particle filtering (sequential monte carlo) ercan engin kuruoğlu, isti-cnr, pisa...
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Particle Filtering(Sequential Monte Carlo)
Ercan Engin KuruoErcan Engin Kuruoğğlu, lu,
ISTI-CNR, PisaISTI-CNR, Pisa
[email protected]@isti.cnr.it
outline
• Review of particle filtering
• Case study: Source separation using Particle Filtering
• Application: separation of independent components in astrophysical images
Special cases
linear observations (h)
Gaussian observation noise (n)
linear state process (f)
Gaussian process noise (v)
Wiener filter Kalman filter
tttmtt nDsHy ,:1
1t t t t x A x v
Kalman filter
• R. Kalman (1960), Swerling (1958)
• In control theory: linear quadratic estimation (LQE).
• Kalman filters are based on linear dynamical systems discretised in the time domain.
• They are modelled on a Markov chain built on linear operators perturbed by Gaussian noise.
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kkk
vHxy
uFxx
1
A
Nonlinear, non-Gaussian case
Extended Kalman Filter
• It was the classical method for non linear state-space systems
– A and H are nonlinear
• Perform first order Taylor expansion
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ttt
uxHy
vxAx
)(
)(1
Unscented Kalman Filter
• We will not discuss it here for the time being• You can read a very clear presentation in http://
cslu.cse.ogi.edu/nsel/ukf/ prepared by Eric Wan• It provides a second order expansion of Taylor
series– Not analytically but through sampling
sequentiality
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We would like to avoid w each time instant and update it sequentially
Resampling strategy
• Deterministic sampling (fixed points with equal spacing)• stratified sampling (random points between fixed
intervals)• Sampling importance sampling (SIS)• Residual resampling• Roughening and editing (adds independent jitter)• For details see:
– A survey of convergence results on particle filtering methods for practitioners by Crisan, D.; Doucet, A. IEEE Transactions on Signal Processing, Volume 50, Issue 3, Mar 2002 Page(s):736 - 746
Proposal distributions• Optimal importance function:
– The posterior itself
• The prior distribution as the importance function:
– Easy to implement– But no information from observation!
• Hybrid importance functions– Somewhere in between
)|p( :1:1 kk yx
Particle Filtering-Summary
• Sequential Monte Carlo technique
• Generalisation of the Kalman filtering to nonlinear/non-Gaussian systems/signals.
• Handles nonstationary signals/systems
equationn observatio :,
equation state :, 11
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vxfx
Basic Particle Filter - SchematicInitialisation
Importancesampling step
Resamplingstep
0k
1 kk
)}(~,{ :0:0i
kki
k xwx
},{ 1:0
Nxik
measurement
ky
Extract estimate, kx :0ˆ
• Importance Sampling step
– For sample and set
– For evaluate the importance weights
– Normalise the importance weights,
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ik
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Applications
• Tracking (Gordon et al.)• Audio restoration (Godsill et al.)• CDMA (Punskaya et al.)• Computer vision (Blake et al.)• Genomics (Haan and Godsill)• Array processing (Reilly et al.)• Financial time series (de Freitas et al.)• sonar (Gustaffson)
Applications: source separation• Ahmed, Andrieu, Doucet, Rayner, “Online non-stationary
ICA using mixture models”, ICASSP 2000.
• Andrieu, Godsill, “A particle filter for model based audio source separation”, ICA 2000.– Source: Gaussian model
– Convolutional mixing
– Audio separation
• Everson, Roberts, “Particle Filters for Non-stationary ICA, Advances in Independent Components Analysis, 2000.– Only the mixing is nonstationary.
• Costagli, Kuruoglu, Ahmed, ICA 2004.
SOURCE SEPARATION
• Model for observations
• Model for the mixing matrix
• Source model
• Importance function
• Resampling strategy
Model for observations
• Assume linear, instantaneous mixing (extension to the convolutional case is possible)
tttmtt nDsHy ,:1
i.i.d. ,,0~ mtn IN
varying-time:tH jittjih ,,, H
Model for the mixing
• In general, time-varying mixing matrix
• In the lack of prior knowledge, we assume
)][( , ,,1 )1( tjitttttt hinj
hvBhAh
hot h 0,0~,,0~ NN Iv
nmtnmt aIBIA ,
Source model
• Gaussian mixtures
• Hidden rv/state
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,,, 2exp
2
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2,,,,,, ,~| tjitjititi jzs N
Evolution of hyperparameters-1
δτδ,-τ~|
and ,τ~
|
:for matrix Evolution
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10,:1,0
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:0:0,
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Evolution of hyperparameters-2
2
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ij,
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μij
,σμ
,σ
N
N
s
ij,t-tijtij
ij
ij
ij,
,σ
,σ
log where
~|
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20, 0
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Particle filtering
• Need to evaluate:
– Can be estimated by Kalman filter– We are left with:
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:1:0:0
:1:0:1:0:0:1:0:0
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it
ittt dwyp i
t:0:0:1:0 θθ~|θ
:0
Choice of importance function
• To be decided on, a choice can be:
– Evaluation of this requires only one step of Kalman Filtering for each particle.
1:11:0 |,|π ttttt py
Resampling strategy
• Sampling importance resampling (SIR)
• Residual resampling
• Stratified sampling
Astrophysical source separationAstrophysical source separation
Observation ModelObservation Model
• n observation channels (30-857 GHz)• H mixing matrix (allowed to be space-varying)• m sources (non-Gaussian and non-stationary)• w space-varying Gaussian noise
Noise
• the noise variance is known for each pixel
Source Model: Mixture of GaussiansSource Model: Mixture of Gaussians
Each source distribution is modelled by a
finite mixture of Gaussians:
A-priori distribution as “importance A-priori distribution as “importance function”function”
Hierarchical structureHierarchical structure
Rao-BlackwellisationRao-BlackwellisationIt is possible to reduce the size of the parameter set
in the Sequential Importance Sampling step:
the mixing matrix H
(re-parametrized into a vector h)
is obtained subsequently through the Kalman Filter:
Simulation results 2
Conclusions
• we introduced a new, general approach to solve the source separation problem in the astrophysical context
• PF provides better results in comparison with ICA, especially in case of SNR < 10 dB
• Non-stationary model, non-Gaussian variables, space-varying noise
• it is possible to exploit the available a-priori information
Computer vision applications
• Now let’s have a look some results obtained using particle filters in computer vision problems