Multi-target PHD filtering: proposition of extensions to the multi-sensor case

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the multi-sensor case

Emmanuel Delande, Emmanuel Duflos, Dominique Heurguier, Philippe Vanheeghe

To cite this version:

Emmanuel Delande, Emmanuel Duflos, Dominique Heurguier, Philippe Vanheeghe. Multi-target PHD filtering: proposition of extensions to the multi-sensor case. [Research Report] RR-7337, INRIA. 2010, pp.64. �inria-00501502v3�

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a p p o r t

d e r e c h e r c h e

N0249-6399ISRNINRIA/RR--7337--FR+ENG

Optimization, Learning and Statistical Methods

Multi-target PHD filtering: proposition of extensions to the multi-sensor case

Emmanuel DELANDE — Emmanuel DUFLOS — Dominique HEURGUIER — Philippe VANHEEGHE

N° 7337

July 2010

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Centre de recherche INRIA Lille – Nord Europe Parc Scientifique de la Haute Borne

EmmanuelDELANDE

∗

,EmmanuelDUFLOS

∗

, Dominique HEURGUIER

†

,Philippe

VANHEEGHE

∗

Theme: Optimization,LearningandStatistialMethods

Équipe-ProjetSequeL

Rapportdereherhe n°7337July201069pages

Abstrat: Common diulties in multi-targettrakingarise from thefat that thesystemstate and

theolletionofmeasurementsareunorderedandtheirsizeevolverandomlythroughtime. Therandom

nite set theory provides apowerfulframework toope withthese issues. This doument fouses more

partiularlyonthePHD(ProbabilityHypothesisDensity)lterproposedbyMahler.

Therst partof this report (up to setion 4) isa synthesis of Mahler'swork and aimsat providing

a thorough desriptionof the onstrution of thesingle-sensor PHD lter. Then, basedon afew leads

provided by Mahler, the seond part (from setion 5) proposes several extensions of this lter to the

multi-sensorase.

Key-words: Multi-target/Multi-sensorTraking,PHD,RandomFiniteSets

∗

LAGISFRECNRS3303-INRIALilleNordEurope(EPISequeL)

†

ThalesCommuniationsFrane

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Résumé: Lepistagemulti-iblesetrouveonfrontéaudoubleproblèmesuivant: l'étatdusystèmeet

laolletiondemesuresnesontpasordonnésetleursdimensionsvarientaléatoirementauoursdutemps.

Danseontexte,l'utilisationdesensemblesaléatoiresnisapporteunadrederésolutionpartiulièrement

pertinent et e travail s'intéresseplus partiulièrement au ltre PHD (Probability Hypothesis Density)

introduitparMahler.

Lapremièrepartie dee rapport (jusqu'à lasetion4) est une synthèse destravauxde Mahleret se

veutpédagogique : ellereprenden détaillaonstrutiondultrePHDmono-apteur. En sebasantsur

les éléments de solutionproposéspar Mahler, la deuxième partie (à partirde la setion 5) propose des

extensionsdultreauasmulti-apteur.

Mots-lés : PistageMulti-apteur/Multi-ible,PHD,EnsemblesAléatoiresFinis

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Contents

1 Introdution 5

2 Multi-sensor/multi-targetBayesian ltering 6

3 Finiteset statistisand generalizedFISST multi-target alulus 8

3.1 Multi-targetstatesandnitesets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.2 Probabilitygeneratingfuntionals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

3.3 Setderivativesandfuntionalderivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9

3.4 Someessentialpropertiesoffuntionalderivatives. . . . . . . . . . . . . . . . . . . . . . . . 9

3.5 Multitargetmomentdensities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

4 Single-sensor/multi-targetPHD ltering 12 4.1 Timeupdateequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

4.2 Single-sensordataupdateequation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

5 Extensionto the multi-sensor ase 17 5.1 Multi-sensordataupdateequation: derivativeform . . . . . . . . . . . . . . . . . . . . . . . 17

5.2 Multi-sensordataupdateequation: ombinationalexpression . . . . . . . . . . . . . . . . . 21

6 Simpliationof the multi-sensordata update equation 24 6.1 Simpliationbytargetspaepartitioning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

6.2 Approximationbyrestritinghypotheses. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

6.3 Approximationbysequentialltering. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.3.1 Produtapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

6.3.2 Myopisequentialapproximation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

6.3.3 Nonmyopisequentialapproximation. . . . . . . . . . . . . . . . . . . . . . . . . . . 36

6.3.4 Elementsofomparisonbetweenthethreeapproximationmethods . . . . . . . . . . 37

7 Conlusion 41 8 Mathematialproofs 42 8.1 Property(13) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.2 Property(14) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.3 Proposition3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

8.4 Theorem3.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

8.5 Proposition3.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

8.6 Theorem3.3. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.7 Theorem3.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44

8.8 Proposition4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

8.9 Theorem4.1. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

8.10 Proposition4.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

8.11 Proposition4.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

8.12 Proposition4.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.13 Theorem4.2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

8.14 Proposition5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

8.15 Proposition5.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

8.16 Lemma5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

8.17 Proposition5.3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58

8.18 Proposition6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

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8.19 Proposition6.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

8.20 Proposition6.4 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

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1 Introdution

ThisdoumentdealswiththeProbabilityHypothesisDensity(PHD) lterasasolutionformulti-target

Bayesianltering. Theaim of thisdoument is to desribethe settingof thesingle-sensor/multi-target

PHD lter as provided by Mahler, and to propose several paths for an extension to the multi-sensor

ase. Although thedesription ofthe single-sensorase (setions2, 3and 4)relies heavily on Mahler's

work on the topi ([1℄, [2℄), the author found nonetheless important to desribe it thoroughly sine it

providesgroundforthedisussionon themulti-sensorase. Pleastenote that thisworkmaybeseenas

unomprehensiveandmayontainmistakessineitmainlyreetstheauthor'smodestunderstandingof

Mahler'sworkonPHDltering.

Setion2providesthegeneralframeworkofmultisensor-multitargetBayesianltering,thatis,itdesribes

theproblem tosolve. Thedenition ofthePHDand abriefdesriptionofthe fundamental stepswhih

allowsthe onstrutionofthePHD-equivalent formulasoutof theBayesianequations arethen givenin

orderto providean insightonthe"logial ow"whihleadstheonstrutionofthesingle-sensor/multi-

targetPHDlter.

Setion 3desribessomebasisontheFiniteSetStatistis(FISST)alulusandgivessometoolswhih

will be requiredto "translate" the Bayesianformulas into their PHD-equivalent (setion4). Note that

some important mathematial proofs are given in setion 8. As other material in these rst setions,

it is strongly based on Mahler's work ([1℄, [2℄). However, the author did not fully grasp several proofs

givenin[1℄andfounditinterestingtoreformulatetheminorderimproveitsunderstandingonthetopi.

Moreover,inludingtheseproofsallowsthisdoumentto beasomprehensiveaspossible.

In setion 5, theextension of thesingle-sensor/multi-targetPHD equation to the multi-sensorase are

presented. Then, a few leads are disussed in order to simplify the Bayesupdate equations in amore

tratable form. Note that this report is arst versionand fouses on theoretial results. Works arein

progresstovalidatetheassumptionsontheproposed extensions.

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2 Multi-sensor/multi-target Bayesian ltering

Inthe generalmultisensor-multitargetBayesianframework,thetimeupdate and dataupdate equations

atstepk+ 1^are^given^by^the^following^formulas:

f_k+1|k(X|Z^(k)) = Z

f_k+1|k(X|W)f_k|k(W|Z^(k))δW ⁽¹⁾

f_k+1|k+1(X|Z^(k+1)) = fk+1(Zk+1|X)f_k+1|k(X|Z^(k))

Rfk+1(Zk+1|W)f_k+1|k(W|Z^(k))δW ⁽²⁾

where:

X ={x1, ..., xn} îsâmulti-targetstate, i.e. anite set ofelements xi ^dened ôn^thesingle-target spaeX^;¹

Zk+1 = {z1, ..., zm} ^is ^the ^urrent multi-sensor observation, i.e. a olletion of measurements zi

produedattimek+ 1^by^all^the^sensors;

Z^(k)=S

t6kZ_tîs^theôlletionôfobservationsuptotimek^;

f_k|k(W|Z^(k))^is^the^urrentmulti-targetposteriordensityinstateW^;

f_k+1|k(X|W)^is^the^urrentmulti-targetMarkovtransitiondensity,fromstateW ^to ^stateX^;

fk+1(Z|X)^is^the^urrentmulti-sensor/multi-targetlikelihoodfuntion.

Although equations (1) and (2) may seem similar to the lassial single-sensor/single-targetBayesians

equations,theyaregenerallyuntratablebeauseofthepreseneoftheset integrals. Integralsin(1)and

(2)areindeed omputedovermulti-targetsets ratherthansingle-targetstates. Beausemulti-targetsets

belongstomulti-targetspaeX=S

n>0Xⁿ^,^toômputeêvery^possible^setône^mustômputeêvery^possible

dimension(i.enumberoftarget)n^,ând ^forêah nômputeêvery^possiblen^-stateôlletion(x1, ..., xn)^.

Likewise, theonstrutionof themulti-targettransitionfuntion f_k+1|k ^or^themulti-sensor/multi-target likelihoodfuntion fk+1 ^may^be^tediousⁱⁿ^many^traking^problems.

Hene theintrodutionofthePHD:

Denition 2.1. ([1 ℄ p.1154) The Probability HypothesisDensity(PHD) isthe densityD_k|k(x|Z^(k)) ^de-

nedonsingle-targetstatespaeX ^whose ^integral R

SD_k|k(x|Z^(k))dxôn âny^regionS⊆ X îs^theêxpeted

numberN_k|k(S) =R

|X∩S|f_k|k(X|Z^(k))δX ^of^targets ^ontainedⁱⁿ ^S.

Although dened on single-state spae X^, ^the ^PHD enapsulates information on both target number and states and therefore providesa nie alternative to umbersome multitarget posteriorf_k|k(X|Z^(k))^.

Furthermore,shouldtheposteriorbeapproximatedasamultitargetPoissonasrequiredlater(seesetion

4.2), Mahler proved ([1℄, theorem 4 p.1166) that the best Poisson approximation - in an information-

theoreti sense-hasanintensityequaltoitsPHDD_k|k(x|Z^(k))^.

1

Thestatexiôfâ^targetîsûsuallyômposedôfîts^position,îts^veloity,êt.

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Assuming that propagating the PHD is suient enough for an aurate estimation of target number

and targetstates, thehallengeis to ndthePHD-equivalent ofBayesformulas(1) and(2) in order to

propagatedensitiesD_k+1|k(x|Z^(k))^andD_k+1|k+1(x|Z^(k+1))^rather^thanmulti-targetdensities. Essentially, theonstrutionof thePHDfollowsfromthefollowingpoints:

1. Themultitargetpriorf_k|k ân^be^seenâsâprobabilitydistributionfΞ ôfâ^random ^nite^set Ξ^;

2. Thanks to Finite Set Statistis (FISST) alulus, f_Ξ ^and ^its multimoment densities an be on- strutedassetderivatives ofitsprobability generating densityfuntional (PGFl)GΞ[h]^;

3. Under ertain assumptions - partiularly on the target motion model - the PGFl GΞ[h] ^an ^be

onstrutedexpliitly;

4. ThePHDD_k|k ^is^the^rst-moment^density^of^themultitargetpriorf_k|k^.

Therefore,underthesameassumptions,thePHDD_k|k ân^beônstrutedêxpliitly. ^Likewise,^the^multi-

targetBayesequationsanbe"translated"inexpliitPHD-equivalentformulas.

Althoughthelosed-formPHD-equivalentofthetimeupdateequation(1)requiresnostrongassumption

on the multitarget posterior(see setion 4.1), this is nottrue for the data update equation (1) , whih

requires the multitarget posterior to be approximately Poisson (see setion 4.2). But, even with this

assumption,thePHD-equivalentremainstratableinthesingle-sensoraseonly. Thatiswhyafewleads

aredisussedin setion5inorderto simplifytheBayesupdateequationinthemulti-sensorase.

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3 Finite set statistis and generalized FISST multi-target alulus

3.1 Multi-target states and nite sets

ThemultitargetstateX ={x1, ..., xn} ^an ^be^formulatedequivalentlybyapointproessΞ ^on^spaeX^.

If wefurther assumethatthe elementsxi âre^distints, ^then^the orrespondingsimple pointproess (or random nite set - RFS) Ξ îs equivalently desribed by the ounting measure NΞ(S) = |Ξ∩S| ôr^the

DiraδΞ(x) =P

w∈Ξδw(x)^. ^From^nowôn,^this âssumption^will ^beônsidered^valid ^so^that multi-target setsandtheposteriordensitiesanbedesribedusingFISSTalulus.

ARFSΞ^isharaterizedbythefamilyofitsJanossydensitiesjΞ,1(x1)^,jΞ,2(x1, x2)...^(see^[6℄^for^a^thorough

desription). Thesedensitiesaresymmetriin allarguments(sine elementsin Ξâreunordered),vanish whenevertwoelementsareequal(sine Ξîs^simple),ândâre^jointlynormalized:

X∞ n=0

1 n!

Z

j_Ξ,n(x1, ..., x_n)dx1...dx_n = 1 ⁽³⁾

The multitarget posterior f_k|k(X|Z^(k)) ân ^then ^be ^desribed ûsing ^the ^Janossy ^densities ôf ^the ôrre-

spondingRFSΞ_k|k^:

f_k|k({x1, ..., xn}|Z^(k)) =jΞk|k,n(x1, ..., xn) ⁽⁴⁾

Beause multi-targetdensities must be desribed as simple RFS in order to use the onvenient FISST

alulusrules asexplained later, an important onsequeneis that f_k|k({x1, ..., xn}|Z^(k)) = 0 ^whenever

twoelementsxi, xjâreêqual. ^Whether^this^might^haveûnwantedônsequenesôn^PHD^lteringⁱⁿêrtain

ases(e.g. verylosetargetsin spaeX⁾^is^aninterestingquestiontobeanswered.

3.2 Probability generating funtionals

Theprobabilitygeneratingfuntionals(PGFls)areageneralizationofthebelief-massfuntionsthat will

proveto beonvenienttoolstoomputebothmulti-targetposteriorsandPHDs.

Denition3.1. ([1 ℄p.1161) LetΞ^beâ^RFSônX ^with^densityfΞândhâreal-valuedfuntionsuhthat

∀x∈ X, 06h(x)61^. ^The ^PGFlGΞ ôf Ξîs^denedâs:

GΞ[h] = Z

h^XfΞ(X)δX ² ⁽⁵⁾

As noted in [3℄, the dention of the argument funtion h ^of ^PGFl G ^is ^simplied ^here. ^The ^gradient

derivation of PGFls in Dirafuntions (see denition 3.2) requires anextension of h âsâ^sumôf ^Dira

funtions; the proper denition is given in [1℄. We will ignore this tehniality here, beause the base

funtions hândg ^we^will^need ^to^derive^the^PHDêquivalentôf ^the^timeând ^dataûpdate êquationsâre

simplereal-valuedfuntionsasdened above.

Notethat thePGFlGΞ^is ^the^funtional generalizationofthebelief-masssetfuntion βΞ^sine:

∀S∈ X, βΞ(S) = Z

1^X_SfΞ(X)δX=GΞ[1S] ⁽⁶⁾

where 1S îs ^the îndiator ^funtion ôn S^. ^Mahler ^provides ^the ^following întuitive interpretation of this funtional. Ifh(x)îs^seenâs^the^detetionprobabilityofasensor'seldofviewonasingletargetspaeX^,

2h^X=Q

x∈Xh(x), h^∅= 1