Particle Filtering(Sequential Monte Carlo)Ercan Engin Kuruolu, ISTI-CNR, Pisakuruoglu@isti.cnr.it

outlineReview of particle filteringCase study: Source separation using Particle FilteringApplication: separation of independent components in astrophysical images

Special caseslinear observations (h) Gaussian observation noise (n) linear state process (f) Gaussian process noise (v)Wiener filterKalman filter

Kalman filterR. 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.

A

Nonlinear, non-Gaussian case

Extended Kalman FilterIt was the classical method for non linear state-space systems

A and H are nonlinearPerform first order Taylor expansion

Unscented Kalman FilterWe will not discuss it here for the time beingYou can read a very clear presentation in http://cslu.cse.ogi.edu/nsel/ukf/ prepared by Eric WanIt provides a second order expansion of Taylor seriesNot analytically but through sampling

sequentialityWe would like to avoid w each time instant and update it sequentially

Resampling strategyDeterministic sampling (fixed points with equal spacing)stratified sampling (random points between fixed intervals)Sampling importance sampling (SIS)Residual resamplingRoughening 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 distributionsOptimal importance function:The posterior itselfThe prior distribution as the importance function:

Easy to implementBut no information from observation!Hybrid importance functionsSomewhere in between

Particle Filtering-SummarySequential Monte Carlo techniqueGeneralisation of the Kalman filtering to nonlinear/non-Gaussian systems/signals.Handles nonstationary signals/systems

Basic Particle Filter - SchematicInitialisationImportancesampling stepResamplingstepmeasurementExtract estimate,

Importance Sampling step

For sample and set

For evaluate the importance weights

Normalise the importance weights,

ApplicationsTracking (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 separationAhmed, 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 modelConvolutional mixingAudio separationEverson, Roberts, Particle Filters for Non-stationary ICA, Advances in Independent Components Analysis, 2000.Only the mixing is nonstationary.Costagli, Kuruoglu, Ahmed, ICA 2004.

SOURCE SEPARATIONModel for observationsModel for the mixing matrixSource modelImportance functionResampling strategy

Model for observationsAssume linear, instantaneous mixing (extension to the convolutional case is possible)

Model for the mixingIn general, time-varying mixing matrix

In the lack of prior knowledge, we assume

Source modelGaussian mixtures

Hidden rv/state

Evolution of hyperparameters-1

Evolution of hyperparameters-2

Particle filteringNeed to evaluate:

Can be estimated by Kalman filterWe are left with:

Choice of importance functionTo be decided on, a choice can be:

Evaluation of this requires only one step of Kalman Filtering for each particle.

Resampling strategySampling importance resampling (SIR)Residual resamplingStratified sampling

Astrophysical source separation

Observation 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

Noisethe noise variance is known for each pixel

Source Model: Mixture of GaussiansEach source distribution is modelled by afinite mixture of Gaussians:

A-priori distribution as importance function

Hierarchical structure

Rao-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

Conclusionswe introduced a new, general approach to solve the source separation problem in the astrophysical contextPF provides better results in comparison with ICA, especially in case of SNR < 10 dBNon-stationary model, non-Gaussian variables, space-varying noiseit is possible to exploit the available a-priori information

Computer vision applicationsNow lets have a look some results obtained using particle filters in computer vision problems