• Beyond the Kalman Filter: Particle filters for tracking applications N. J. Gordon Tracking and Sensor Fusion Group Intelligence, Surveillance and Reconnaissance Division Defence Science and Technology Organisation PO Box 1500, Edinburgh, SA 5111, AUSTRALIA.
[email protected] N.J. Gordon : Lake Louise : October 2003 – p. 1/47
Contents
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– General PF discussion – History – Review – Tracking applications – Sonobuoy – TBD – DBZ
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What is tracking? – Use models of the real world to – estimate the past and present – predict the future – Achieved by extracting underlying information from sequence of noisy/uncertain observations – Perform inference on-line – Evaluate evolving sequence of probability distributions
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Recursive filter System model xt = ft (xt`1 ; ›t )
$
p (xt xt`1 )
yt = ht (xt ; t )
$
p (yt xt )
Measurement model
Information available y1:t = (y1 ; : : : ; yt ) p(x0 ) Want p (x0:t+i y1:t ) and especially p (xt y1:t )
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Recursive filter Prediction p (xt y1:t`1 ) =
Z
p (xt xt`1 ) p (xt`1 y1:t`1 ) dxt`1
Update p (xt y1:t ) =
p (yt y1:t`1 ) =
Z
p (yt xt ) p (xt y1:t`1 ) p (yt y1:t`1 ) p (yt xt ) p (xt y1:t`1 ) dxt
Alternatively ...
p (x0:t y1:t ) = p (x0:t`1 y1:t`1 )
p (xt y1:t ) =
Z
p (yt xt ) p (xt xt`1 ) p (yt y1:t`1 )
p (x0:t y1:t ) dx0:t`1
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Tracking : what are the problems?
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– On-line processing – Target manoeuvres – Missing measurements – Spurious measurements – Multiple objects and/or sensors – Finite sensor resolution – Prior constraints – Signature information
Nonlinear/non-Gaussian models
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Analytic approximations – EKF and variants : linearisation, Gaussian approx, unimodal – Score function moment approximation : Masreliez (75), West (81), Fahrmeir (92), Pericchi, Smith (92) – Series based approximation to score functions : Wu, Cheng (92)
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Numerical approximations – Discrete grid: Pole, West (88) – Piecewise pdf: Kitagawa (87), Kramer, Sorenson (88) – Series expansion: Sorenson, Stubberud (68) – Gaussian mixtures: Sorenson, Alspach (72), West (92) – Unscented filter: Julier, Uhlman (95)
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Monte Carlo Approximations – Sequential Importance Sampling (SIS): Handschin & Mayne, Automatica, 1969. – Improved SIS: Zaritskii et al., Automation and Remote Control, 1975. – Rao-Blackwellisation: Akashi & Kumamoto, Automatica, 1977...
) Too computationally demanding 20-30 years ago
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Sequential Monte Carlo (SMC) – SMC methods lead to estimate of the complete probability distribution – Approximation centred on the pdf rather than compromising the state space model – Known as Particle filters, SIR filters, bootstrap filters, Monte Carlo filters, Condensation etc
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Why random samples? – nonlinear/non-Gaussian
– constraints
– whole pdf – moments/quantiles
– association hypotheses independent over time
– HPD interval
– multiple models trivial
– re-parameterisation
– scalable - how big is 1 – parallelisable
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Comparison Kalman filter
Particle filter
– analytic solution
– sequential MC solution
– restrictive assumptions
– based on simulation
– deduce state from measurement
– no modelling restrictions
– KF “optimal”
– predicts measurements from states
– EKF “sub-optimal”
– optimal (with 1 computational resources)
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Book Advert Sequential Monte Carlo methods in practice Editors: Doucet, de Freitas, Gordon Springer-Verlag (2001) – Theoretical Foundations – Efficiency Measures – Applications : – Target tracking, missile guidance, image tracking, terrain referenced navigation, exchange rate prediction, portfolio allocation, in-situ ellipsometry, pollution monitoring, communications and audio engineering.
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Useful Information Books – “Sequential Monte Carlo methods in practice”, Doucet, de Freitas, Gordon, Springer, 2001. – “Monte Carlo strategies in scientific computing”, Liu, Springer, 2001. – “Beyond the Kalman filter : Tracking applications of particle filters”, Ristic, Arulampalam, Gordon, Artech House, 2003? Papers – “On sequential Monte Carlo sampling methods for Bayesian filtering”, Statistics in Computing, Vol 10, No 3, pgs 197-208, 2000. – IEEE Trans. Signal Processing special issue, February 2002. Web site – www.cs.ubc.ca/ nando/smc/index.html (includes software)
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Particle Filter – Represent uncertainty over x1:t using diversity of weighted particles n
i x1:t ; wti
oN
i=1
– Ideally: i x1:t ‰ p(x1:t j y1:t ) =
where p(y1:t ) =
Z
p(x1:t ; y1:t ) p(y1:t )
p(x1:t ; y1:t )dx1:t
– What if we can’t sample p(x1:t j y1:t )?
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Particle Filter - Importance Sampling
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– Sample from a convenient proposal distribution q(x1:t j y1:t ) – Use importance sampling to modify weights Z
p(x1:t j y1:t )f (x1:t )dx1:t = ı
Z
p(x1:t j y1:t ) q(x1:t j y1:t )f (x1:t )dx1:t q(x1:t j y1:t )
N X
i wti f (x1:t )
i=1
where i x1:t ‰q(x1:t j y1:t )
wti =
p(x1:t j y1:t ) q(x1:t j y1:t )
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Particle Filter - Importance Sampling
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– Pick a convenient proposal – Define the un-normalised weight: w~ti =
p(x1:t ; y1:t ) q(x1:t j y1:t )
– Can then calculate approximation to p(y1:t )
p(y1:t ) ı
N X
w~ti
i=1
– Normalised weight is
wti
p(x1:t ; y1:t ) 1 w~ti p(x1:t j y1:t ) = = PN = q(x1:t j y1:t ) q(x1:t j y1:t ) p(y1:t ) ~ti i=1 w N.J. Gordon : Lake Louise : October 2003 – p. 17/47
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Particle Filter - SIS – To perform Sequential Importance Sampling, SIS q(x1:t j y1:t ) , q(x1:t`1 j y1:t`1 ) q(xt j xt`1 ; yt ) | {z }| {z } Keep existing path
Extend path
– The un-normalised weight then takes the appealing form
w~ti
i ;y jy p(x1:t t 1:t`1 ) = i jy ) q(x1:t 1:t
=
i i p(x1:t`1 j y1:t`1 ) p(xti ; yt j x1:t`1 )
i i ; yt ) q(x1:t`1 j y1:t`1 ) q(xti j xt`1
i =wt`1
i i p(yt j xt`1 )p(xti j xt`1 )
|
i q(xti j xt`1 ; yt ) {z Incremental weight
} N.J. Gordon : Lake Louise : October 2003 – p. 18/47
Illustration of SIS
•
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Illustration of SIS - data conflict
•
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SIS Problem : Whatever the importance function, degeneracy is observed (Kong, Liu and Wong 1994). i with respectively – Introduce a selection scheme to discard/multiply particles x0:t high/low importance weights i ; w ) onto the equally – Resampling maps the weighted random measure (x0:t t i ; N `1 ) weighted random measure (x0:t P – Scheme generates Ni children such that N i=1 Ni = N and satisfies E(Ni ) = Nwki
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Illustration of SIR
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Illustration of SIR
•
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Ingredients for Particle filter – Importance sampling function (i) ) – Prior p(xt j xt`1 (i) ; yt ) – Optimal p(xt j xt`1
– UKF, linearised EKF, : : : – Redistribution scheme – Multinomial – Deterministic – Residual – Stratified – Careful initialisation procedure (for efficiency) N.J. Gordon : Lake Louise : October 2003 – p. 24/47
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Improvements to SIR To alleviate degeneracy problems many other methods have been proposed – Local linearisation (Doucet, 1998; Pitt & Shephard, 1999) using the EKF to estimate the importance distribution or UKF (Doucet et al, 1999) – Rejection methods (Müller, 1991; Hürzeler & Künsch, 1998; Doucet, 1998; Pitt & Shephard, 1999) – Auxiliary particle filters (Pitt & Shephard, 1999) – Kernel smoothing (Gordon, 1993; Liu & West, 2000; Musso et al, 2000) – MCMC methods (Müller, 1992; Gordon & Whitby, 1995; Berzuini et al, 1997; Gilks & Berzuini, 1999; Andrieu et al, 1999) – Bridging densities : (Clapp & Godsill, 1999)
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Auxiliary SIR - ASIR – Introduced by Pitt and Shephard 1999. – Use importance sampling function q(xt ; i j y1:t ) – Auxiliary variable i refers to index of particle at time t ` 1 – Importance distribution chosen to satisfy i i q(xt ; i j y1:t ) / p(yt j —it )p(xt j xt`1 )wt`1 i – —it is some characterisation of xt given xt`1
i i ) – eg, —it = E(xt j xt`1 ) or —it ‰ p(xt j xt`1
– This gives j
wtj
ij
/ wt`1
i p(yt j xtj )p(xtj j xt`1 )
q(xtj ; i j j y1:t )
=
p(yt j xtj ) j
p(yt j —ti )
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ASIR – Naturally uses points at t ` 1 which are “close” to measurement yt – If process noise is small then ASIR less sensitive to outliers than SIR – This is because single point —it characterises p(xt j xt`1 ) well – But if process noise is large then ASIR can degrade performance – Since a single point —it does not characterise p(xt j xt`1 )
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Regularised PF - RPF – Resampling introduced to reduce degeneracy – But, also reduces diversity – RPF proposed as a solution – Uses continuous Kernel based approximation p^(x1:t j y1:t ) ı
N X i=1
wti Kh
xt `
xti
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RPF
•
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RPF – Kernel K(.) and bandwidth h chosen to minimise MISE MISE(p) ^ =E
Z
fp(x ^ t j y1:t ) ` p(xt j y1:t )g2 dxt
– For equally weighted samples, optimal choice is Epanechnikov kernel
Kopt
nx +2 (1` k x k2 ) if k x k< 1 2cnx = 0 otherwise
– Optimal bandwidth can be obtained as a function of underlying pdf – Assume this is Gaussian with unit covariance matrix hopt
p nx `1 1=(nx +4) = 8N(nx + 4)(2 ı) cnx
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MCMC Moves – RPF moves are blind – Instead, introduce Metropolis style acceptance step o n R R R R – Resampled xk and created x1:k = xk ; x1:k`1
– Resampled xkR and then sampled from a proposal distribution o n R P P R P xk ‰ q(: j xk ) and created x1:k = xk ; x1:k`1
– Assume q(: j :) symmetric x1:k =
x P
1:k x R 1:k
¸ = min
= min
with probability ¸ otherwise 1;
1;
R P jy p(x1:k 1:k )q(xk R jy P p(x1:k 1:k )q(xk
j xkP )) j xkR )
R ) p(yk j xkP )p(xkP j xk`1
R ) p(yk j xkR )p(xkR j xk`1
!
!
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Tracking dim targets – Detection and tracking of low SNR targets - better not to threshold the sensor data ! – The concept: track-before-detect – Conventional TBD approaches: - Hough transform (Carlson et al, 1994) - dynamic programming (Barniv, 1990; Arnold et al, 1993) - maximum likelihood (Tonissen, 1994) – The performance improved by 3-5 dB in comparison to the MHT (thresholded data).
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Recursive Bayesian TBD – Drawbacks of conventional TBD approaches: batch processing; prohibit or penalise deviations from the straight line motion; require enormous computational resources. – A recursive Bayesian TBD (Salmond, 2001), implemented as a particle filter – no need to store/process multiple scans – target motion - stochastic dynamic equation – valid for non-gaussian and structured background noise – the effect of point spread function, finite resolution, unknown and fluctuating SNR are accommodated – target presence and absence explicitly modelled – Run Demo
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Mathematical formulation – Target state vector xk = [xk x_k yk y_k Ik ]T : – State dynamics: xk+1 = fk (xk ; vk ); – Target existence - two state Markov chain, Ek 2 f0; 1g with TPM 1 ` Pb Pb : ˝= Pd 1 ` Pd
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Mathematical formulation (Cont’d) – Sensor model: 2D map, image of a region ´x ˆ ´y . – At each resolution cell (i ; j) measured intensity: (i;j) (i;j) h (x ) + w if target present k k k zk(i;j) = w (i;j) if target absent k where
2 2 ´x ´y Ik (i ´x ` xk ) + (j´y ` yk ) (i;j) hk (xk ) ı exp ` 2 2ı˚ 2˚2 N.J. Gordon : Lake Louise : October 2003 – p. 35/47
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Example – 30 frames and target present in 7 to 22 – SNR = 6.7 dB (unknown) – 20 ˆ 20 cells Frame 2
Frame 7
Frame 12
Frame 17
Frame 22
Frame 27
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Particle filter output (6 states)
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Figures suppressed to reduce file size.
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Particle filter output (6 states)
16 TRUE PATH ESTIMATED
1
14
0.9 0.8
12 0.6
10 y
PROBABILITY
0.7
0.5
8 0.4 0.3
START
6
0.2
4
0.1 0
5
10
15 FRAME NUMBER
20
25
30
2 0
2
4
6
8
10
12
14
x
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PF based TBD - Performance – Derived CRLBs (as a function of SNR) – Compared PF-TBD to CRLBs – Detection and Tracking reliable at 5 dB (or higher)
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Sonobuoy and Submarine – Noisy bearing measurements from drifting sensors – Uncertainty in sensor locations – Sensor loss – High proportion of spurious bearing measurements – Run demo
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Blind Doppler – Blind Doppler zone to to filter out ground clutter + ground moving targets – A simple EP measure against any CW or pulse Doppler radar – Causes track loss – Aided by on-board RWR or ESM
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Tracking with hard constraints – Prior information on sensor constraints – Posterior pdf is truncated (non-Gaussian) – Example : – 2-D tracking with CV model – (r; „; r_) measurements – pd < 1 – EKF and Particle Filter with identical gating and track scoring
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EKF (a)
(b)
Range−Rate [km/h]
400 50
40
200 0 −200 −400 −600
0
30
50
100
150
T I M E [s] (c) 1
20
Track Score
Y [km]
−800
10
0.8 0.6 TRACK LOSS
0.4 0.2
0 −15
0 −10
−5
0
X [km]
5
10
15
0
50
100
150
T I M E [s] N.J. Gordon : Lake Louise : October 2003 – p. 43/47
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Particle Filter (a)
(b)
Range−Rate [km/h]
400 50
40
200 0 −200 −400 −600
0
30
50
100
150
T I M E [s] (c) CLOUD OF PARTICLES AT t=108 s
20
1
Track Score
Y [km]
−800
10
0.8 0.6 0.4 0.2
0
0 −10
0
X [km]
10
0
50
100
150
T I M E [s] N.J. Gordon : Lake Louise : October 2003 – p. 44/47
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Track continuity
1
Probability of Track Maintenance
Probability of Track Maintenance
1
0.8
0.6
0.4
0.2
0.8
0.6
PF EKF
0.4
0.2 PF EKF
0
0
20
40
60
80
Doppler blind zone limit [km/h]
T=2s
100
120
0
0
20
40
60
80
100
120
Doppler blind zone limit [km/h]
T=4s
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Problems hindering SMC – Convergence results – Becoming available (Del Moral, Crisan, Chopin, Lyons, Doucet ...) – Communication bandwidth – Large particle sets impossible to send – Interoperability – Need to integrate with varied tracking algorithms – Computation – Expensive so look to minimise Monte Carlo – Multi-target problems – Rao-Blackwellisation
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Final comments – Sequential Monte Carlo methods – “Optimal” filtering for nonlinear/non Gaussian models – Flexible/Parallelizable – Not a black-box: efficiency depends on careful design. – If the Kalman filter is appropriate for your application use it – A Kalman filter is a (one) particle filter
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