Engineering Statistics forMulti-Object Tracking

Ronald MahlerLockheed Martin NE&SS Tactical Systems

Eagan, Minnesota, USA651-456-4819 / ronald.p.mahler@lmco.com

IEEE Workshop on Multi-Object TrackingMadison WI

June 21, 2003

This work was supported by the U.S. Army Research Office under contracts DAAH04-94-C-0011 and DAAG55-98-C-0039, and by the U.S. Air Force Office of Scientific Research under contract FF49620-01-C-0031. The content does not necessarily reflect the position or the policy of the Government. No official endorsement should be inferred.

Short Tutorial

Top-Down vs. Bottom-Up Data Fusion

usual “bottom-up” approach to data fusion

diversedata

ill-characterizeddata data association

tracking

I.D.

datafusion

heuristically integrate apatchwork of optimal orheuristic algorithms thataddress specific functions(mostly done by operators)

“top-down,” system-level approach

multiple sensors, multipletargets = single system

specify states of system= multisensor-multitarget

Bayesian-probabilisticformulation of problem

reformulate in “Statistics 101,“engineering-friendly” terms

requires random set theory

low-level functions subsumed in optimally integrated algorithms

fuzzy logic

detection

sensor mgmt

• Systematic probabilistic foundation for multisensor, multitarget problems • Preserves the “Statistics 101” formalism that engineers prefer• Corrects unexpected difficulties in multitarget information fusion• Potentially powerful new computational techniques (1st-order moment filtering)• Unifies much of multisource, multitarget, multiplatform data fusion

detection

tracking

identification

performance estima- tion

information theory

control theory

sensor manage- ment

group targets

Bayes

fuzzy logic

Dempster- Shafer

rules

Our Topic: “Finite-Set Statistics” (FISST)

FISST

unified Bayesiandata fusion

MATHEMATICSOF

DATA FUSION

I.R. Goodman, Ronald. P.S. Mahlerand Hung T. Nguyen

Kluwer Academic Publishers

Theory and Decision Library

Random SetsTheory and Applications

John Goutsias, Ronald. P.S. MahlerHung T. NguyenEditors

Springer

SPIE Milestone Series Volume MS 124

Volune 97

ATR Working GroupUSAF Correlation Symp.SPIE AeroSense Conf.Nat'l Symp. on Data FusionIEEE Conf. Dec. & Contr.Optical Discr. Alg's Conf.ONR Workshop on TrackingU. Wisconsin

Invited PresentationsScientific Workshops1995 ONR/ARO/LM WorkshopInvited Session: SPIE AeroSense'99

An Introduction toMultisource-Multitarget Statistics

and its Applications

Ronald MahlerMarch 15, 2000

book (out of print)hardcover proceedings monograph

HANDBOOKOF

MULTISENSORDATA FUSION

Edited byDAVID L. HALLJAMES LLINAS

book chapter 14

MDA/POETAFITNRaDHarvardJohns HopkinsU. MassachusettsNew Mexico StateIDC2002

Lockheed Martin

FISST-Related Books and Monographs

finite-setstatistics

scientificperformanceestimation forLevel 1 fusion

robust SAR ATR against

ground targets

unifiedcollection& controlfor UAVs

BMD technology

BMD technology

anti-air multisourceradar fusion

robustINTELL fusion

scientificperformance

estimation for Levels 2,3,4 data fusion

robust joint tracking and NCTI using HRRR

and track radar

basic researchin data fusion,

tracking, & NCTI

AFRLAFRL

AFRL

AFRLAFOSR

MRDEC

LMTS

MDA

MDA

ARO

robust multitargetASW /ASUW

fusion & NCTI

LMTS

= basic research project = applied research project

Current FISST Technology Transition= completed research project

robust ASWacoustictarget ID

SSC/ONR

AFOSR = Air Force Office of Scientific ResearchAFRL = Air Force Research LaboratoryARO = Army Research OfficeLMTS = Lockheed Martin Tactical SystemsMDA = Missile Defense AgencyMRDEC = Army Missile Research, Dev &Eval CenterONR = Office of Naval ResearchSSC = SPAWAR Systems Center

Topics

1. Introducing “Dr. Dogbert,” Critic of FISST

2. What is “engineering statistics”?

3. Finite-set statistics (FISST)

4. Multitarget Bayes Filtering

5. Brief introduction to applications

multitarget calculusmultitarget statisticsmultitarget modeling

Dr. Dogbert, Critic of FISST

What is extraordinary aboutthis assertion is not the

ignorance it displaysregarding FISST but, rather,

the ignorance it displaysregarding Bayes’Rule!

The multisource-multitarget engineering problems addressed by FISST actually require nothing more complicated than Bayes’ rule…

…which means that FISST is of meretheoretical interest at best. At worst,it is nothing more than pointlessmathematical obfuscation.

The Bayesian Iceberg

Bayes’ rule

non-standard application

Nothing deep here!

The optimality and simplicity of the Bayesian framework can be taken for granted only within the confines of standard applications addressed by standard textbooks. When one ventures out of these confines one must exercise proper engineering prudence—which includes verifying that standard textbook assumptions still apply. This is especially true in multitarget problems!

How Multi- and Single-Object Statistics Differ

Bayes’ rule

I sit astride amountaintop!

multitarget maximum likelihood estimator (MLE) is defined but multitarget

maximum a posteriori(MAP) estimator is not!

multitarget Shannonentropy cannot be

defined!

need explicit methods for constructing provably true

multisensor-multitargetlikelihood functions fromexplicit sensor models!

need explicit methods forconstructing provably true

multitarget Markov transition densities fromexplicit motion models!

new multitarget state estimators must be defined

and proved optimal!

multitarget expected values are not defined!

multitarget L2

(i.e., RMS distance)metric cannot be

defined!

Dr. Dogbert (Ctd.)

I, the great Dr. Dogbert , have discovered a powerful new research strategy! Strip off the mathematics that makes FISST systematic, general, and rigorous, copy its main ideas, change notation, and rename it Dogbert Theory!

a mere change of name and notation does not constitute

a great advance.

It is easy to claim invention of a simple and yet elastically all-subsuming theory if

one does so by neglecting, avoiding, or glossing over the specifics required to actually substantiate puffed-up boasts

of generality and “first principles”.

This is a great advance because it’s much simpler: It doesn’t need all that ugly math (Blech!) that no one understands anyway!

Dogbert Theory is straightforward precisely because it avoids specifying explicit general methods and found-ations—and yet is still fully general and derived from first principles!

• Engineering statistics, like all engineering mathematics, is a tool and not an end in itself. It must have two (inherently conflicting) characteristics: – Trustworthiness: Constructed upon systematic, reliable mathematical

foundations, to which we can appeal when the going gets rough. – Fire and forget: These foundations can be safely neglected in most

applications, leaving a parsimonious and serviceable mathematical machinery in their place.

What is “Engineering Statistics”?

• The “Bar-Shalom Test” defines the dividing line between the serviceable and the simplistic:– “Things should be as simple as possible—but no simpler!”

– Y. Bar-Shalom, quoting Einstein

• If foundations are:– so mathematically complex that they cannot be taken for granted in

most situations, then they are shackles and not foundations!– so simple that they repeatedly lead us into engineering blunders, then

they are simplistic, not simple!

Single-Sensor / Target Engineering Statistics

• Progress in single-sensor, single-target applications has been greatly facilitated by the existence of a systematic, rigorous, and yet practical engineering statistics that supports the development of new concepts in this field.

zx

observations

states

Zk = hk(x)+Wk

Xk+1= k(x)+Vk

sensor measurement model

target Markov motion model

pk(S)=Pr(ZkS)pk+1|k(S)=Pr(Xk+1S)

probability mass function

probability mass function

dxintegro-differentialcalculus

d

E[Z]E[X]

expectations &other moments

||z1-z2||||x1-x2||

miss distance

miss distance

^ xk|kBayes-optimalstate estimators

Bayes-optimal filtering

fk|k(x|Zk)fk+1|k(x|Zk)

posterior & prior densities

fk(z|x)fk+1|k(y|x)

likelihood function

Markov transition density

ZX

randomobservation

vector

randomstate vector

What is Single-Target Bayes Statistics?

Classical statistics: unknown state-vector x of the target is a fixed parameter

Bayesian statistics: unknown state is always a random vector X even if we do not

explicitly know either it or its distribution f0(X) (the prior). The

single-sensor, single-target likelihood function is a cond. prob. density

* equalities temporarily assume discrete state & observationspaces for ease of argument

f0(x) = Pr(X = x)

X

state space, X

randomstate-vector

prior distribution* of X

randomobservation-vector

observation space, Z

Z

Lz(x) = f(z|x) = Pr(Z=z|X=x)sensor likelihood function*

Single-Target Bayes Statistics: Distributions

specify single-target state space and its measure structureXZ

specify single-sensor/target measurement space and

its measure structurerandom variables

(e.g. random vectors)

f(z|x) = Pr(Z=z|X=x)signifies a random draw from a single measurement space signifies a random draw

from a single state-space

f0(x) = Pr(X=x)

interim targetstate-change

Xk|k=x

target

random target state

predictedrandom state

1. time-update step via prediction integral

The Recursive Bayes Filter

Xk+1|k=y

k

true Markov transition density

Xk+1|k = k(x)+ Vk (motion model)

fk+1|k(y|x) = fV (y- k(x)) k

fk+1|k(y|Zk) = fk+1|k(y|x) fk|k(x|Zk)dx

prediction of posterior

to time of new datum zcurrent posterior, conditioned

on old data-stream

Zk = {z1,…, zk}

assuming fk+1|k(y|x,Zk) = fk+1|k(y|x)

The Recursive Bayes Filter

sensor

zk+1observation

2. data-update step via Bayes’ rule

k

Xk+1|k

Xk+1|k+1

data-updatedrandom state

predictedrandom state

true likelihood function

Zk = hk(x)+ Wk (measurement model)

fk(z|x) = fW (z- hk(x)) k

fk+1|k+1(x|Zk+1) fk+1(zk+1|x) fk+1|k(x|Zk)

new posterior, conditioned

on all data Zk+1 ={z1,…, zk+1}prior = predicted

posterior

assuming fk+1(z|x,Zk) = fk+1(z|x)

Xk+1|k+1

Xk+1|k

xk+1|k+1 =^

current best estimate of xk+1

3. estimation step via Bayes-optimal state estimator

The Recursive Bayes Filter

argsupx fk+1|k+1(x|Zk+1)maximum a posteriori (MAP){

x fk+1|k+1(x|Zk+1)dxexpected a posteriori (EAP)

f(z|x)

without “true” likelihood forboth target &background,Bayes-optimal claim is hollow

“over-tuned” likelihoods may benon-robust w/r/tdeviations between model & reality

sensor models

Basic Issues in Single-Target Bayes Filtering

poor motion-model selection leads to performance degradation

without “true” Markov density, good model selection is wasted

motion models fk+1|k(x|y)

without guaranteedconvergence, approx.filter may be unstable, non-robust

poorly chosen state estimators may have hidden inefficiencies (erratic, inaccurate, slowly-convergent, etc.)

state estimation/convergence

xk|k^

filtering equations are computationally nasty

“textbook” approaches create computationalinefficiencies: e.g,numerical instabilitydue to central finite-difference schemes

comput-ability fk|k(y|Zk)

Bayes-Optimal State Estimatorspre-specifiedcost function

C(x,y) x(z1,…, zm)^state estimator: familyof state-valued func-tions of the obser-vations z1,…, zm, m>0

state estimator x is “Bayes-optimal” with respect to

cost C if it minimizes the Bayes risk

^

without a Bayes-optimal estimator, we aren’t Bayes-optimal!

RC(x,m) C(x, x(z1,…, zm)) f(z1,…, zm |x) f0(x)dz1 dzmdx^^

Bayes risk = expected value of the total posterior cost

True Likelihood Functions & Markov Densities

Step 1:construct

model

Step 2:use probability-mass function

to describe model statistics

Step 3:construct “true”density function

that faithfullydescribes

model

Bayesian modelingOBSERVATIONS

measurement model

randomobser-vation

deterministicmeasurement

model

sensornoise

process

Zk = hk(x) + Wk

probability-mass function

pk(S|x) = Pr(Zk S)= probability that observation is in S

true likelihood function

fk(z|x) = pk

z= Radon-Nikodým derivative of pk

motion model

deterministicmotion model

randomstate of

moved target

motionnoise

process

Xk+1|k = k+1|k(x) + Vk

MOTION

probability-mass function

= probability that moved target is in S

pk+1|k(S|x) = Pr(Xk+1|k S)

true Markov transition density

fk+1|k(y|x) =pk+1|k

y= Radon-Nikodým derivative of pk+1|k

Finite-Set Statistics (FISST)

• One would expect that multisensor, multitarget applications rest upon a similar engineering statistics

• Not so, despite long existence of point process theory (PPT) • FISST is in part an “engineering friendly” version of PPT

ZX

multisensor-multitarget observations

multitarget states

randomobservation-

set

randomstate-set

? k = hk(X) Ck

k+1= k(X)Bk

multitarget measurement model

multitarget Markov motion model

? k(S)=Pr(kS)k+1|k(S)=Pr(k+1 S)

belief-mass function

belief-mass function

?XFISST integro-

differential calculus

fk(Z|X)fk+1|k(Y|X)multitarget Markov

transition density

? fk|k(X|Z(k))fk+1|k(X|Z(k))multitarget posterior &

prior densities

Bayes-optimal filtering

?

Bayes-optimalmultitarget state estimators

^ Xk|k

D(z)D(x)PHDs &

other moments

??multi-observation

miss distance

d(Z1,Z2)d(X1,X2)multitarget

miss distance

FISST, Isystematic, probabilistic framework for modeling uncertainties in data

statistical

random setmodel

common probabilistic framework: random closed sets,

radar report

likelihood function, f(z|x)

1)|( zxz df

generalized likelihood function, (z|x)

zxz d)|(

random setmodel

fuzzy

English-languagereport

imprecise datalinkattribute

contingent rule

random setmodel

poorly character-ized likelihood

Observation = “Gustav is NEAR the tower”

interpretations of “NEAR”

p1

p2

p3

p4

(random set model of natural language statement)

probabilitiesfor each interpretation

Dealing With Ill-Characterized Data

FISST, II

sensors

targets

diverse observations

reformulate multi-object problem as generalized single-object problem

sensor”“meta-

“meta-observation”

“meta- target”

random

obser-vation-

set

random state-set

multisensor state

X*k = {x*1,…,x*s}

multisensor-multitargetobservation

Z = {z 1,…,z m}

multitarget state

X = {x1,…,xn}

FISST, III

single-sensor/target

sensortargetvector observation, zvector state, x

derivative, dpZ /dzintegral, f(x) dx

prob.-mass func., pZ(S)likelihood, fZ(z|x)prior PDF, f0(x)

information theoryfiltering theory

multisource-multitarget “Statistics 101”

meta-sensormeta-targetfinite-set observation, Zfinite-set state, X

set derivative, /Zset integral, f(Z) Z

belief-mass func., (S)multitarget likelihood, f(Z|X)multitarget prior PDF, f0(X)

multitarget information theorymultitarget filtering theory

multi-sensor/target

Almost-parallelWorlds

Principle (APWOP)

detection

tracking

identification

performance estima- tion

information theory

control theory

sensor manage- ment

group targets

Bayes

fuzzy logic

Dempster- Shafer

rules

FISST, IV

FISST

unified Bayesiandata fusion

systematic, rigorous foundation for:- ill-characterized data & likelihoods- multitarget filtering and estimation- multigroup filtering & sensor management- computational approximation

Classical statistics: unknown state-set X of the target is a fixed parameter

What is Bayes Multitarget Statistics?

Bayesian statistics: unknown state-set is always a random finite set even if we do not

explicitly know either it or its distribution f0(X) (the prior). The

multisensor-multitarget likelihood function is a cond. prob. density

* equalities temporarily assume discrete state & observationspaces for ease of argument

f0(X) = Pr( = X)

state space, X

randomstate-set

multitarget prior distribution* of

randomobservation-set

observation space, Z

multisensor-multitarget likelihood function*

LZ(X) = f(Z|X) = Pr(=Z|=X)

Multitarget Bayes Statistics: Distributionsmultitarget likelihoods and distributions must have a specific form

strictly speaking: re-writing f0(X) as a family of ordinary densities

(one for each choice of target number) is not Bayesian because then it does not describe a random draw from a single multitarget state-space:

,...),...,(,...,),(),(),( 10212

011

00

0 nnffff xxxxx

specify multitarget state space and its measure structure (e.g.: finite subsets of single-target state space). Subtleissues are involved here!

X*Z*specify multisensor-multitarget

measurement space and itsmeasure structure—e.g., finite

subsets of Z = Z1 Zs

union of the measurement spaces of the individual sensors)

random variables(e.g. random sets)

f(Z|X) = Pr(=Z|=X)signifies a random draw from a single multisensor, multi-target measurement space

signifies a random draw from a single multitarget state-space

f0(X) = Pr(=X)

Point Processes / Random Finite Setsa point process is a random finite multi-set

copies copies

11 ,...,,...,,...,1 n

nn

xxxxfinite multi-set(= set with repetition of elements)

)(...)(11 xx xx nn

Dirac mixture density

counting measure

)(...)(11 SS

nn xx

SdS yyxx )()(

)(...)( ),(),( 11TT

nn xx simple counting measure on pairs

),(...),( ),(),( 11xx xx nn

simple Dirac mixture density on pairs

)},(),...,,{( 11 nn xx finite set of pairs

Set Integral (Multi-Object Integral)

Set integral:

sum over all possiblenumbers of targets

sum over all multi-objectstates with n objects

nn

n ddffYYf yyyy 11

1 }),...,({)()(

Set Derivative (Multi-Object Derivative)

][][lim][

0

hGghGh

g

G

Frechét derivative of a functional G[h] (of a function-variable h(x)):

cts. linear, is ][hg

Fg

Functional derivative (from physics)

][][lim][

0

hGhGh

G

x

x

Dirac delta function

][)()(11

Sn

n

n

GSS

X1

xx xx

Set derivative of a set-function (S) = G[1S] :

X = {x1,…,xn}

Probability Law of a Random Finite Set

][E][ hhGprobability generating

functional (p.g.fl.)

X

X hhx

x)( h = bounded real-valued test function

)Pr(][)( SGS S 1belief-massfunction

probability that all targets are in S

Choquet capacity theorem

multitargetprobability

distribution(= Janossy densities)

xx

gg

GG

X = {x1,…,xn})"Pr(")( )( X

XXf

fk|k(X|Z(k))multitarget posterior

Multitarget Posterior Density Functions

fk|k(|Z(k)) (no targets)

fk|k({x}|Z(k) (one target)

fk|k({xx2}|Z(k)) ( two targets)

…

fk|k({xxn}|Z(k)) (n targets)

multitarget state

measurement-stream

Z(k)Z ,...,Zk

multisensor-multitarget measurements: Zk = { zzm(k)}

individual measurementscollected at time k

fk|k(X|Z(k))X = 1normality condition

fk|k(X|Z(k)) fk+1|k+1(X|Z(k+1))

randomobservation-

sets Z producedby targets

randomstate-set multi-target motion

state space

observation space

fk+1|k(X|Z(k))

five targets three targets

The Multitarget Bayes Filter

k|k k+1|k+1k+1|k

random

state-set

multitargetBayes filter

Zk+1

f(Z|X)

what does “true”multisensor-multitargetlikelihood even mean?

sensor models

how do we constructmultisensor- multitargetmeasurementmodels andmultisensor-multitargetlikelihoodfunctions?

Issues in Multitarget Bayes Filtering

multitarget motionmodels must accountfor changes andcorrelations in targetnumber and motions

how do we constructmultitarget motion models and true multitargetMarkov densities?

what does “true” multitarget Markovdensity even mean?

fk1|k(X|Y)motion models

multitargetanalogs of usualBayes-optimalestimators arenot even defined!

must define newmultitarget stateestimators andshow well-behaved

Xk|k

^state estimation

new mathematicaltools even moreessential than insingle-target case

fk|k(Y|Z(k))comput-ability

True Multitarget Likelihoods & Markov Densities

Step 1:construct

model

Step 2:use belief-

mass functionto describe

model statistics

Step 3:construct “true”density function

that faithfullydescribes

model

Bayesian modelingOBSERVATIONS

multitarget meas. model

k = Tk(X) Ck

randomobser-vation

target-generated

measurements

clutterprocess

belief-mass function

k(S|X) = Pr(k S)= probability that observation is in S

true likelihood function

)|()|( XZ

XZf kk

= set derivative of k

MOTION

multitarget motion model

random stateof old

targets

randomstate of

new targetsnew

targets

k+1|k = k+1|k(X) Bk

belief-mass function

= probability that moved target is in S

k+1|k(S|X) = Pr(k+1|k S)

true Markov transition density

= set derivative of k+1|k

)|()|( |1|1 X

YXYf kk

kk

state

no-target state position states (km)

posteriorprobability 1/2 f(x) = 1/4 km -1

0 1 2

Some problems go away if continuous states are discretized into cells, BUT: - approach is no longer comprehensive - power of continuous mathematics is lost

New multitarget state estimators must be defined and proved optimal, consistent, etc.!

Failure of Classical Bayes Estimators

simple posterior distribution

compare f() = 1/2 f(x) = 1/4 km -1

naïveMAP

naïveEAP

X f(X) X = F() + x f(x)dx

= 1/2 ( + 1 km) ?

}incommensurable!

SARfeatures

LROfeatures

MTIfeatures

ELINTfeatures

SIGINTfeatures

Application 1: Poorly Characterized Data

interpretationby operators

sent out on coded datalink

?alphanumeric formatmessages corrupted

by unknowablevariations

(“fat-fingering”,fatigue, overloading,

differences intraining & ability)

model “ambiguous data” as random closed subsets ofobservation space

Af XSAWvague imprecise contingent general(fuzzy) (Dempster- (rules) Shafer)

choose specificmodeling technique

Modeling Ill-Characterized Data

(|x) = Pr( x ) practical implementationvia “matching models”

CAUTION!

partiallyheuristic

single- or multi-target Bayes filters

(|x) generalized (i.e., unnormalized)likelihood function

CAUTION!

non-Bayesian

IMA/SAR/22.0F/29.0F/14.9F/ Y / - / Y / - / - / T / 6 / - //

SAR line from formatted datalink message

Generalized Likelihood of Datalink Message Line

convertfeatures tofuzzy sets

g1 g2 g3 g4 g5 g6 g7 g8 g9 g10 g11

compare tofuzzy templates

in databasef1,x f2,x f3,x f4,,x f5,x f6,x f7,x f8,x f9,x f10,x f11,x

compute attri-bute likelihoods

1 2 3 4 5 6 7 8 10 11

generalized likelihoodof SAR message line

SAR(SAR|x) = 1(x) 2(x)10(x) 11(x)

'' operation models statistical correlation intermediate between perfect correlation and independence

ab = a + b - abab

Application 2: Poorly Characterized Likelihoods

statistically uncharacterizable“extended operatingconditions” (EOCs)

wet mud

dents

irregular placement of standard equipment (e.g. grenade launchers)

turret articulations

non-standard equipment (e.g. hibachi welded on motor)

Modeling Ill-Characterized Likelihoods

nominal likelihoodfunction

)|()( czfc iiz

intensity value in pixel

likelihood value

z

best-guess error bar

somewhat hedged error bar

more hedged error bar

very hedged error bar

)()(

)()(1 cc

ccNzz

zz

random error bar onimage likelihood

(interval arithmetic)

random error bar onpixel likelihood

(random interval)

})()( cc i

ziz

))(Pr()(, cxxic zz

fuzzy error bar onimage likelihood

ordinarymetricsmultisensor-

multitargetalgorithm

mathematical information produced by algorithm(multitarget Kullback-Leibler)

miss distance

I.D. miss

track purity

multitarget missdistance

Application 3: Scientific Performance Estimation

Multitarget Miss Distance

ground truth targets, G

multi-target trackerdata

n tracks, X

Oliver Drummond et. al. have proposed optimal truth-to-track association for evaluating multitarget tracker algorithms

find best association xi gi

between truths gi and tracks xi

Wasserstein distance rigorously and tractably generalizes intuitive approach

“natural” approach works only when numbers of truths and tracks are equal:

iii

XGd

xg

1

1min),(

Application 4: Group-Target Filtering

group target 2 (tank brigade)

group target 1 (armored infantry)

individualtargets

detect (i.e., this target group is actually a group target)

track (e.g., this group target is

moving towards objective A)

classify (e.g., this group target is an armored brigade)

fk|k(Xk|Z(k)) fk+1|k+1(Xk+1|Z(k+1))multi-group

Bayes filter

randomobservation-

sets Z producedby targets ingroup targets

randomgroup-

set, k|k

multi-group motion

state space

observation space

fk+1|k(Xk+1|Z(k))

five groups three groups

Approximate 1st-Order Multi-Group Bayes Filter

randomgroup-set, k+1|k+1

1st-moment“group PHD”

filter

Dk|k(g,x|Z(k)) Dk+1|k(g,x|Z(k)) Dk+1|k+1(g,x|Z(k+1))

multiple sensors on multiple platforms

- data- attributes- language- rules

multisensor-multitarget observation

all sensors on all platforms regarded as single system

all targetsregardedas singlesystem (possiblyundetected &low-observable)

multitarget system

all observations regarded as single observation

predicted multitarget system

sensor and

platformcontrols

to bestdetect, track, &

identify alltargets

Application 5: Sensor Management

fk|k(X|Z(k)) fk+1|k+1(X|Z(k+1)) fk+1|k(X|Z(k))

Multi-Sensor / Target Mgmt Based on MHC’s

k|k k+1|k+1k+1|k

multitargetBayes filter

use multi-hypothesis correlator (MHC) as filter part of sensor mgmt

observation space

single-target state space

multitarget motion

*|kk *

|1 kk *1|1 kk

multisensor motion

MHC-likealgorithm

(step k|k)

prediction

MHC-likealgorithm(step k+1|k)

MHC-likealgorithm(step k+1|k+1)

data-update

optimize geometric O.F.:multisensor FoV & predicted tracks

),( *1

ik

iDp xx

FoV of ith sensor

multisensor FoV)),(1()),(1(1)( *

1*1

11 s

ksDkDD ppp xxxxx

optimize Csiszár information objective function

Summary / Conclusions

• “Finite-set statistics” (FISST) provides a systematic, Bayes-probabilistic foundation and unification for multisource, multitarget data fusion

• Is an “engineering friendly” formulation of point process theory, the recognized theoretical foundation for stochastic multi-object systems

• Has led to systematic foundations for previously murky problems: e.g. group-target fusion

• Is leading to new computational approaches, e.g. PHD filter; multistep look-ahead sensor management

• Currently being investigated by other researchers:– Defense Sciences Technology Office (Australia), U. Melbourne

(Australia), Swedish Defense Research Agency, Georgia Tech Research Institute, SAIC Corp.