Random Set/Point Process in Multi-Target Tracking Ba-Ngu Vo EEE Department University of Melbourne Australia SAMSI,

Random Set/Point Random Set/Point Process in Multi-Target Process in Multi-Target TrackingTracking Ba-Ngu VoBa-Ngu Vo

EEE Department EEE Department University of University of MelbourneMelbourneAustraliaAustraliahttp://www.ee.unimelb.edu.au/staff/bv/

SAMSI, RTP, NC, USA, 8 September 2008

Collaborators (in no particular order):

Mahler R., Singh. S., Doucet A., Ma. W.K., Panta K., Clark D., Vo B.T., Cantoni A., Pasha A., Tuan H.D., Baddeley A., Zuyev S., Schumacher D.

The Bayes (single-target) filterThe Bayes (single-target) filter

Multi-target trackingMulti-target tracking

System representationSystem representation

Random finite set & Bayesian Multi-target filteringRandom finite set & Bayesian Multi-target filtering

Tractable multi-target filtersTractable multi-target filters

Probability Hypothesis Density (PHD) filterProbability Hypothesis Density (PHD) filter

Cardinalized PHD filterCardinalized PHD filter

Multi-Bernoulli filterMulti-Bernoulli filter

ConclusionsConclusions

OutlineOutline

The Bayes (single-target) Filter The Bayes (single-target) Filter

state-vector

target motion

state space

observation space

xk

xk-1

zk-1

zk

fk|k-1(xk| xk-1)

Markov Transition Density Measurement Likelihood

gk(zk| xk)

Objective

measurement history (z1,…, zk)posterior (filtering) pdf of the state

pk(xk | z1:k)

System Model

state-vector

target motion

state space

observation space

xk

xk-1

zk-1

zk

Bayes filter

pk-1(xk-1 |z1:k-1) pk|k-1(xk| z1:k-1) pk(xk| z1:k)prediction data-update

pk-1(xk-1| z1:k-1) dxk-1

fk|k-1(xk| xk-1) gk(zk| xk)K-1 pk|k-1(xk| z1:k-1)


pk-1(. |z1:k-1) pk|k-1(. | z1:k-1) pk(. | z1:k)prediction data-update

Bayes filter

N(.;mk-1, Pk-1) N(.;mk|k-1, Pk|k-1) N(.;(mk, Pk )

Kalman filter

i=1

N{wk|k-1, xk|k-1} i=1

N(i) (i) {wk, xk } i=1 N(i) (i) {wk-1, xk-1}

(i) (i)Particle filter

state-vector

target motion

state space

observation space

xk

xk-1

zk-1

zk

fk|k-1(xk| xk-1)

gk(zk| xk)



observation produced by targets

target motion

state space

observation space

5 targets 3 targetsXk-1

Xk

Objective: Jointly estimate the number and states of targets

Challenges:

Random number of targets and measurements

Detection uncertainty, clutter, association uncertainty


System RepresentationSystem Representation

0

0

1

1

X

0

0

1

1

X

1

1'

0

0

X

1

1'

0

0

X

Estimate is correct but estimation error ???

TrueMulti-target state

EstimatedMulti-target state

|| ' || 2X X

How can we mathematically represent the multi-target state?

2 targets 2 targets

Usual practice: stack individual states into a large vector!

Problem:

Remedy: use( ')

min || ' || 0perm X

X X

1

1'

0

0

X

1

1'

0

0

X

True

Multi-target state

?X ?X

EstimatedMulti-target State

2 targetsno target

1

1'

0

0

X

1

1'

0

0

X

True

Multi-target state

0

0X

0

0X

EstimatedMulti-target State

2 targets1 target


What are the estimation errors?

Error between estimate and true state (miss-distance)

fundamental in estimation/filtering & control

well-understood for single target: Euclidean distance, MSE, etc

in the multi-target case: depends on state representation

For multi-target state:

vector representation doesn’t admit multi-target miss-distance

finite set representation admits multi-target miss-distance: distance between 2 finite sets

In fact the “distance”

is a distance for sets not vectors

( ')min || ' || 0

perm XX X ( ')

min || ' || 0perm X

X X


observation produced by targets

target motion

state space

observation space

5 targets 3 targetsXk-1

Xk

Number of measurements and their values are (random) variables

Ordering of measurements not relevant!

Multi-target measurement is represented by a finite set


RFS & Bayesian Multi-target FilteringRFS & Bayesian Multi-target Filtering

targets target set

observed set

observations

X

Z

Need suitable notions of density & integration

pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1) prediction data-update

1| 1 1: 1( | ) ( | )k k k k k k kK g Z X p X Z

1| 1 1: 1( | ) ( | )k k k k k k kK g Z X p X Z

Reconceptualize as a generalized single-target problem [Mahler 94]

Bayesian: Model state & observation as Random Finite Sets [Mahler 94]


S

N(S) = | S|

point process or random counting measure

random finite set or random point pattern

state space E

state space E

Belief “density” of

f : F(E) [0,)

(T ) = T f (X)X

Belief “distribution” of (T ) = P(T ) , T E

E

Probability density of p : F(E) [0,)

P (T ) = T p (X)(dX)

Probability distribution of P (T ) = P(T ) , T F(E)

F(E)

Collection of finite

subsets of E State space

Mahler’s Finite Set Statistics (1994)

Choquet (1968)

T T

Conventional integral Set integral

Vo et. al. (2005)

Point Process Theory (1950-1960’s)


x x’

X’

xdeath

creation

X’

x

spawn

motion

Multi-target Motion ModelMulti-target Motion Model

fk|k-1(Xk|Xk-1 )Multi-object transition

density

Xk = Sk|k-1(Xk-1)Bk|k-1(Xk-1)k

Evolution of each element x of a given multi-object state Xk-1

Multi-target Observation ModelMulti-target Observation Model

gk(Zk|Xk)

Multi-object likelihood

Zk = k(Xk)Kk(Xk)

x z

x

likelihood

misdetection

clutter

state space observation space

Observation process for each element x of a given multi-object state Xk


Computationally intractable in general

No closed form solution

Particle or SMC implementation

[Vo, Singh & Doucet 03, 05, Sidenbladh 03, Vihola 05, Ma et al. 06]

Restricted to a very small number of targets

)()|()|( 1:111| dXZXpXXf skkkkk )()|()|( 1:111| dXZXpXXf skkkkk

)()|()|(

)|()|(

1:11|

1:11|

dXZXpXZg

ZXpXZg

skkkkk

kkkkkkk

)()|()|(

)|()|(

1:11|

1:11|

dXZXpXZg

ZXpXZg

skkkkk

kkkkkkk

Multi-target Bayes FilterMulti-target Bayes Filter

Multi-target Bayes filter

Particle Multi-target Bayes FilterParticle Multi-target Bayes Filter

AlgorithmAlgorithm

for i =1:N, % Initialise => Sample: Compute:

end;normalise weights;for k =1: kmax ,

for i =1:N, % Update => Sample:

Update:end;normalise weights;resample;MCMC step;

end;

( )

( )1:

1

( | ) ( ),ik

Ni

k k k k kXi

p X Z w X

( )

( )1:

1

( | ) ( ),ik

Ni

k k k k kXi

p X Z w X

( )0 0 0~ ( )iX q X

( )0

( )0 0 0 0

1

( ) ( ),i

Ni

Xi

p X w X

( )0

( )0 0 0 0

1

( ) ( ),i

Ni

Xi

p X w X

( ) ( ) ( )0 0 0 0 0( ) ( )i i iw p X q X

( ) ( ) ( ) ( ) ( ) ( ) ( )1 | 1 1 1 1:( | ) ( | ) ( | , )i i i i i i i

k k k k k k k k k k k k kw w g Z X f X X q X X Z

( ) ( )1 1:~ ( | , )i i

k k k k kX q X X Z


Multi-target Bayes filter: very expensive!

single-object Bayes filter

multi-object Bayes filter

state of system: random vector

first-moment filter(e.g. -- filter)

state of system: random set

first-moment filter(“PHD” filter)

Single-object

Multi-object

)()|()|( 1:111| dXZXpXXf skkkkk )()|()|( 1:111| dXZXpXXf skkkkk

)()|()|(

)|()|(

1:11|

1:11|

dXZXpXZg

ZXpXZg

skkkkk

kkkkkkk

)()|()|(

)|()|(

1:11|

1:11|

dXZXpXZg

ZXpXZg

skkkkk

kkkkkkk

The PHD FilterThe PHD Filter

x0 state space

v PHD (intensity function) of a RFS

S

v(x0) = density of expected

number of objects

at x0

The Probability Hypothesis DensityThe Probability Hypothesis Density

v(x)dx = expected number

of objects in SS

= mean of, N(S), the

random counting

measure at S

The PHD FilterThe PHD Filter

state space

vk vk-1

PHD filter

vk-1(xk-1|Z1:k-1) vk(xk|Z1:k) vk|k-1(xk|Z1:k-1) PHD

prediction

PHD

update

Multi-object Bayes filter

pk-1(Xk-1|Z1:k-1) pk(Xk|Z1:k) pk|k-1(Xk|Z1:k-1)prediction update

Avoids data association!

PHD PredictionPHD Prediction

vk|k-1(xk |Z1:k-1) = k|k-1(xk, xk-1) vk-1(xk-1|Z1:k-1)dxk-1 k(xk) intensity from

previoustime-step

term for spontaneousobject births

= intensity of k

k|k-1(xk, xk-1) = ek|k-1(xk-1) fk|k-1(xk|xk-1) + k|k-1(xk|xk-1)

Markovtransitionintensity

probabilityof objectsurvival

term for objectsspawned by

existing objects= intensity of Bk(xk-1)

Markov transition density

predictedintensity

Nk|k-1 = vk|k-1 (x|Z1:k-1)dxpredicted expected number of objects

(k|k-1)(xk) k|k-1(xk, x)(x)dx k(xk)

vk|k-1 k|k-1vk-1

PHD UpdatePHD Update

vk(xk|Z1:k) zZk Dk(z) + k(z)

pD,k(xk)gk(z|xk) + 1 pD,k(xk)]vk|k-1(xk|Z1:k-1)

Dk(z) = pD,k(x)gk(z|x)vk|k-1(x|Z1:k-1)dx Nk= vk(x|Z1:k)dx

Bayes-updated intensity

predicted intensity (from previous time)

intensity offalse alarms

sensor likelihood function

probabilityof detection

expected number of objects

measurement

vk kvk|k-1

(k)(x) =zZk

<k,z,> + k(z) k,z(x)

+ 1 pD,k(x)](x) [

Particle PHD filterParticle PHD filter

| 1 1( )k k k k kv v

Particle approximation of vk-1 Particle approximation of vk

state space

[Vo, Singh & Doucet 03, 05], [Sidenbladh 03], [Mahler & Zajic 03]

The PHD (or intensity function) vk is not a probability density

The PHD propagation equation is not a standard Bayesian recursion

Sequential MC implementation of the PHD filter

Need to cluster the particles to obtain multi-target estimates

Particle PHD filterParticle PHD filter

AlgorithmAlgorithm

Initialise;for k =1: kmax ,

for i =1: Jk , Sample: ; compute: ;

end;

for i = Jk +1: Jk +Lk-1 , Sample: ; compute: ;

end;for i =1: Jk +Lk-1 ,

Update: ;end;

Redistribute total mass among Lk resampled particles;end;

ki

k p ~ )(x)(

)(1)(

)()(

1| ikk

ikk

k

ikk pJ

wx

x

ki

k q~)(x)(

),()(

)(1

)(1

)(1|)(

1| ikk

ik

ik

ikkki

kk q

ww

x

xx

)(1|

1

)(1|

)(,

)(,)()(

1 )()(

)()(1 i

kkZz

LJ

j

jkk

jzkk

izki

Di

k wwz

pwk

kk

x

xx

Convergence: [Vo, Singh & Doucet 05], [Clark & Bell 06], [Johansen et. al. 06]

Gaussian Mixture PHD filterGaussian Mixture PHD filter

Closed-form solution to the PHD recursion exists for linear Gaussian multi-target model

vk-1( . |Z1:k-1) vk(. |Z1:k) vk|k-1(. |Z1:k-1)

1| kk k

1| kk k{wk-1, mk-1, Pk-1} i=1

Jk-1(i) (i) (i) {wk|k-1, mk|k-1, Pk|k-1} i=1Jk|k-1(i) (i) (i) {wk, mk, Pk } i=1

Jk(i) (i) (i)

PHD filter

Gaussian Mixture (GM) PHD filter [Vo & Ma 05, 06]

Gaussian mixture prior intensity Gaussian mixture posterior intensities at all subsequent times

Extended & Unscented Kalman PHD filter [Vo & Ma 06]

Jump Markov PHD filter [Pasha et. al. 06]

Track continuity [Clark et. al. 06]

Cardinalised PHD FilterCardinalised PHD Filter

Drawback of PHD filter: High variance of cardinality estimate

Relax Poisson assumption: allows arbitrary cardinality distribution

Jointly propagate: intensity function & probability generating function of cardinality.

More complex PHD update step (higher computational costs)

CPHD filter [Mahler 06,07]

vk-1(xk-1|Z1:k-1) vk(xk|Z1:k) vk|k-1(xk|Z1:k-1) intensity

prediction

intensity

update

pk-1(n|Z1:k-1) pk(n|Z1:k) pk|k-1(n|Z1:k-1) cardinality

prediction

cardinality

update

Gaussian Mixture CPHD FilterGaussian Mixture CPHD Filter

{wk-1, xk-1} i=1Jk-1(i) (i) {wk|k-1, xk|k-1} i=1

Jk|k-1(i) (i) {wk, xk } i=1 Jk(i) (i)intensity

prediction

intensity

update

cardinality

prediction

cardinality

update {pk-1(n)}n=0

{pk|k-1(n)} n=0

{pk(n)}n=0

Particle CPHD filter [Vo 08]

Closed-form solution to the CPHD recursion exists for linear Gaussian multi-target model

Gaussian mixture prior intensity Gaussian mixture posterior intensities at all subsequent times [Vo et. al. 06, 07]

Particle-PHD filter can be extended to the CPHD filter

CPHD filter DemonstrationCPHD filter Demonstration

10 20 30 40 50 60 70 80 90 1000

5

10

Time

Car

dina

lity

Sta

tistic

s

True

Mean

StDev

10 20 30 40 50 60 70 80 90 1000

5

10

Time

Car

dina

lity

Sta

tistic

s

True

Mean

StDev

1000 MC trial average1000 MC trial average

GMCPHD filter

GMPHD filter



Comparison with JPDA: linear dynamics, Comparison with JPDA: linear dynamics, vv = 5, = 5, = 10, = 10, 4 targets,4 targets,

Sonar imagesSonar images


MeMBer FilterMeMBer Filter

{(rk-1, pk-1)} i=1

Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1

Mk|k-1(i) (i)

{(rk, pk )} i=1

Mk(i) (i)

prediction

update

Valid for low clutter rate & high probability of detection



(Multi-target Multi-Bernoulli ) MeMBer filter [Mahler 07], biased

Approximate predicted/posterior RFSs by Multi-Bernoulli RFSs

Cardinality-Balanced MeMBer filter [Vo et. al. 07], unbiased

Cardinality-Balanced MeMBer FilterCardinality-Balanced MeMBer Filter

{(rk-1, pk-1)} i=1

Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1

Mk|k-1(i) (i)

{(rk, pk )} i=1

Mk(i) (i)

prediction

update

{(rP,k|k-1, pP,k|k-1)} {(r,k, p,k)} (i) (i) (i) (i)

i=1

Mk-1

i=1

M,k

rk-1pk-1, pS,k(i) (i)

fk|k-1(|), pk-1 pS,k(i)

pk-1, pS,k(i)

term for object births

Cardinality-Balanced MeMBer filter [Vo et. al. 07]

{(rk-1, pk-1)} i=1

Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1

Mk|k-1(i) (i)

{(rk, pk )} i=1

Mk(i) (i)

prediction

update

{(rL,k, pL,k)} {(rU,k,(z), pU,k(z))} (i) (i)

zZki=1

Mk|k-1

1 pk|k-1, pD,k(i)

pk|k-1(1 pD,k)(i)

1 rk|k-1 pk|k-1, pD,k(i) (i)

rk|k-1(1 pk|k-1, pD,k)(i)(i)


rk|k-1(1 rk|k-1) pk|k-1, pD,kgk(z|)

1 rk|k-1 pk|k-1, pD,k(i) (i)

rk|k-1 pk|k-1, pD,kgk(z|)(i)(i)

i=1

Mk|k-1

(1 rk|k-1pk|k-1, pD,k)2(i) (i)

(i)(i) (i)

i=1

Mk|k-1

(z)

1 rk|k-1(i)

rk|k-1 pk|k-1(i)(i)

i=1

Mk|k-1

pD,kgk(z|)

rk|k-1pk|k-1, pD,kgk(z|)

1 rk|k-1(i)

(i)(i)

i=1

Mk|k-1



Closed-form (Gaussian mixture) solution [Vo et. al. 07],

Particle implementation [Vo et. al. 07],

{(rk-1, pk-1)} i=1

Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1

Mk|k-1(i) (i)

{(rk, pk )} i=1

Mk(i) (i)

prediction

update

{wk-1, xk-1} j=1

Jk-1(i,j) (i,j)j=1Jk|k-1(i,j) (i,j){wk|k-1, xk|k-1 } {wk, xk } j=1

Jk(i,j) (i,j)

{wk-1, mk-1, Pk-1} j=1

Jk-1(i,j) (i,j) (i,j) {wk|k-1, mk|k-1, Pk|k-1} j=1Jk|k-1(i,j) (i,j) (i,j) {wk, mk, Pk } j=1

Jk(i,j) (i,j) (i,j)

More useful than PHD filters in highly non-linear problems

Performance comparisonPerformance comparison

Example:Example: 10 targets max on scene, with births/deaths 4D states: x-y position/velocity, linear Gaussian

observations: x-y position, linear Gaussian

-1000 -800 -600 -400 -200 0 200 400 600 800 1000-1000

-800

-600

-400

-200

0

200

400

600

800

1000

x-coordinate (m)

y-co

ord

inat

e (m

)

/start/end positions

Dynamics constant velocity model:

v = 5ms-2, survival probability:

pS,k = 0.99,

Observations additive Gaussian noise:

=10m,

detection probability: pD,k = 0.98,

uniform Poisson clutter:c = 2.5x10-6m-2

10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Time

Car

dina

lity

Sta

tistic

s

True

Mean

StDev

10 20 30 40 50 60 70 80 90 1000

5

10

15

20

Time

Car

dina

lity

Sta

tistic

s

Cardinality-BalancedRecursion

Mahler’sMeMBerRecursion


Gaussian implementationGaussian implementation

Gaussian implementationGaussian implementation


CPHD Filter has better performance

10 20 30 40 50 60 70 80 90 1000

10

20

30

Time

OS

PA

Loc

(m

)(c=

300,

p=

1)

10 20 30 40 50 60 70 80 90 1000

100

200

300

Time

OS

PA

Car

d (m

)(c=

300,

p=

1)

10 20 30 40 50 60 70 80 90 1000

100

200

300

Time

OS

PA

(m

)(c=

300,

p=

1)

GM-CBMeMBer

GM-PHDGM-CPHD

GM-MeMBer

Particle implementationParticle implementation


CB-MeMBerFilter has better performance

10 20 30 40 50 60 70 80 90 1000

100

200

300

Time

OS

PA

(m

)(c=

300,

p=

1)

SMC-CBMeMBer

SMC-PHDSMC-CPHD

SMC-MeMBer

10 20 30 40 50 60 70 80 90 1000

50

100

Time

OS

PA

Loc

(m

)(c=

300,

p=

1)

10 20 30 40 50 60 70 80 90 1000

100

200

300

Time

OS

PA

Car

d (m

)(c=

300,

p=

1)

Concluding RemarksConcluding Remarks

Thank You!

Random Finite Set frameworkRandom Finite Set framework

Rigorous formulation of Bayesian multi-target filteringRigorous formulation of Bayesian multi-target filtering

Leads to efficient algorithmsLeads to efficient algorithms

Future research directions Future research directions

Track before detectTrack before detect

Performance measure for multi-object systemsPerformance measure for multi-object systems

Numerical techniques for estimation of trajectoriesNumerical techniques for estimation of trajectories

For more info please see http://randomsets.ee.unimelb.edu.au/

ReferencesReferences

• D. Stoyan, D. Kendall, J. Mecke, Stochastic Geometry and its Applications, John Wiley & Sons, 1995• D. Daley and D. Vere-Jones, An Introduction to the Theory of Point Processes, Springer-Verlag, 1988. • I. Goodman, R. Mahler, and H. Nguyen, Mathematics of Data Fusion. Kluwer Academic Publishers, 1997.• R. Mahler, “An introduction to multisource-multitarget statistics and applications,” Lockheed Martin

Technical Monograph, 2000.• R. Mahler, “Multi-target Bayes filtering via first-order multi-target moments,” IEEE Trans. AES, vol. 39, no. 4,

pp. 1152–1178, 2003.• B. Vo, S. Singh, and A. Doucet, “Sequential Monte Carlo methods for multi-target filtering with random finite

sets,” IEEE Trans. AES, vol. 41, no. 4, pp. 1224–1245, 2005,.• B. Vo, and W. K. Ma, “The Gaussian mixture PHD filter,” IEEE Trans. Signal Processing, IEEE Trans.

Signal Processing, Vol. 54, No. 11, pp. 4091-4104, 2006. • R. Mahler, “A theory of PHD filter of higher order in target number,” in I. Kadar (ed.), Signal Processing,

Sensor Fusion, and Target Recognition XV, SPIE Defense & Security Symposium, Orlando, April 17-22, 2006

• B. T. Vo, B. Vo, and A. Cantoni, "Analytic implementations of the Cardinalized Probability Hypothesis Density Filter," IEEE Trans. SP, Vol. 55, No. 7, Part 2, pp. 3553-3567, 2007.

• D. Clark & J. Bell, “Convergence of the Particle-PHD filter,” IEEE Trans. SP, 2006.• A. Johansen, S. Singh, A. Doucet, and B. Vo, "Convergence of the SMC implementation of the PHD filter,"

Methodology and Computing in Applied Probability, 2006. • A. Pasha, B. Vo, H. D Tuan and W. K. Ma, "Closed-form solution to the PHD recursion for jump Markov

linear models," FUSION, 2006.

• D. Clark, K. Panta, and B. Vo, "Tracking multiple targets with the GMPHD filter," FUSION, 2006. • B. T. Vo, B. Vo, and A. Cantoni, “On Multi-Bernoulli Approximation of the Multi-target Bayes Filter," ICIF,

Xi’an, 2007.

See also: http://www.ee.unimelb.edu.au/staff/bv/publications.html

1 1

1/

( )( )

1

{ ,..., }, { ,..., }

0, 0

1( , ) min ( , ) ( ) ,

( , ),

m n

pm

c p pi i

i

X x x Y y y

m n

d X Y d x y c n m m nn

d Y X

( )

( , ) min(|| ||, )c

m n

d x y x y c

permutation

1 1

1/

( )( )

1

{ ,..., }, { ,..., }

0, 0

1( , ) min ( , ) ( ) ,

( , ),

m n

pm

c p pi i

i

X x x Y y y

m n

d X Y d x y c n m m nn

d Y X

( )

( , ) min(|| ||, )c

m n

d x y x y c

permutation

Optimal Subpattern Assignment (OSPA) metric

[Schumacher et. al 08]

Fill up X with n - m dummy points located at a distance greater than c from any points in Y

Calculate pth order Wasserstein distance between resulting sets

Efficiently computed using the Hungarian algorithm

Representation of Multi-target stateRepresentation of Multi-target state

Gaussian Mixture PHD PredictionGaussian Mixture PHD Prediction

vk-1(x) = wk-1N(x; mk-1, Pk-1)

i=1

Jk-1

(i) (i) (i)

vk|k-1(x) = [pS,kwk-1N(x; mS,k|k-1, PS,k|k-1) +i=1

Jk-1

(i) (i) (i)wk-1w,kN(x; m,k|k-1, P,k|k-1)] + k(x) (i) (i,l) (i,l)

l=1

J,k

(l)

Gaussian mixture posterior intensity at time k-1:

Gaussian mixture predicted intensity to time k:

k|k-1vk-1 mS,k|k-1 = Fk-1mk-1

PS,k|k-1 = Fk-1 Pk-1 Fk-1 + Qk-1

(i) (i)

T(i)(i)

(i,l)

(i,l) (l)

m,k|k-1 = F,k-1mk-1 + d,k-1

P,k|k-1 = F,k-1 Pk-1 (F,k-1 )T + Q,k-1

(l)

(l)

(l) (i)

(i) (l)

Gaussian Mixture PHD UpdateGaussian Mixture PHD Update

vk|k-1(x) = wk|k-1N(x; mk|k-1, Pk|k-1)i=1

Jk|k-1

(i) (i) (i)

Gaussian mixture predicted intensity to time k:

Gaussian mixture updated intensity at time k:

vk(x) =i=1

Jk|k-1

(i) (i) N(x; mk|k(z), Pk|k) + (1 pD,k)vk|k-1(x)

zZk

(i)

(j)

(i)

j=1

Jk|k-1pD,k wk|k-1qk (z) + k(z)

pD,kwk|k-1qk (z)

(j)

Pk|k = (IKk Hk )Pk|k-1

(i) (i)(i)

Kk = Pk|k-1Hk (Hk Pk|k-1Hk + Rk )1(i) (i) (i)T T

mk|k(z) = mk|k-1 + Kk (zHk mk|k-1 )(i) (i)(i)(i)

qk(z) = N(z; Hkmk|k-1, HkPk|k-1Hk + Rk )T(i)(i)(i)

kvk|k-1

vk|k-1(xk) = pS,k(xk-1) fk|k-1(xk|xk-1) vk-1(xk-1)dxk-1 k(xk) intensity from

previoustime-step

intensity of spontaneous

object births k

probabilityof survival

Markov transition density

predictedintensity

pk|k-1(n) = p,k(n - j) k|k-1[vk-1,pk-1](j)

probability of n - j spontaneous births

predictedcardinality

j=0

n

probability of j surviving targets

Cardinalised PHD PredictionCardinalised PHD Prediction

Cjl <pS,k ,vk-1> j <1 pS,k ,vk-1> l-j

l=j

<1,vk-1>lpk-1 (l)

vk(xk) = vk|k-1(xk)k, Zk(xk)

predicted intensity

updated intensity

zZk

k,z(xk)<k[vk|k-1, Zk], pk|k-1>

<k[vk|k-1, Zk\{z}], pk|k-1>1

0<k[vk|k-1, Zk], pk|k-1>

<k[vk|k-1, Zk], pk|k-1>0

1

(1pD,k(xk))

predicted cardinality distribution

k[vk|k-1, Zk](n)pk|k-1(n)

updated cardinality distribution

0

<k[vk|k-1, Zk], pk|k-1>pk(n) = 0

Cardinalised PHD UpdateCardinalised PHD Update

k[v, Z](n) = pK,k(|Z|–j) (|Z|–j)! Pj+u

esfj({<v,k,z>: zZk})

<1 pD,k ,v >n-(j+u)

<1,v >n

n

j=0

min(|Z|,n)u

SS Z,|S|=jesfj(Z) = likelihood

functionprob. of

detectionclutter intensity

pD,k(xk)gk(z|xk)<1,k>/k(z)clutter cardinality distribution

Mahler’s MeMBer FilterMahler’s MeMBer Filter

{(rk-1, pk-1)} i=1

Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1

Mk|k-1(i) (i)

{(rk, pk )} i=1

Mk(i) (i)

prediction

update

Valid for low clutter rate & high probability of detection



(Multi-target Multi-Bernoulli ) MeMBer filter [Mahler 07]

Approximate predicted/posterior RFSs by Multi-Bernoulli RFSs

Biased in Cardinality (except when probability of detection = 1)

{(rk-1, pk-1)} i=1

Mk-1(i) (i){(rk|k-1, pk|k-1)} i=1

Mk|k-1(i) (i)

{(rk, pk )} i=1

Mk(i) (i)

prediction

update

1 rk|k-1 pk|k-1, pD,k(i) (i)

rk|k-1 pk|k-1(i)(i)

i=1

Mk|k-1

vk|k-1 =~

(1 rk|k-1pk|k-1, pD,k)2(i) (i)

rk|k-1(1 rk|k-1) pk|k-1(i)(i) (i)

i=1

Mk|k-1

vk|k-1 = (1)

1 rk|k-1(i)

rk|k-1 pk|k-1(i)(i)

i=1

Mk|k-1

vk|k-1 =~*

{(rL,k, pL,k)} {(rU,k,(z), pU,k(z))} (i) (i)

zZki=1

Mk|k-1

(z) vk|k-1, pD,kgk(z|)

vk|k-1, pD,kgk(z|)(1)

~

1 pk|k-1, pD,k(i)

pk|k-1(1 pD,k)(i)

vk|k-1, pD,kgk(z|)

vk|k-1 pD,kgk(z|)

~*

~*

1 rk|k-1 pk|k-1, pD,k(i) (i)

rk|k-1(1 pk|k-1, pD,k)(i)(i)



Linear Jump Markov PHD filter [Pasha et. al. 06]

-6 -4 -2 0 2 4 6

x 104

-6

-4

-2

0

2

4

6x 10

4

x coordinate (in m)

y co

ordi

nate

(in

m)

Aircraft 1 start of flight at k= 1;end of flight at k=90



Payload 1 separates from Aircraft 1at k= 31; end of flight at k=100

Payload 2 separates from Aircraft 2at k= 44; end of flight at k=88

Extensions of the PHD filterExtensions of the PHD filter

10 20 30 40 50 60 70 80 90 100

-5

0

5

x 104

time step

x co

ordi

nate

(in

m) PHD filter estimates

True tracks

10 20 30 40 50 60 70 80 90 100

-5

0

5

x 104

time step

y co

ordi

nate

(in

m)

Example: 4-D, Linear JM target dynamics with 3 modelsExample: 4-D, Linear JM target dynamics with 3 models

4 targets, birth rate= 3x0.05, death prob. = 0.01, 4 targets, birth rate= 3x0.05, death prob. = 0.01, clutter rate = clutter rate = 4040

Extensions of the PHD filterExtensions of the PHD filter

What is a Random Finite Set (RFS)?What is a Random Finite Set (RFS)?

The number of points is random,

The points have no ordering and are random

Loosely, an RFS is a finite set-valued random variable

Also known as: (simple finite) point process or random point pattern

Pine saplings in a Finish forest [Kelomaki & Penttinen]

Childhood leukaemia & lymphoma in North Humberland [Cuzich & Edwards]

Random Set/Point Process in Multi-Target Tracking Ba-Ngu Vo EEE Department University of Melbourne Australia SAMSI,

Documents

target true multitarget

targets x

multitarget measurement

single target

multitarget case

multitarget tracking

object state x

element x