A Gaussian Mixture PHD filter for Extended Target Tracking€¦ · FUSION 2010, Edinburgh 1(14) A...

FUSION 2010, Edinburgh 1(14)

A Gaussian Mixture PHD filter forExtended Target Tracking

Karl Granström, Christian Lundquist,Umut Orguner

Division of Automatic ControlDepartment of Electrical Engineering

Linköping University, Sweden

K. Granström, C. Lundquist, U. Orguner

A Gaussian Mixture PHD filter for Extended Target Tracking, FUSION 2010

AUTOMATIC CONTROLREGLERTEKNIK

LINKÖPINGS UNIVERSITET

Definition: Extended Target Tracking 2(14)

• Extended targets often modelled as having extension or shape.

• All targets have extensions and shapes.• Estimate target size and shape depending on...

1. ...sensor/target setup.2. ...application.

Definition:Extended targets are targets that potentially give riseto more than one measurement per time step.





Definition: Extended Target Tracking 2(14)

• Extended targets often modelled as having extension or shape.

• All targets have extensions and shapes.• Estimate target size and shape depending on...

1. ...sensor/target setup.2. ...application.

Definition:Extended targets are targets that potentially give riseto more than one measurement per time step.





Contribution 3(14)

• Multiple extended targets, clutteredmeasurements⇒ difficult DA.

• GM-PHD-filter works well forcluttered measurements[Vo and Ma, IEEE TSP 2006].

• PHD filter for extended targets,Poisson assumption[Mahler, FUSION 2009].

−5 0 5 10 15 20−5

0

5

10

15

20

S

Meas.

Main contribution:Implementation of GM-PHD-filter for extended targets.Minor contribution:Simple method for measurement set partitioning.





Contribution 3(14)



• PHD filter for extended targets,Poisson assumption[Mahler, FUSION 2009]. −5 0 5 10 15 20

−5

0

5

10

15

20

S T1

T2

True meas.

Clutter






Contribution 3(14)



• PHD filter for extended targets,Poisson assumption[Mahler, FUSION 2009]. −5 0 5 10 15 20

−5

0

5

10

15

20

S T1

T2

True meas.

Clutter






Target tracking setup 4(14)

• RFS of targets Xk = {x(i)k }

Nx,ki=1

x(i)k+1 = Fkx(i)k + Gkw(i)k , w(i)

k ∈ N (0, Qk) .

• RFS of measurements Zk = {z(j)k }

Nz,kj=1

z(j)k = Hkx(i)k + e(j)k , e(j)k ∈ N (0, Rk) .

• Generated measurements per target per time step

N(i)m,k ∈ POIS (βD) .

• Effective probability of target detection

pD,eff =(

1− e−βD)

pD.





Target tracking setup 4(14)

• RFS of targets Xk = {x(i)k }

Nx,ki=1

x(i)k+1 = Fkx(i)k + Gkw(i)k , w(i)

k ∈ N (0, Qk) .

• RFS of measurements Zk = {z(j)k }

Nz,kj=1

z(j)k = Hkx(i)k + e(j)k , e(j)k ∈ N (0, Rk) .

• Generated measurements per target per time step

N(i)m,k ∈ POIS (βD) .

• Effective probability of target detection

pD,eff =(

1− e−βD)

pD.





Measurement Pseudo Likelihood 5(14)

• Prediction of PHD-intensity performed identically to[Vo and Ma, 2006].

• vk|k−1 (x|Z) is predicted PHD-intensity. Corrected PHD-intensity

vk|k (x|Z) = LZk (x) vk|k−1 (x|Z) ,

where measurement pseudo-likelihood is given by

LZk (x) =1−(

1− e−γ(x))

pD (x) +

e−γ(x)pD (x) ∑p∠Zk

ωp ∑W∈p

γ (x)|W|

dW· ∏

z∈W

φz (x)λkck (z)

.

[Mahler, 2009]





Additional assumption 6(14)

• Six assumptions made in derivation of GM-PHD

[Vo and Ma, 2006].

• Additional assumption:The expected number of generated measurements γ (x) can beapproximated as functions of the means of the individualGaussian components

γ(j) , γ(

m(j)k|k−1

).





Partitioning the measurements 7(14)

• In each time step Zk must be partitioned.

• A partition p is a division of Zk into cells W.

• Important since more than one measurement can stem from thesame target.





Partitioning the measurements — example 8(14)

Partition the measurement set Zk ={

z(1)k , z(2)k , z(3)k

}

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

z(1)kz

(2)k

z(3)k

Z

x

y

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

W 11

p1

xy

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

W 21

W 22

p2

x

y

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

W 31

W 32

p3

x

y

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

W 41

W 42

p4

x

y

−0.5 0 0.5 1 1.5−0.5

0

0.5

1

1.5

W 51

W 52

W 53

p5

x

y





Partitioning method 9(14)

• Measurements belong to same cell W if distance is “small”.• “Small” measured by Mahalanobis distance

(z(1)k − z(2)k )TR−1k (z(1)k − z(2)k ) < δPG

for some gate probability PG.

• With Rk = σ2e I2 this reduces to∥∥∥z(1)k − z(2)k

∥∥∥2< σe

√δPG = di.

• Let {dmi }

Ndi=1 be set of measurement to measurement distances.

• Good partitions for di corresponding to

σe√

δ0.30 ≤ dmi < σe

√δ0.80





Partitioning method 9(14)

• Measurements belong to same cell W if distance is “small”.• “Small” measured by Mahalanobis distance

(z(1)k − z(2)k )TR−1k (z(1)k − z(2)k ) < δPG

for some gate probability PG.• With Rk = σ2

e I2 this reduces to∥∥∥z(1)k − z(2)k

∥∥∥2< σe

√δPG = di.

• Let {dmi }

Ndi=1 be set of measurement to measurement distances.

• Good partitions for di corresponding to

σe√

δ0.30 ≤ dmi < σe

√δ0.80





Partitioning method — example 10(14)

0 100 200 300

50

100

150

200

250

300

Measurements

X [m]

Y [m

]

0 100 200 3000

100

200

300

d = 14.9927

X [m]

Y [m

]

0 100 200 3000

100

200

300

d = 26.3491

X [m]

Y [m

]

0 100 200 3000

100

200

300

d = 43.5245

X [m]

Y [m

]

0 100 200 3000

100

200

300

d = 59.5143

X [m]

Y [m

]

0 100 200 3000

100

200

300

d = 62.9965

X [m]Y

[m]





Simulation — Measurements 11(14)

0 10 20 30 40 50 60 70 80 90 100−1000

−500

0

500

1000

Time

X [

m]

True vs measured

0 10 20 30 40 50 60 70 80 90 100−1000

−500

0

500

1000

Time

Y [

m]

True vs measured

• True target track crossing at time k = 56.

• New target birth and target spawned at time k = 66.





Simulation — Tracking performance 12(14)

0 10 20 30 40 50 60 70 80 90 100−400

−200

0

200

400

600

800

1000

Time

X [m

]

True vs extracted

0 10 20 30 40 50 60 70 80 90 100−1000

−500

0

500

Time

Y [m

]

True vs extracted

0 10 20 30 40 50 60 70 80 90 100−400

−200

0

200

400

600

800

1000

Time

X [m

]

True vs extracted

0 10 20 30 40 50 60 70 80 90 100−1000

−500

0

500

Time

Y [m

]

True vs extracted

0 10 20 30 40 50 60 70 80 90 1001

1.5

2

2.5

3

3.5

4

Time

No. ta

rgets

True vs extracted

0 10 20 30 40 50 60 70 80 90 1000

10

20

30

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Time

No. ta

rgets

True vs extracted

Extended target GM-PHD Standard GM-PHD





Conclusions 13(14)


The suggested implementation handles...

• ...unknown number of targets.

• ...extended target measurements.

• ...cluttered measurement sets.

Matlab code used for simulations in paper available onlinehttp://www.control.isy.liu.se/publications/doc?id=2299





The End 14(14)

Thank you for listening!

Any questions?

Matlab code used for simulations in paper available onlinehttp://www.control.isy.liu.se/publications/doc?id=2299





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A Gaussian Mixture PHD filter for Extended Target Tracking€¦ · FUSION 2010, Edinburgh 1(14) A...

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