basedonastatisticallyandphenomenologicallyrigorousRFS
3) “Rigorousmathematics”thatproceedsfromfaultymath- measurement model. Specifically, they are the consequence
ematicaland/orphenomenologicalassumptionsisspuri- of a mere change of mathematical notation. This will then
ously rigorous. naturally lead us, in Sections VII and VIII, to a summary of
2
RFS multitarget measurement modeling and, in Section IX, where f(ζ)= f(ζ|ξ)·f (ξ)dξ and where ‘ ·dξ’ denotes
0
to a discussion of the rigorous meaning of the term “Bayes- the integration Rconcept for X. In Bayes filteRring theory, if
optimal” in a multitarget context. ζ = ζ was collected at time t , then the prior f (ξ)
k+1 k+1 0
Wewillthenturntothefollowingquestion: SincetheMTA is the predicted distribution
and RFS theories both involve MTA’s, how are they related?
In Section X, we will describe a mathematical connection f (ξ|ζ )= f (ξ|ξ′)·f (ξ′|ζ )dξ′ (3)
k+1|k 1:k Z k+1|k k|k 1:k
betweenRFSandMTAlikelihoods. Thiswillleadustofinally
deduce that: where f (ξ|ξ′) is the Markov state-transition density,
k+1|k
• MHT’s are heuristic approximationsof the actual Bayes- and in which case the posterior distribution is f(ξ|ζ) =
optimal approach, namely the multitarget Bayes filter. fk+1|k+1(ξ|ζ1:k+1).
We will conclude with two final questions: Is there an Onecandeterminethe“best”valueof ξ usingsomeBayes-
MTA-oriented multitarget tracker that is provably Bayes- optimal state estimator, such as the maximum a posteriori
optimal? Ifso,isitcomputationallytractable? Theaffirmative (MAP) estimate (if it exists):
answer to these questions—the generalized labeled multi- ˆξ(ζ)=argsup f(ξ|ζ). (4)
Bernoulli (GLMB) filter of Vo and Vo—is the subject of
ξ
Section XI. This discussion will produce a theoretically
An estimator is Bayes-optimal if it minimizes the Bayes risk
grounded tracking interpretation of MTA’s:
• An MTA is a purely mathematical entity—namely, the R (ˆξ)= C(ˆξ(ζ),ξ)·f(ξ|ζ)·f(ζ)dξdζ (5)
C
index of one possible weighted hypothesis about which Z
track labels exist in the scene and which track distribu- forsome costfunction C(ξ,ξ′) definedonstates ξ,ξ′ ([14],
tions correspond to those labels.
pp. 54-63). This is the only meaning of “Bayes-optimal.”
Evenso,thispapershouldnotbeconstruedasadenigration
of MHT’s. When the MHT was introduced by Reid in
III. MEASUREMENT-TO-TRACK ASSOCIATIONS (MTA’S)
1979 [12], computer processing was primitive by today’s
standards. Reid addressed this difficulty by using MTA’s to The MTA approach presumes the “small target” sensor
decomposethemultitargettrackingproblemintoacoordinated model. A detection process (such as thresholding) is applied
system of extended Kalman filters. Since then, theoretical toasensorsignature,resultinginaset Z ofpointdetections.
and practical advances by his successors have made MHT’s Everytargetisassumedtobedistantenoughthatitgeneratesat
the workhorsesofmultitargettracking. Quite understandably, mosta singledetection,butcloseenoughthatdifferenttargets
however, limited computing power, combined with a lack of produce distinct detections.
suitable mathematical theory, also made it difficult to adhere Suppose,then,thatattime tk wehave n predictedtarget
to proper levels of theoretical rigor. Now that computional tracks with state-set X = {x1,...,xn} ⊆ X with |X| = n,
and mathematicaltools are sufficiently mature, it is important and associated track distributions f(x|1),...,f(x|n). From
to move from conventional wisdom to scientific clarity. these tracks, we collect measurements Z ={z1,...,zm}⊆Z
with |Z|=m.
II. BAYESIAN ANALYSIS An MTA is a hypothesis about which tracks in X gen-
Let X be a space whose elements ξ are the states of the erated which measurements in Z. That is, assume that the
physical entities of interest. Here, ξ ∈ X should uniquely measurements in Z′ ⊆ Z were generated by the tracks in
and exhaustively model and correspondto the actualphysical X′ ⊆ X. Then the excess tracks in X −X′ are “missed
states of the system. detections”andtheexcessmeasurementsin Z−Z′ are“false
The physical entities are observed by a sensor with mea- detections” or “clutter.” In addition, there is a bijection (one-
surements ζ ∈ Z. The goal is to estimate ξ based on ζ. to-one and onto) function γ :X′ ↔Z′, which specifies that
In a Bayesian analysis, the two are related by a likelihood x ∈ X′ generates γ(x) ∈ Z′. If X′ = ∅ then Z′ = ∅
function (measurement distribution) and all of the measurements in Z are false detections.
Stated with greater mathematical specificity, an MTA is a
L (ξ)=f(ζ|ξ), (1)
ζ 4-tuple α˜ = (ν,X′,Z′,γ) such that: (a) ν is an integer
which gives the probability (or probability density) that with 0≤ν ≤min{n,m}; (b) X′ ⊆X with |X′|=ν; (c)
measurement ζ will be collected if an entity with state Z′ ⊆Z with |Z′|=ν; and (d) γ :X′ ↔Z′ is a bijection
ξ is present. In particular the normalization condition if X′ 6=∅ and the null map if otherwise.
f(ζ|ξ)dζ =1 must be true for every ξ ∈X. According to Section II, α˜ = (ν,X′,Z′,γ) cannot be a
R In classical statistics, the unknown state ξ is assumed to validstaterepresentationsinceitdependsonthemeasurement-
be a nonrandom constant. In Bayesian statistics, however, ξ space parameters Z′ and γ. In particular, α˜ cannot be
is assumed to a random variable, the statistical behavior of specified without knowing, ahead of time and for any time,
which is characterized by some prior probability distribution what measurement-set Z will be collected.
f0(ξ). If a measurement ζ is collected, then the posterior To address this conundrum, choose orderings x1,...,xn
probability distribution of ξ, conditioned on ζ, is and z ,...,z of the elements of X and Z, respectively.
1 m
f(ζ|ξ)·f (ξ) Then redefine an MTA to be a function α : {1,...,|X|} →
f(ξ|ζ)= 0 (2)
f(ζ) {0,1,...,|Z|} suchthat α(i)=α(i′)>0 impliesthat i=i′.
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In this case X′ = {xi ∈ X| α(i) > 0} and γ(xi) = zα(i) V. MTA’SAND BAYESIAN ANALYSIS
if α(i)>0. MTA’s can be applied to multitarget tracking in numerous
Thisstrategemdoesnotcompletelyresolvetheconundrum, ways: single-hypothesis trackers, hypothesis-based MHT’s,
sincethecardinality |Z| of Z stillmustbeknownaheadof track-based MHT’s, etc. [15]. Regardless of the approach,
timeandforanytime. Onecansidestepthisdifficultybyagain soonerorlaterthefollowingquestionmustbeanswered: How
redefininganMTA,thistimeasapair (m,α˘(m)) where m≥ many targets are present, and what are their states?
0 isanintegerand α˘(m) isafunction α˘(m) :{1,...,|X|}→ ToanswerthisquestioninaBayesianfashion,weconstruct
{0,1,...,m} such that α˘(m)(i) = α˘(m)(i′) > 0 implies the posterior distribution
i=i′. Theunknownstatevariable m istherebyconceptually
p(m,α˘ |Z)∝f˘(Z|m,α˘ )·p (m,α˘ ), (11)
disengaged from the known measurement cardinality |Z|. (m) (m) 0 (m)
on MTA’s where L (m,α˘ ) = f˘(Z|m,α˘ ) is the
Z (m) (m)
likelihoodfunctionand p (m,α˘ ) isapriorontheMTA’s.
IV. THELIKELIHOOD OFAN MTA 0 (m)
The most probable MTA is then:
Inwhatfollows,let(a) Lz(x)=f(z|x) bethesingle-target (mˆ,αˆ(mˆ))=arg max p(m,α˘(m)|Z). (12)
likelihoodfunction(measurementdistribution);(b) pD(x) be m,α˘(m)
the (state-dependent) probability of detection; and (c) κ(z) Given this, the estimated number of targets is the number
be the intensity function of a Poisson clutter process, with of i’s such that αˆ (i) > 0. Also, for any i such that
(mˆ)
λ = κ(z)dz the clutter rate (expected number of clutter αˆ (i) > 0, we get the corresponding estimated target state
(mˆ)
measuRrements in each frame) and c(z) = κ(z)/λ the byupdating f(x|i) usingthemeasurement z andthe
αˆ(mˆ)(i)
schluotwtenr stphaattiatlhedis“tgrilboubtailona.ssoTchieantioinn lEikqe.li(h7o.3o2d)”ooff[3a]nitMwTaAs sinTghlee-tlairkgeelithloikoedlifhuonocdtioLnziαˆn(mˆE)q(i.)((x11))=isf(zαˆ(mˆ)(i)|x).
α : {1,...,|X|} → {0,1,...,|Z|} is ℓ (α) = e−λκZ if
|X|=∅ and, if otherwise1, Z|X f˘(Z|m,α˘(m))=δ|Z|,m·f(Z|α˘(|Z|)), (13)
where f(Z|α) is as in Eq. (10). Note that
clutter misseddetections detections f˘(Z|m,α˘ )δZ =1 for all (m,α˘ ).
(m) (m)
ℓ (α)=e−λκZ−Zα ℓ(∅|i) ℓ(z |i) (6) R Eq. (13) conceptually disengages m (unknown state
Z|X α(i)
z }| {iz:αY(i)=}0| {zi:αY(i)>0}| { parameter) from |Z| (known measurement parameter). The
MTA approach can thereby be endowed with a Bayesian
where Z = {z | 1 ≤ i ≤ |X|, α(i) > 0}; where κZ = formulation. However, Eq. (13) seems somewhat peculiar—
α α(i)
κ(z) if Z 6=∅ and κZ =1 otherwise; where contrived, even—because it implies that m is always a
x∈Z
Q nonrandom constant. This is at variance with the usual
Bayesian presumption that unknown state variables are ran-
ℓ(z|i)= p (x )·f(z|x)·f(x|i)dx (7)
Z D i dom variables. It is in perfect agreement, however, with
the classical-statistics presumption that they are nonrandom
is the probability (density) that the i’th track generates z, constants. We therefore conclude that:
giventhedegreetowhichthetrackcanbedetected;andwhere
• The MTA approach is not entirely consistent with
the probability that it is undetected is:
Bayesian statistics.
ℓ(∅|i)= (1−pD(x))·f(x|i)dx. (8) VI. MTA’S AND PHENOMENOLOGY
Z
Beyond this purely mathematical incongruity, one must
As an example, suppose that there is no clutter and no address a more serious physical one:
missed detections: λ = 0 and pD = 1. Then MTA’s α • However formulated, is an MTA actually a physical
reduce to permutations π : {1,...,n} ↔ {1,...,n} and Eq. entity? Thatis,isitphenomenologicallysensicaltoclaim
(6) reduces to ([3], Eq. (7.29)) thatcertain measurementsoriginatedwith certain tracks?
This seems doubtful. An MTA is a heuristic extrapolation
ℓ (π)=ℓ(z |1)···ℓ(z |n). (9)
Z|X π(1) π(n) of the following special case to general multitarget scenarios.
Suppose that we have a sensor with no missed or false
The likelihood function ℓZ|X(α) is not normalized. Its detections. Further suppose that we have n targets that
normalization is easily seen to be: are well-separated with respect to the noise resolution of this
sensor, as specified by Lz(x) = f(z|x). In this case it
ℓ(z |i)
f(Z|α)=ℓˆZ|X(α)=cZi:αY(i)>0c(zα(i))·α((1i)−ℓ(∅|i)). mseeeamsusresemlfe-netvsidezn1t,.t.h.,aztnthesruechisthaatperzmπ0u(tia)tioonrigiπn0atedofwtihthe
(10) xi for all i = 1,...,n—because there is only a small
probability that z could have originated with any target
π0(i)
other than x . Expressed in termsof association likelihoods,
1Note: Eq.(7.32)of[3]isageneralization ofEq.(G.238),p.739of[8], i
whereitwasimplicity assumedthat f(x|i)=δxi(x). ℓZ|X(π0)=ℓ(zπ0(1)|1)···ℓ(zπ(m0)|m) ismaximalforallπ.
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When all targets are very close together, however, it be- • The finite set Z of measurements was generated by
comes statistically impossible to maintain that any particular the finite set X of tracks, with probability (density)
measurement was generated by any particular target. Such a f(Z|X) that this is the case.
claim becomes even more difficult to maintain if the sensor This insight is useless unless we can also answer the
has missed and false detections. To furtherinsist that there is following question:
a “Bayes-optimal” MTA is to impose a phenomenologically • What is the concrete formula for f(Z|X)?
spurious stucture upon the modeling of the physical system.
These issues immediately lead us to finite-set statistics,
Worse, by imposing physically extraneous information we
in which L (X) = f(Z|X) is known as the multitarget
Z
potentially insert a hidden statistical bias into our analysis.
likelihood function or multitarget measurement distribution.
There is an additional issue:
Finite-setstatisticswillnotbedescribedatlengthhere. We
• Is an MTA a valid state representation of a multitarget instead direct interested readers to the books [8], [3], [10],
system to begin with?
[13] and the variously oriented overviews [2], [7], [9], [5],
This seems dubious. First, the MTA concept is specific to [18], [15]. Also, a systematic investigation of “finite point
a particular sensor measurement model—one in which some processes” versus RFS’s, in the multitarget tracking context,
detection process is applied to a sensor signature, resulting in can be found in [6].
a set of pointdetections. If the sensor has some other model,
however—for example, a signature such as a pixelized image VII. THE MULTITARGETLIKELIHOOD FUNCTION
or a rotating-radar range-bin amplitude trace—then the MTA
Let Z = {z ,...,z } with |Z| = m and X =
1 n
concept is completely meaningless.
{x ,...,x } with |X| = n. Then f(Z|X) was derived
1 n
Second, to apply MTA theory we must choose a priori
in Eq. (12.139) of [8], and reiterated in Eq. (7.21) of [3]:
orderings of the elements of both Z and X. Generally
p (x )·f(z |x )
speaking,suchorderingshavenophenomenologicalbasis. By f(Z|X)=κ(Z)·(1−p )X D i α(i) i
imposing them, we potentially introduce additional unknown D Xα i:αY(i)>0κ(zα(i))·(1−pD(xi))
biases into our analysis. (14)
Third and most importantly, a multitarget system is by where the summation is taken over all MTA’s α :
definition an ensemble consisting of an unknown number of {1,...,|X|}→{0,1,...,|Z|};where (1−p )X = (1−
D x∈X
targetswithunknownstates. Thusitsstaterepresentationmust pD(x)) if X 6=∅ and (1−pD)X =1 otherwise;Qandwhere
be based on the single-target states x1,...,xn of those the distribution of the Poisson clutter process is
targets for n ≥ 0—and on nothing else (since, otherwise,
phenomenologically extraneous information is potentially in- κ(Z)=e−λκZ =e−λ κ(z). (15)
troduced). How do we proceed? zY∈Z
One frequently proposed approach is to employ concate- Note that Eq. (14) does not functionally depend on the
nated vectors x = (x ,...,x ) (along with the null particular orderings chosen for the elements of Z and X.
1 n
vector φ for n = 0). Such an approach is conceptually Eq.(14)mightseemto requireMTAtheorysinceitinvolves
questionable [18], [15]. Estimation error is an important MTA’s. The important point to understand, however, is that
aspectofmultitargettracking. Itmustbepossibletocompute the MTA’s occurring in Eq. (14)) do not, as in MTA theory,
the distance between the “ground truth” multitarget state arisefromheuristicintuition. Rathertheyarisefromachange
and a multitarget tracker’s estimate of it. As an example, ofnotationinamathematicallyrigorousRFSderivationbased
let x 6= x be Euclidean states. Then (x ,x ) and onastatisticallyandphenomenologicallyrigorousRFSmodel.
1 2 1 2
(x ,x ) are two possible vector representations of a two- The purpose of the next section is to demonstrate this claim.
2 1
target system with states x ,x . But the Euclidean distance
1 2
k(x ,x ) − (x ,x )k = k(x − x ,x − x )k is not 0. VIII. THE RFSINTERPRETATIONOF MTA’S
1 2 2 1 1 2 2 1
Likewise, what is the distance between the two-target state The demonstration consists of the following steps: (a) the
(x1,x2) and single-target state (y)? Or between (y) and RFSmeasurementmodel Σ;(b)thebeliefmeasure βΣ(T) of
the no-target state φ? Σ; (c) the probabilitygeneratingfunctional(p.g.fl.) G [g|X]
Σ
A multitarget state is more correctly modeled as a finite of Σ; (d) the derivation of the set-theoretic formula for
set {x1,...,xn} with n≥0; and there are well-defined and f(Z|X) from GΣ[g|X] using the general product rule for
computationally tractable metrics for finite sets, such as the functionalderivatives;(e) the derivation of Eq. (14) from this
optimal sub-pattern assignment (OSPA) metric (see Section formulaviaachangeofnotation;and(f)theRFSinterpretation
6.2 of [3]). of MTA’s.
At the same time, the fact that finite sets are unordered
doesnotmean—asisoftenasserted—thattheycannotbeused A. The RFS Measurement Model
to model temporally-connected tracks. This is because, in
This is
general,each x hasauniqueidentifyinglabel—seeSections
i Σ=C∪Υ(x )∪...∪Υ(x ) (16)
II and XI. 1 n
Consequently,thefollowingistheonlyphenomenologically were C is the Poisson clutter RFS; Υ(x) with |Υ(x)|≤1
legitimate claim that can be ventured about the relationship istherandommeasurement-setgeneratedbyatargetwithstate
between measurements and tracks: x;and Σ isthetotalmeasurement-RFSfortheentirescene.
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B. The Belief Measure of Σ E. The MTA Formula for f(Z|X)
If Z has the Fell-Matheron topology then the statistics of Because of Eq. (30), the only surviving terms in the sum-
Σ are completely characterized by its belief measure mationinEq.(29)arethoseforwhich W ,...,W areeither
1 n
emptyorsingleton;and W contributesafactortotheproduct
β (T|X)=Pr(Σ⊆T|X) (17) i
Σ inEq.(28)onlyifitisasingleton. Thusforagivenchoiceof
for all closed subsets T ⊆ Z. If target and clutter measure- W ,...,W , define α:{1,...,n}→{0,1,...,m} implicitly
1 n
ments are generatedindependently,then C,Υ(x1),...,Υ(xn) by {zα(i)}=Wi if Wi 6=∅ and α(i)=0 otherwise. Note
are independent and that α is an MTA in the sense of Section III. Conversely
βΣ(T|X)=βC(T)·β(T|x1)···β(T|xn) (18) if we are given α, define Wi = {zα(i)} if α(i) > 0 and
W =∅ if otherwise. Either way, W =Z−Z where
i 0 α
where, if 1 (x) is the indicator function of subset T,
T
Z =W ∪...∪W ={z |α(i)>0}. (31)
β(T|x) = βΥ(x)(T) (19) α 1 n α(i)
Thusthere is a one-to-onecorrespondencebetween MTA’s α
= 1−p (x)+p (x) 1 (z)·f(z|x)dz(20)
D D Z T and lists W1,...,Wn of mutuallydisjointempty or singleton
subsets of Z. Furthermore, only those i’s with α(i) > 0
βC(T) = exp(cid:18)Z (1T(z)−1)·κ(z)dz(cid:19). (21) contributeafactortothe product. Consequently,Eq.(29) can
be rewritten as:
C. The p.g.fl. of Σ
f(Z|X) = e−λ κZ−Zα(1−p )X (32)
Substituting test functions 0 ≤ g(z) ≤ 1 for 1 (x), we D
T Xα
get the p.g.fl. of Σ ([3], Eq. (7.19)):
· pD(xi)·Lzα(i)(xi)
GΣ[g|X]=GC[g]·G[g|x1]···G[g|xn] (22) i:αY(i)>0 1−pD(xi)
where
from which Eq. (14) immediately follows.
G[g|x] = 1−p (x)+p (x)·L (x) (23)
D D g
G [g] = eκ[g−1] (24)
C F. The RFS Interpretation of MTA’s
Lg(x) = g(z)·f(z|x)dz (25) This leads us to the following inferences:
Z
1) TheMTA’s α inEq.(14)donotarisefromheuristicin-
κ[g−1] = (g(z)−1)·κ(z)dz. (26) tuition. Rather,theyaretheconsequenceofachangeof
Z
mathematicalnotation—i.e.,as a mathematicallyequiv-
D. The Set-Theoretic Formula for f(Z|X) alent way of rewriting the purely set-theoretic formula
of Eq. (29).
The multitarget likelihood function f(Z|X) is [9], [8]:
2) Eq. (14) involves all possible MTA’s, with no MTA
δG δ
f(Z|X)= Σ[0|X]= G [g|X] (27) having a greater impact on the value of f(Z|X)
Σ
δZ (cid:20)δZ (cid:21)
g=0 than any other. Thus Eq. (14) does not assign any
where “δ/δZ” is the functional derivative with respect to phenomenologicalrealitytoMTA’sasisolatedentities.
Z. When applied to Eq. (22), the general product rule for 3) Also, f(Z|X) does not functionally depend on par-
functional derivatives (see [3], Eq. (3.68)) yields: ticular orderings of the elements of Z or X. Thus
the potential statistical biases associated with the MTA
n
δG δ
Σ[g|X]= eκ[g−1]κW0 G[g|xi] approach, as identified in Section VI, cannot occur.
δZ δW
W0⊎W1X⊎...⊎Wn=Z iY=1 i
(28)
IX. MULTITARGETBAYES OPTIMALITY
where the summation is taken over all mutually disjoint (and
possiblyempty)subsets W ,W ,...,W of Z whoseunion Thismaterialreiterates the discussion in Section 5.3of [3].
0 1 n
is Z. Setting g =0, from Eq. (28) we get Suppose that f0(X) is the multitarget prior distribution and
that we have collected a measurement-set Z. Then as per
f(Z|X) = e−λ κW0(1−pD)X (29) Section II, the multitarget posterior distribution is
W0⊎W1X⊎...⊎Wn=Z
f(Z|X)·f (X)
· n hδWδiG[g|xi]ig=0 f(X|Z)= f(Z|Y)·f00(Y)δY (33)
iY=1 1−pD(xi)  where now the integral isRthe set integral ([3], Section 3.3)
 
where ∞ 1
f(X)δX =f(∅)+ f({x ,...,x })dx ···dx .
δWδ G[g|xi] g=0 = pD(xi)1·Lz(xi) iiff WW=={∅z} . Z nX=1n!Z 1 n 1 (3n4)
(cid:2) 1−pD(x(cid:3)i)  1−pD0(xi) if |W|>1 Also as per Section II, a multitarget state estimator is a
(30) function Xˆ(Z) of the measurements Z whose values

6
are finite state-sets. It is Bayes-optimal if it maximizes the distribution: f (X|Z )=f(X|p ) for some
k+1|k 1:k k+1|k
multitarget Bayes risk p .
k+1|k
2) If the predicted distribution is f (X|Z ) is
k+1|k 1:k
RC(Xˆ)=Z C(Xˆ(Z),X)·f(X|Z)·f(Z)δXδZ (35) a GLMB distribution f(X|pk+1|k) then so is the
new posterior distribution: f (X|Z ) =
k+1|k+1 1:k+1
with respect to some cost function C(X,Y) defined on
f(X|p ) for some p .
k+1|k+1 k+1|k+1
multitarget state-sets X,Y. This is the only meaning of
3) An arbitrary labeled multitargetdistribution f(X) can
the term “Bayes-optimal” in a multitarget context. The joint
be approximated by a GLMB distribution that has the
multitarget (JoM) and marginal multitarget (MaM) estimators same PHD and cardinality distribution as f(X) [11].2
(seeSection14.5of[8])havebeenshowntobeBayes-optimal;
Thusif we choose f (X)=f(X|p ) then f (X) is
and the former has been shown to be statistically consistent. 0 k+1|k 0
not a heuristic choice—it is the actual predicted distribution:
f(X|p )=f (X|Z ). Asimilarclaimcanbemade
X. RELATIONSHIP BETWEENMTA AND FISST k+1|k k+1|k 1:k
for the time-update step. If f (X|Z ) is approximated
k|k 1:k
Assume a priori that n targets are known to exist and heuristically as some a priori distribution f (X), then
−
arestatisticallyindependent. Thenthemultitargetdistribution the corresponding predicted distribution f (X|Z ) is
k+1|k 1:k
that describes them is almostnevercorrectbecause f (X) isnotanactualposterior
−
distribution. But if we instead choose f (X) = f(X|p )
f (X)=δ f(x |1)···f(x |n) (36) − k|k
0 |X|,n π(1) π(n)
then f (X) is not a heuristic choice, since in this case it is
Xπ −
theactualposteriordistribution: f(X|p )=f (X|Z )—
where X = {x ,...,x } with |X| = n and where the k|k k|k 1:k
1 n which in turn means that f (X|Z ) is the actual
summation is taken over all permutations π on 1,...,n. k+1|k 1:k
predicted distribution.
If f(Z|X) is defined as in Eq. (14), then the following
Properties 1 and 2 state that the family of GLMB distribu-
equation, Eq. (7.48) of [3], establishes the basic relationship
tionsisanexactclosed-formsolutionofthemultitargetBayes
between RFS theory and MTA theory:
filter. It follows that, when restricted to GLMB distributions,
RFStheory MTAtheory the labeled multitarget Bayes filter can be equivalently re-
placedbya filter onthe parametersalone—i.e.,bythe GLMB
f(Z|X)·f (X)δX = ℓ (α). (37)
zZ }| 0 { z Z}||X { filter (invented by Vo and Vo in 2011 [17], [16]):
Xα
...→p →p →p →...
Thatis,theprobability(density) f(Z) thatthemeasurement- k|k k+1|k k+1|k+1
set Z willbecollectedfrom n independenttracksisthesame The GLMB filter is not only provably Bayes-optimal but is a
thing as the total (unnormalized) likelihood of association truemultitargettracker. Thatis,becauseitexplicitlyaccounts
between Z and X. This demonstrates that (Remark 15, for target labeling (see Section XI-A), it inherently incorpo-
p. 173 of [3]): ratesaprovablyBayes-optimaltrack-managementscheme. A
• The MHT approach is a heuristic approximation. chapter-lengthdiscussion of the GLMB filter can be found in
For, the optimal approach to multitarget tracking is the [3]. See also [4] for a formulation based solely on p.g.fl.’s.
multitarget Bayes filter: What follows is a brief overview of the GLMB filter.
It is organized as follows: (a) labeled RFS’s; (b) GLMB
... → f (X|Z )→f (X|Z )
k|k 1:k k+1|k 1:k distributions;(c)theGLMBfilter;(d)acomparisonofMHT’s
→ fk+1|k+1(X|Z1:k+1)→... andtheGLMBfilter;and(e)adiscussionof“unlabeled”exact
closed-form filters.
At time t in the measurement-update step
k+1
f (X|Z ) → f (X|Z ), the multitarget
k+1|k 1:k k+1|k+1 1:k+1
A. Labeled RFS’s
likelihood function is L (X) = f(Z |X). More
Zk+1 k+1
importantly, the correct prior distribution f (X) in Eq. TracklabelinginanRFScontextwasfirstaddressedin1997
0
(37) is the predicted multitargetdistribution f (X|Z ). in [1], pp.135,196-197. However, the first implementations
k+1|k 1:k
Consequentlyanyapriorichoiceof f (X),suchasEq.(36), of RFS filters did not take track labels into account. Later
0
is a heuristic approximation. implementations,such as Gaussian mixtureCPHD filters ([3],
pp. 244-250), addressed labeling heuristically. The subject
XI. THEGLMBFILTER was not addressed systematically until the 2011 and 2013
“labeled RFS” (LRFS) papers [17], [16] by Vo and Vo.
Suppose,instead,that f (X) ischosennon-heuristically—
0
We address labeling via the following change of notation.
specifically, that it is a generalized labeled multi-Bernoulli
(GLMB) distribution (to be defined shortly). Let a GLMB Single-target states are now assumed to have the form ˚x =
distribution be denoted as f(X|p) where p is a parameter-
2Thecardinality distribution pΞ(n) andPHD DΞ(x) ofRFS Ξ are:
vector (also to be defined shortly).
ThethreemostimportantpropertiesofGLMBdistributions pΞ(n) = fΞ(X)δX
are as follows: Z|X|=n
1) Ifthepreviousposteriordistribution f (X|Z ) isa DΞ(x) = fΞ({x}∪X)δX.
k|k 1:k Z
GLMB distribution f(X|p ) then so is the predicted
k|k
7
(x,ℓ) ∈ ˚X where x is a conventional target state (e.g., C. The GLMB Filter
kinematic variables and, if appropriate, target type) and ℓ is The following intuitive overview of the GLMB filter is
a track label.3 The integral on ˚X is defined by adapted from Section 15.4.2 of [3]. Suppose that:
1) Every label ℓ in L has the form ℓ = (k,i)
f˚(˚x)d˚x= f˚(x,ℓ)dx, (38)
Z Z where tk with k ≥ 0 is the time that the track
Xℓ∈L was created; and where integer i ≥ 1 distinguishes
where f˚(x,ℓ)dx=0 forallbutafinitenumberof ℓ. The the track from any other track created at time tk. Let
correspoRnding set integral is L0:k ={0,1,...,k}×{1,...} be the set of all possible
labels for targets existing at time t .
k
f˚(X˚)δX˚ (39) 2) ThelabeledmultitargetMarkovdensities f˚ (X˚|X˚′)
k+1|k
Z
have the following form (see Section 15.4.7 of [3]
1
= f˚({˚x ,...,˚x })d˚x ···d˚x (40) for more details). (a) Persisting targets are governed
1 n 1 n
nX≥0n!Z by the labeled version of the standard multi-Bernoulli
1 motion model where, in particular, the single-target
= (41) labeled Markov density has the form f(x,ℓ|x′,ℓ′) =
n!
nX≥0 (ℓ1,..X.,ℓn)∈Ln δℓ,ℓ′ ·f(x|x′)—i.e., every track retains its label during
· f˚({(x ,ℓ ),...,(x ,ℓ )})dx ···dx . a time update. (b) The target-appearancedistribution is
1 1 n n 1 n
Z GLMB distribution with |O|=1.
3) The multitarget likelihood functions L (X˚) =
Z
Now let f (Z|X˚) are the labeled versions of the standard
k
X˚={(x ,ℓ ),...,(x ,ℓ )} (42) multitarget likelihood functions (see Section 15.4.5 of
1 1 n n
[3] for more details).
be a finite subsetof ˚X. Theset of labelsof the targetsin X˚ 4) Theinitialdistribution f˚ (X˚) isaGLMBdistribution.
0|0
is denoted as
Let A denote the set of MTA’s α : L →
X˚L ={ℓ1,...,ℓn}. (43) {0,1,...,|ZZj|}. Abbreviate A = A ×..j. ×A 0:j and
j Z1:k Z1 Zk
Given this, X˚ is a labeled multitarget state-set if |X˚L| = α1:k =(α1,...,αk). Then the following are true:
|X˚|—i.e.,ifitselementshavedistinctlabels. AnRFS ˚Ξ⊆X 1) Let Z1:k−1 : Z1,...,Zk−1 be the time-sequence of
is a labeled RFS (LRFS) if measurement-setsattime tk−1. Thenthe time-updated
distribution at time t is GLMB of the form:
k
|˚ΞL|=|˚Ξ| (44)
f˚ (X˚|Z ) (47)
k|k−1 1:k−1
for all realizations ˚Ξ = X˚ of ˚Ξ. The distribution of an
|LX˚RLF|S6=˚Ξ|X˚|m.uTsthuhsa,vefotrheexfaomllpolwe,inagPporiospsoenrtyR:FSf˚Ξ(˚ΞX˚)of=˚X0 iisf = δ|X˚|,|X˚L|α1:k−1X∈AZ1:k−1ωαk|1k:k−−11(X˚L)·(˚sαk|1k:k−−11)X˚
not an LRFS. where, if X˚L = {ℓ1,...,ℓn} with |X˚L| = n then
ωk|k−1({ℓ ,...,ℓ }) is the weight of the hypothesis:
α1:k−1 1 n
B. GLMB Distributions a) there are n tracks with distinct labels ℓ ,...,ℓ ;
1 n
A multitarget probability distribution f˚(X˚) on ˚X is a and
b) their respective track distributions
generalized labeled multi-Bernoulli (GLMB) distribution if it
are sk|k−1 (x) = ˚sk|k−1(x,ℓ ),...,
has the following distribution and p.g.fl [17], [16]: ℓ1,α1:k−1 α1:k−1 1
sk|k−1 (x)=˚sk|k−1(x,ℓ ); and
f˚(X˚) = δ|X˚L|,|X˚| ωo(X˚L) ˚so(x,ℓ) (45) c) thℓens,αe1:dk−is1tributionsα1a:kr−o1se asna consequence of the
oX∈O (x,Yℓ)∈X˚ time-history α1:k−1 of MTA’s; and
G˚ [˚h] = ω (L) ˚h(x,ℓ)·˚s (x,ℓ)dx(46) d) for each i = 1,...,n, the track distribution
f˚ o Z o sk|k−1 (x) will be of two types:
oX∈OLX⊆L ℓY∈L ℓi,α1:k−1
i) the distribution of a track that persisted from
where (a) O is a finite set of indices o; (b) so,ℓ(x) = the previoustime t , and which thus arose
k−1
˚so(x,ℓ) with so,ℓ(x)dx = 1 for each o,ℓ is the from the previous time-history α ; or
1:k−1
track distributionRcorresponding to the track label ℓ and the ii) thedistributionofanewly-appearingtrack,and
index o; (c) ωo(L) ≥ 0 for all finite L ⊆ L; and which therefore does not depend on α .
1:k−1
(d) ω (L) = 1. The last condition implies
o∈O L⊆L o 2) Let Z1:k : Z1,...,Zk be the time-sequence of
that Pωo(LP) > 0 for only a finite number of pairs o,L. measurement-sets at time t . Then the measurement-
Note that f˚(X˚) = f˚(X˚|p) where the parameter-vector is k
updated distribution at time t is GLMB of the form:
k
p=(ωo(L),so,ℓ(x))o∈O,ℓ∈L∈L.
f˚ (X˚|Z ) (48)
k|k 1:k
ho3oTdsh,eℓsZy|mXb(oαl),“bℓu”tihsarseppurrepvoioseudslyhebreeetnoudseendotteoldabenelost.e Tashseocmiaetaionninglikweliil-l = δ|X˚|,|X˚L| ωkα|1k:k(X˚L)·(˚skα|1k:k)X˚
always beclear fromcontext. α1:kX∈AZ1:k
8
where, if X˚L = {ℓ1,...,ℓn} with |X˚L| = n, then rather than GLMB LRFS’s ˚Ξ of ˚X with p.g.fl.’s as in Eq.
ωkα|1k:k({ℓ1,...,ℓn}) is the weightof the hypothesisthat: (46). Here O is an index set, D(x) and so,i(x) are
a) there are n tracks with distinct labels ℓ1,...,ℓn; respectively a PHD and a spatial distribution on X, νo ≥0
and is an integer, 0 ≤ qo,i ≤ 1, and ωo ≥ 0 with oωo = 1.
b) their respective track distributions are given by ThePoissonfactor eD[h−1] isaheuristicmodeloPf“unknown
sk|k (x) = ˚sk|k (x,ℓ ), ..., sk|k (x) = targets” (i.e., undetected target births [22, Def. 1]).
ℓ1,α1:k α1:k 1 ℓn,α1:k Unlabeled exact closed-form filters are both theoretically
˚sk|k (x,ℓ ); and
α1:k n and practically redundant. They are inherently inferior to the
c) these distributions arose as a consequence of the
GLMB filter because they are not true multitarget trackers.
time-history α of MTA’s; and
1:k Furthermore,labelingpermitsabigdecreaseincomputational
d) for each i = 1,...,n, the track distribution
complexity in the prediction step of the GLMB filter [16].
sk|k (x) will be of two types:
ℓi,α1:k Thisdecreaseisunavailableforunlabeleddistributions,andin
i) the distributionofa trackthatwasnotdetected
particular for those as in Eq. (49). Indeed, severe and purely
and which therefore arose from the previous
ad hoc approximations [22, Eqs. (61,73)], [21] are necessary
time-history α ; or
1:k−1 to address this difficulty—thereby inviting skepticism about
ii) thedistributionofatrackthatwasdetectedand
the “exactness” part of any “exact closed-form” claim.
which therefore arises from the current time-
The “unknown targets” model in Eq. (49) is theoretically
history α .
1:k questionable. By implication, the “known targets” must be
modeled by the summation in Eq. (49). Given this, Eq. (49)
D. The GLMB Filter and MHT implies that the unknown-target RFS and known-target RFS
Like many MHT-type algorithms, the GLMB filter propa- are statistically independent—an impossibility, since the two
gatestime-historiesofMTA’s. Amajorconceptualdifference, are inherently correlated but the latter is non-Poisson.
however, is that in the GLMB filter an MTA α : L → Finally, the following claim must be addressed:
j 0:j
{0,1,...,|Zj|} inanMTAtime-sequence α1:k =(α1,...,αk) • “...[the 2013 Vo-Vo GLMB filter paper [16] ] shows that
is not a representation of the multitarget state. Rather, it is: the labelled case can be handled within the unlabelled
• an index of a weighted hypothesisabout(a) which labels framework by incorporating a label element in to the
exist in the scene; and, (b) which track distributions underlying state space” [22, p. 1675].
correspond to those labels. This assertion is manifestly untrue. An RFS filter on X
In particular, no attempt is made to estimate the best MTA cannot be converted to an LRFS filter on ˚X simply by
at any given time-step. Rather, the GLMB filter estimates substituting (x,ℓ) whenever x occursinthefilterequations.
the best state-set using an approximation of a Bayes-optimal Thisisbecausesuchsubstitutionsdonotforbidstate-setswith
multitarget state estimator. non-distinctlabels—i.e., |X˚|>|X˚L| becomespossible. For
The baseline computationalcomplexity of the GLMB filter example, Poisson RFS’s on ˚X are not LRFS’s—with the
isroughlythesameasthatoftrack-orientedMHT: itiscombi- consequencethat the factor eD[h−1] in Eq. (49) is inherently
natorialinboth m (thecurrentnumberofmeasurements)and non-LRFS. Thus it is not possible to use Eq. (49) as the
n (the current number of tracks). However, one can greatly basis of a theoretically rigorous LRFS filter—and thereby of
decrease complexity using statistical sampling methods. The a theoretically rigorous true multitarget tracker.
Gibbs sampler is a computationally efficient special case
of the Metropolis-Hasting MCMC algorithm which, in this XII. CONCLUSIONS
application, has an exponential convergence rate. Vo and
This paper has addressed the following claims made about
Vo have used it—together with a merging of the time-update
multi-hypothesistrackers(MHT’s),aswellasthefundamental
pk|k → pk+1|k and measurement-update pk+1|k →pk+1|k+1 conceptuponwhich they are based, the measurement-to-track
into a joint update pk|k → pk+1|k+1— to devise an im- association (MTA):
plementation of the GLMB filter with computational order
1) Claim 1: MHT’s are not only theoretically rigorous,
O(mn2) [19]. This results in an at least two orders of
buttheoreticallyrigorouswithintheBayesianframework
magnitude computational improvement, as compared to the
(“Bayes-optimal”).
original GLMB filter implementation described in [20].
2) Claim 2: “The RFS model of the multiple target state
isanapproximation,becausetheBayesposteriorRFSis
E. “Unlabeled”Exact Closed-Form Filters not exact, but is an approximation based on the earlier
After the GLMB filter was introduced in 2011 [17], a invocations of the PHD approximation used to close
few authors began investigating unlabeled exact closed-form the Bayesian recursion. The Bayes posterior RFS is an
filters. First and most notably, [22, Thms. 1,2] employed approximation even before the PHD approximation is
“hybrid Poisson and multi-Bernoulli” RFS’s Ξ of X with invoked. Therightmodelofthemultitargetstate isthat
p.g.fl.’s of the form used in the multi-hypothesis tracker (MHT) paradigm,
not the RFS paradigm.”
νo
G [h]=eD[h−1] ω (1+q s [h−1]), (49) 3) Claim 3: The RFS approach—and specifically the
Ξ o o,i o,i
oX∈O iY=1 generalizedlabeledmulti-Bernoulli(GLMB)filterofVo
9
and Vo [17], [16], [20]—is essentially a mathematically [12] D.B.Reid,“AnAlgorithmforTrackingMultipleTargets,”IEEETrans.
obfuscated reinvention of MHT. Auto.Contr.,24(6):843-854,1979.
[13] Ristic,B.,ParticleFiltersforRandomSetModels,Springer,NewYork,
Claim 1 can be ascribed to uncritical acceptance of unex- 2013.
aminedconventionalwisdom. Specifically,inthispaperithas [14] H. van Trees, Detection, Estimation, and Modulation Theory, Part I:
Detection, Estimation, and Linear Modulation Theory, John Wiley &
been demonstrated that:
Sons,NewYork,1968.
1) MTA’s are not phenomenologically real. Rather, they [15] B.-N. Vo, M. Mallick, Y. Bar-Shalom, S. Coraluppi, R. Osborne III,
are purely mathematical entities arising from a change R. Mahler, and B.-T. Vo, “Multitarget Tracking,” in J. Webster (ed.),
Wiley Encyclopedia of Electrical and Electronics Engineering, Wiley,
of notation in the RFS derivation of the multitarget
NewYork,2015.
likelihood function for the “standard” multitarget mea- [16] B.-T. Vo and V.-N. Vo, “Labeled random finite sets and multi-object
surement model. conjugate priors,”IEEETrans.Sign.Proc.,61(13):3460-3475,2013.
[17] B.-T. Vo and B.-N. Vo, “A random finite set conjugate prior and
2) The MHT/MTA approach is neither theoreticially rig-
applicationtomulti-targettracking,”Proc.2011Int’lConf.onIntelligent
orous nor strictly Bayesian. It is, rather, an intuitive- Sensors, Sensor Networks, and Information Processing (ISSNIP2011),
heuristic approximation of the Bayes-optimal approach Adelaide, Australia, 2011.
[18] B.-N. Vo, B.-T. Vo, and D. Clark, “Bayesian multiple target filtering
to multitarget tracking, the multitarget Bayes filter.
usingrandomfinitesets,”Chapter3inMallick,M.,Krishnamurthy,V.,
3) TheGLMBfilter,likeMHTalgorithms,employsMTA’s. and Vo, B.-N., (eds.), Integrated Tracking, Classification, and Sensor
Unlike them, however, it is a provably Bayes-optimal Management, Wiley,NewYork,2013.
[19] B.-N.Vo,B.-T.Vo,andHungGiaHoang,“Anefficientimplementation
exact closed-form solution of the labeled multitarget
of the generalized labeled multi-Bernoulli filter,” IEEE Trans. Sign.
Bayes filter, which can be considerably faster than Proc.,accepted forpublication.
conventionalcombinatorial algorithms. [20] B.-N. Vo, B.-T. Vo, and D. Phung, “The Bayes multi-target tracking
filter,”IEEETrans.Sign.Proc.,62(24):6554-6567, 2014.
As for Claims 2 and 3, they appear attributable to a [21] J. Williams, “Hybrid Poisson and multi-Bernoulli filters,” Proc. 15’th
superficial understanding of the finite-set statistics literature. Int’lConf.onInformation Fusion,Singapore, July9-12,2012.
[22] J.Williams,“Marginalmulti-Bernoullifilters: RFSderivationofMHT,
For example, the first sentence of Claim 2 repeats a common
JIPDA, and association-based MeMber,” IEEE Trans. Aerospace &
misconception: that the “RFS modelof the multitargetstate” Electronic Systems,51(3):1664-1687.
is the same thing as the “PHD approximation” of that state.
To the contrary, the actual “RFS model of the multitarget
state” is theevolvingrandomfinite multitargetstate-set Ξ .
k|k
The PHD filter results when we assume that Ξ is,
k+1|k
approximately and for every k ≥0, a Poisson RFS ΞPoiss .
k+1|k
This ΞPoiss isanapproximation,notamodel. Furthermore,
k+1|k
itisonlythesimplestofaseriesofincreasinglymoreaccurate
RFS approximations: i.i.d.c., multi-Bernoulli, labeled multi-
Bernoulli,andgeneralizedlabeledmulti-Bernoulli[9],[2],[3].
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