Particle Filtering(Sequential Monte Carlo)

Ercan Engin KuruoErcan Engin Kuruoğğlu, lu,

ISTI-CNR, PisaISTI-CNR, Pisa

kuruoglu@isti.cnr.itkuruoglu@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.

kkk

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

ttt

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

111

11

11111

11

,|

||

,|

1

|

|

||

|

kkk

kkkkk

kkkkkk

kkkk

kkkk

kkkk

q

ppw

xqqq

ppw

xqq

ppw

Yxx

xxxY

YxxYx

xxY

YYx

xxY

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

kkkk

kkkk

nxhy

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,

Ni ,,1 )|(~~1

ikk

ik xxpx

),(~1:0:0

ik

ik

ik xxx

Ni ,,1

)~|( ikk

ik xypw

N

j

jk

ik

ik www

1

/~

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

2

,,

2,,,

1 ,,

,,, 2exp

2

1)(

tzj

tzjtjn

q tzj

tzjtj

j

jj

j j

j

swsp

iZ wtiZ ,p ;

2,,,,,, ,~| tjitjititi jzs N

Evolution of hyperparameters-1

δτδ,-τ~|

and ,τ~

|

:for matrix Evolution

i1-tm,l,

i1-tm,l,1,,,,

10,:1,0

1,,

:0:0,

Ui

tmli

tml

q

l

iql

i

titii

kl

itti

ττ

τ

kzlzp

z

i

iD

τ

Evolution of hyperparameters-2

2

11,,

20,

~|

,0~0

ij

ij,

μij,t-tijtij

μij

,σμ

,σ

N

N

s

ij,t-tijtij

ij

ij

ij,

,σ

,σ

log where

~|

,0~

211,,

20, 0

N

N

Particle filtering

• Need to evaluate:

– Can be estimated by Kalman filter– We are left with:

.,.~|

|||,

:1:0:0

:1:0:1:0:0:1:0:0

Nttt

tttttttt

yhp

ypyhpyhp

N

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