Target tracking suing multiple auxiliary particle filtering

Target tracking using multiple auxiliary

particle filtering

Luis Ubeda-Medina?, Angel F. Garcıa-Fernandez†, Jesus Grajal?

?Universidad Politecnica de Madrid, Spain

†Aalto University, Finland

20th International Conference on Information Fusion, 2017.

July 10-13, 2017. Xi’an, China.

Outline

Multiple Filtering

Multiple Particle Filter

The Multiple Auxiliary Particle Filter

Simulations and results

Conclusions

Outline

Multiple Filtering

Conclusions

Bayesian filtering

• Estimate the state Xk of the dynamic system,

computing its posterior PDF, p(Xk |z1:k)

• ... given the dynamic and measurement models

Xk = f(Xk−1,wk−1)

zk = h(Xk , vk)

• ... using a two step recursion:

• prediction

p(Xk |z1:k−1) =

ˆp(Xk |Xk−1)p(Xk−1|z1:k−1)dXk−1

• and update

p(Xk |z1:k) ∝ p(zk |Xk)p(Xk |z1:k−1)

Bayesian filtering

zk = h(Xk , vk)

• prediction

p(Xk |z1:k−1) =

• and update

Bayesian filtering

zk = h(Xk , vk)

• prediction

p(Xk |z1:k−1) =

• and update

Bayesian filtering

zk = h(Xk , vk)

• prediction

p(Xk |z1:k−1) =

• and update

Bayesian filtering

zk = h(Xk , vk)

• prediction

p(Xk |z1:k−1) =

• and update

p(Xk |z1:k) ∝ p(zk |Xk)p(Xk |z1:k−1)4

Multiple filtering

• Nonlinearities in the dynamic and measurement models

can make it hard to compute the posterior PDF,

specially for high-dimensional state spaces (the curse of

dimensionality)

• Multiple filtering tries to alleviate the curse of

dimensionality, considering the state can be partitioned

into t components

Xk =[(xk1)T , (xk2)T , ..., (xkt )T

]T• ... and instead computing the marginal posterior PDF of

each component (lower dimension)

p(xkj |z1:k) =

ˆp(Xk |z1:k)dXk

−{j}

Xk−{j} =

[(xk1)T , ..., (xkj−1)T , (xkj+1)T , ..., (xkt )T

Multiple filtering

dimensionality)

into t components

Xk =[(xk1)T , (xk2)T , ..., (xkt )T

• ... and instead computing the marginal posterior PDF of

p(xkj |z1:k) =

ˆp(Xk |z1:k)dXk

−{j}

Xk−{j} =

Multiple filtering

dimensionality)

into t components

Xk =[(xk1)T , (xk2)T , ..., (xkt )T

]T• ... and instead computing the marginal posterior PDF of

p(xkj |z1:k) =

ˆp(Xk |z1:k)dXk

−{j}

Xk−{j} =

Multiple filtering

• Given the following assumptions:

• The dynamic model can be expressed as

p(Xk |Xk−1) =t∏

p(xkl |xk−1l )

• posterior independence

p(Xk |z1:k) =t∏

p(xkl |z1:k)

Multiple filtering

p(Xk |Xk−1) =t∏

p(xkl |xk−1l )

p(Xk |z1:k) =t∏

p(xkl |z1:k)

Multiple filtering

p(Xk |Xk−1) =t∏

p(xkl |xk−1l )

p(Xk |z1:k) =t∏

p(xkl |z1:k)

Multiple filtering

• The predicted density can be expressed as

p(Xk |z1:k−1) =t∏

p(xkl |z1:k−1)

• So that the marginal posterior for xkj becomes

p(xkj |z1:k) ∝ˆ

p(zk |Xk)p(Xk |z1:k−1)dXk−{j}

= p(xkj |z1:k−1)

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j}

• The main difficulty is computing the “marginal likelihood”

l(xkj ) ,ˆ

p(zk |Xk)p(Xk−{j}|z

1:k−1)dXk−{j}

Multiple filtering

p(Xk |z1:k−1) =t∏

p(xkl |z1:k−1)

p(xkj |z1:k) ∝ˆ

= p(xkj |z1:k−1)

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j}

l(xkj ) ,ˆ

1:k−1)dXk−{j}

Multiple filtering

p(Xk |z1:k−1) =t∏

p(xkl |z1:k−1)

p(xkj |z1:k) ∝ˆ

= p(xkj |z1:k−1)

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j}

l(xkj ) ,ˆ

1:k−1)dXk−{j}

Outline

Multiple Filtering

Conclusions

• First approach to multiple particle filtering.

• Approximate each marginal posterior PDF with a

different PF using N weighted particles

p(xkj |z1:k) ≈N∑i=1

ωkj ,iδ(xkj − xkj ,i )

• weights are computed according to the principle of

importance sampling

ωkj ,i ∝

p(xkj ,i |z1:k)

qj(xkj ,i |z1:k)

• with the importance sampling function being the prior PDF

qj(xkj |z1:k) ∝ p(xkj |xk−1j )

importance sampling

ωkj ,i ∝

p(xkj ,i |z1:k)

qj(xkj ,i |z1:k)

importance sampling

ωkj ,i ∝

p(xkj ,i |z1:k)

qj(xkj ,i |z1:k)

importance sampling

ωkj ,i ∝

p(xkj ,i |z1:k)

qj(xkj ,i |z1:k)

qj(xkj |z1:k) ∝ p(xkj |xk−1j ) 9

• First order approximation of the “marginal likelihood”

• Compute Xk−{j}

Xk−{j} =

• where

xkl ≈N∑i=1

ωk−1l,i · x

k|k−1l,i

• Assuming the approximation

p(Xk−{j}|z

1:k−1) ≈ δ(

Xk−{j} − Xk

−{j}

)• We approximate the “marginal likelihood” as

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j} ≈ p(zk |xkj , Xk−{j})

Xk−{j} =

• where

xkl ≈N∑i=1

ωk−1l,i · x

k|k−1l,i

p(Xk−{j}|z

1:k−1) ≈ δ(

Xk−{j} − Xk

−{j}

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j} ≈ p(zk |xkj , Xk−{j})

Xk−{j} =

]T• where

xkl ≈N∑i=1

ωk−1l,i · x

k|k−1l,i

p(Xk−{j}|z

1:k−1) ≈ δ(

Xk−{j} − Xk

−{j}

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j} ≈ p(zk |xkj , Xk−{j})

Xk−{j} =

]T• where

xkl ≈N∑i=1

ωk−1l,i · x

k|k−1l,i

p(Xk−{j}|z

1:k−1) ≈ δ(

Xk−{j} − Xk

−{j}

• We approximate the “marginal likelihood” as

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j} ≈ p(zk |xkj , Xk−{j})

Xk−{j} =

]T• where

xkl ≈N∑i=1

ωk−1l,i · x

k|k−1l,i

p(Xk−{j}|z

1:k−1) ≈ δ(

Xk−{j} − Xk

−{j}

ˆp(zk |Xk)p(Xk

−{j}|z1:k−1)dXk

−{j} ≈ p(zk |xkj , Xk−{j})

Outline

Multiple Filtering

Conclusions

• MAPF takes advantage of auxiliary filtering. This is, use

the current measurement at time k, zk , to improve the

way samples are drawn for the importance sampling

function.

• MAPF uses an auxiliary PF to approximate the marginal

posterior PDF of each component of the partition of the

state.

• MAPF uses the approximation of the “marginal

likelihood” of MPF.ˆ

1:k−1)dXk−{j} ≈ p(zk |xkj , Xk

−{j})

function.

state.

−{j})

function.

state.

−{j})

• MAPF indirectly obtains samples from p(xkj |z1:k) using an

auxiliary variable aj .

• Compute µkj,i , a characterization of xkj given xk−1

j,i , such as

µkj,i = E[xkj |xk−1j,i ]

• Sample aj,i according to

λj,i ∝ p(zk |µkj,i , X

k−{j})ω

k−1i

• Using the index aj thus allows to draw particles that are prone

to obtain a higher likelihood with the current measurement zk .

• The importance sampling function of MAPF therefore draws

samples in a higher dimension from

qj(xkj , aj |z1:k) ∝ p(zk |µkj,aj , X

k−{j})p(xkj |xk−1j,aj

)ωk−1j,aj

j,i , such as

k−{j})ω

k−1i

)ωk−1j,aj

j,i , such as

k−{j})ω

k−1i

)ωk−1j,aj

j,i , such as

k−{j})ω

k−1i

)ωk−1j,aj

j,i , such as

k−{j})ω

k−1i

)ωk−1j,aj

Outline

Multiple Filtering

Conclusions

Target dynamics

• 8 target trajectories were generated according to an

independent nearly-constant velocity model.

0 20 40 60 80 100 120

x position [m]

ositio

Sensor model

• A nonlinear measurement model is considered. Each sensor

receives amplitude range-dependent measurements.

zk+1i = hi (Xk+1) + vk+1

hi (Xk+1) =

√√√√ t∑j=1

SNR(dk+1j ,i )

SNR(dk+1j ,i ) =

SNR0 dk+1j ,i ≤ d0

SNR0d2

(dk+1j,i )2

dk+1j ,i > d0

Compared filters

• Jointly Auxiliary PF (JA) [1]

• Parallel Partition PF (PP) [2]

• Auxiliary PP PF (APP) [3]

• Multiple PF (MPF) [4]

• Multiple Auxiliary PF (MAPF)

[1] M. R. Morelande, “Tracking multiple targets with a sensor network,” in Proceedings of the 9th InternationalConference on Information Fusion (FUSION), 2006.

[2] A. F. Garcıa-Fernandez, J. Grajal, and M. Morelande, “Two-layer particle filter for multiple target detection andtracking,” IEEE Transactions on Aerospace and Electronic Systems, vol. 49, no. 3, pp. 1569–1588, 2013.

[3] L. Ubeda-Medina, A. F. Garcıa-Fernandez, and J. Grajal, “Generalizations of the auxiliary particle filter formultiple target tracking,” in Proceedings of the 17th International Conference on Information Fusion (FUSION),2014.

[4] M. F. Bugallo, T. Lu, and P. M. Djuric, “Target Tracking by Multiple Particle Filtering,” IEEE AerospaceConference, pp. 1–7, 2007.

Compared filters

Tracking 2 targets

50 100 150 200 250 300 350 400 450 500

Number of particles

positio

• MAPF is the best filter, closely followed by APP

• A remarkably small number of particles is needed for MAPF

to obtain good tracking results

Tracking 2 targets

50 100 150 200 250 300 350 400 450 500

Number of particles

positio

• MAPF is the best filter, closely followed by APP

• A remarkably small number of particles is needed for MAPF

to obtain good tracking results 18

Tracking 6 targets

50 100 150 200 250 300 350 400 450 500

number of particles

• The performance improvement of MAPF is bigger in this

higher-dimensional scenario.

• JA acutely suffers the curse of dimensionality, as it considers

the whole state in the sampling procedure.

Tracking 6 targets

50 100 150 200 250 300 350 400 450 500

number of particles

• The performance improvement of MAPF is bigger in this

higher-dimensional scenario.

• JA acutely suffers the curse of dimensionality, as it considers

the whole state in the sampling procedure. 19

Tracking 8 targets

50 100 150 200 250 300 350 400 450 500

number of particles

• MAPF outperforms the rest of the filters, this time followed

by MPF.

Tracking 1 to 8 targets, 100 particles

1 2 3 4 5 6 7 8

number of targets

positio

• Multiple filters such as MAPF and MPF remarkably deal to

increases in dimensionality.

• Overall, for 100 particles, MAPF is the best performing filter,

followed by APP and MPF.

Tracking 1 to 8 targets, 100 particles

1 2 3 4 5 6 7 8

number of targets

positio

• Multiple filters such as MAPF and MPF remarkably deal to

increases in dimensionality.

• Overall, for 100 particles, MAPF is the best performing filter,

followed by APP and MPF. 21

Tracking 8 targets (zoom). Eq. execution time (I)

50 100 150 200 250 300 350 400 450 500

number of particles

mean e

xecution tim

50 100 150 200 250 300 350 400 450 500

number of particles

positio

• MAPF and APP have a higher computational cost.

• Considering a different number of particles for each filter such

that they all have similar computational cost, MAPF is still

the best performing filter.

Tracking 8 targets (zoom). Eq. execution time (I)

50 100 150 200 250 300 350 400 450 500

number of particles

mean e

xecution tim

50 100 150 200 250 300 350 400 450 500

number of particles

positio

• MAPF and APP have a higher computational cost.

• Considering a different number of particles for each filter such

that they all have similar computational cost, MAPF is still

the best performing filter. 22

Tracking 8 targets (zoom). Eq. execution time (II)

50 100 150 200 250 300 350 400 450 500

number of particles

mean e

xecution tim

50 100 150 200 250 300 350 400 450 500

number of particles

positio

• This behavior also holds for different computational costs.

Tracking 8 targets (zoom). Eq. execution time (III)

50 100 150 200 250 300 350 400 450 500

number of particles

mean e

xecution tim

50 100 150 200 250 300 350 400 450 500

number of particles

positio

• This behavior also holds for different computational costs.

Outline

Multiple Filtering

Conclusions

• Multiple particle filtering shows a remarkable

performance in high-dimensional nonlinear systems.

• In this paper, we have formalized the use of auxiliary

filtering within the multiple particle filtering framework.

• We have demonstrated through simulations in an MTT

scenario with nonlinear measurements that the MAPF

can outperform the MPF as well as other MTT

algorithms.

Conclusions

algorithms.

Conclusions

algorithms.

Thank you

Further questions:

[email protected]

Target tracking suing multiple auxiliary particle filtering

Engineering

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