Beyond the Kalman FilterParticle Filters for Tracking Applications
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Transcript of Beyond the Kalman FilterParticle Filters for Tracking Applications
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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.
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Contents
– 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)
Measurement model
yt = ht(xt ; �t) $ p (yt xt)
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) =∫
p (xt xt`1) p (xt`1 y1:t`1) dxt`1
Update
p (xt y1:t) =p (yt xt) p (xt y1:t`1)
p (yt 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 (yt xt) p (xt xt`1)
p (yt y1:t`1)
p (xt y1:t) =
∫
p (x0:t y1:t) dx0:t`1
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Tracking : what are the problems?
– 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 completeprobability distribution
– Approximation centred on the pdf rather thancompromising the state space model
– Known as Particle filters, SIR filters, bootstrapfilters, Monte Carlo filters, Condensation etc
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Why random samples?
– nonlinear/non-Gaussian
– whole pdf
– moments/quantiles
– HPD interval
– re-parameterisation
– constraints
– association hypotheses
independent over time
– multiple models trivial
– scalable - how big is1
– parallelisable
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Comparison
Kalman filter
– analytic solution
– restrictive assumptions
– deduce state from
measurement
– KF “optimal”
– EKF “sub-optimal”
Particle filter
– sequential MC solution
– based on simulation
– no modelling
restrictions
– predicts measurements
from states
– optimal (with 1 compu-
tational 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
{
x i1:t ; wit
}N
i=1
– Ideally:
x i1:t ‰ p(x1:t j y1:t) =p(x1:t ; y1:t)
p(y1:t)
where
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
– Sample from a convenient proposal distribution q(x1:t j y1:t)
– Use importance sampling to modify weights
∫
p(x1:t j y1:t)f (x1:t)dx1:t =∫p(x1:t j y1:t)q(x1:t j y1:t)
q(x1:t j y1:t)f (x1:t)dx1:t
ıN∑
i=1
w it f (xi1:t)
where
x i1:t ‰q(x1:t j y1:t)
w it =p(x1:t j y1:t)q(x1:t j y1:t)
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Particle Filter - Importance Sampling
– Pick a convenient proposal
– Define the un-normalised weight:
~w it =p(x1:t ; y1:t)
q(x1:t j y1:t)
– Can then calculate approximation to p(y1:t)
p(y1:t) ıN∑
i=1
~w it
– Normalised weight is
w it =p(x1:t j y1:t)q(x1:t j y1:t)
=p(x1:t ; y1:t)
q(x1:t j y1:t)1
p(y1:t)=
~w it∑Ni=1 ~w
it
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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)︸ ︷︷ ︸
Keep existing path
q(xt j xt`1; yt)︸ ︷︷ ︸
Extend path
– The un-normalised weight then takes the appealing form
~w it =p(x i1:t ; yt j y1:t`1)q(x i1:t j y1:t)
=p(x i1:t`1 j y1:t`1)q(x i1:t`1 j y1:t`1)
p(x it ; yt j x i1:t`1)q(x it j x it`1; yt)
=w it`1p(yt j x it`1)p(x it j x it`1)q(x it j x it`1; yt)
︸ ︷︷ ︸
Incremental weight
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SIS
Problem : Whatever the importance function, degeneracy is observed (Kong, Liu and Wong
1994).
– Introduce a selection scheme to discard/multiply particles x i0:t with respectively
high/low importance weights
– Resampling maps the weighted random measure (x i0:t ; wt) onto the equally
weighted random measure (x i0:t ; N`1)
– Scheme generatesNi children such that∑Ni=1 Ni = N and satisfies
E(Ni ) = Nwik
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Ingredients for Particle filter
– Importance sampling function
– Prior p(xt j x(i)t`1)
– Optimal p(xt j x(i)t`1; yt)
– UKF, linearised EKF, : : :
– Redistribution scheme
– Multinomial
– Deterministic
– Residual
– Stratified
– Careful initialisation procedure (for efficiency)
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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
q(xt ; i j y1:t) / p(yt j—it)p(xt j x it`1)w it`1
– —it is some characterisation of xt given x it`1– eg, —it = E(xt j x it`1) or —it ‰ p(xt j x it`1)
– This gives
w jt / w ij
t`1p(yt j x jt )p(x
jt j x i
j
t`1)
q(x jt ; ij j y1:t)
=p(yt j x jt )p(yt j—i jt )
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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
p̂(x1:t j y1:t) ı
N∑
i=1
w itKh(xt ` x
it
)
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•RPF
– Kernel K(.) and bandwidth h chosen to minimise MISE
MISE(p̂) = E[∫ fp̂(xt j y1:t)` p(xt j y1:t)g2 dxt]– For equally weighted samples, optimal choice is Epanechnikov kernel
Kopt =
nx+22cnx(1` k x k2) if k x k< 1
0 otherwise
– Optimal bandwidth can be obtained as a function of underlying pdf
– Assume this is Gaussian with unit covariance matrix
hopt =[8N(nx + 4)(2
pı)nx c`1nx
]1=(nx+4)
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MCMC Moves
– RPF moves are blind
– Instead, introduce Metropolis style acceptance step
– Resampled xRk and created xR1:k ={
xRk ; xR1:k`1
}
– Resampled xRk and then sampled from a proposal distribution
xPk ‰ q(: j xRk ) and created xP1:k ={
xPk ; xR1:k`1
}
– Assume q(: j :) symmetric
x1:k =
xP1:k with probability ¸
xR1:k otherwise
¸ =min
(
1;p(xP1:k j y1:k)q(xRk j xPk ))p(xR1:k j y1:k)q(xPk j xRk )
)
=min
(
1;p(yk j xPk )p(xPk j xRk`1)p(yk j xRk )p(xRk j xRk`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 _xk yk _yk 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:
z(i ;j)k =
h(i ;j)k (xk) + w
(i ;j)k if target present
w(i ;j)k if target absent
where
h(i ;j)k (xk) ı
´x´y Ik2ı˚2
exp
{
`(i´x ` xk)
2 + (j´y ` yk)2
2˚2
}
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•Example
– 30 frames and target present in 7 to 22
– SNR = 6.7 dB (unknown)
– 20ˆ 20 cells
Frame 2
Frame 7
Frame 12
Frame 17
Frame 22
Frame 27
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Particle filter output (6 states)
Figures suppressed to reduce file size.
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Particle filter output (6 states)
5 10 15 20 25 300
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
FRAME NUMBER
PR
OB
AB
ILIT
Y
0 2 4 6 8 10 12 142
4
6
8
10
12
14
16
x
y
TRUE PATHESTIMATED
START
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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
−15 −10 −5 0 5 10 15
0
10
20
30
40
50
X [km]
Y [k
m]
(a)
0 50 100 150
0
0.2
0.4
0.6
0.8
1
T I M E [s]
Tra
ck S
core
(c)
0 50 100 150
−800
−600
−400
−200
0
200
400
T I M E [s]
Ran
ge−
Rat
e [k
m/h
]
(b)
TRACKLOSS
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Particle Filter
−10 0 10
0
10
20
30
40
50
X [km]
Y [k
m]
(a)
0 50 100 150
−800
−600
−400
−200
0
200
400
T I M E [s]
Ran
ge−
Rat
e [k
m/h
]
(b)
0 50 100 150
0
0.2
0.4
0.6
0.8
1
T I M E [s]
Tra
ck S
core
(c)
CLOUD OF PARTICLESAT t=108 s
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Track continuity
0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1
Doppler blind zone limit [km/h]
Pro
babi
lity
of T
rack
Mai
nten
ance
PF EKF
T=2s
0 20 40 60 80 100 1200
0.2
0.4
0.6
0.8
1
Doppler blind zone limit [km/h]
Pro
babi
lity
of T
rack
Mai
nten
ance
PF EKF
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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