Particle Filtering (Sequential Monte Carlo)
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Transcript of Particle Filtering (Sequential Monte Carlo)
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Particle Filtering(Sequential Monte Carlo)Ercan Engin Kuruolu, ISTI-CNR, [email protected]
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outlineReview of particle filteringCase study: Source separation using Particle FilteringApplication: separation of independent components in astrophysical images
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Special caseslinear observations (h) Gaussian observation noise (n) linear state process (f) Gaussian process noise (v)Wiener filterKalman filter
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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
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Nonlinear, non-Gaussian case
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Extended Kalman FilterIt was the classical method for non linear state-space systems
A and H are nonlinearPerform first order Taylor expansion
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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
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sequentialityWe would like to avoid w each time instant and update it sequentially
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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
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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
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Particle Filtering-SummarySequential Monte Carlo techniqueGeneralisation of the Kalman filtering to nonlinear/non-Gaussian systems/signals.Handles nonstationary signals/systems
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Basic Particle Filter - SchematicInitialisationImportancesampling stepResamplingstepmeasurementExtract estimate,
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Importance Sampling step
For sample and set
For evaluate the importance weights
Normalise the importance weights,
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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)
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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.
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SOURCE SEPARATIONModel for observationsModel for the mixing matrixSource modelImportance functionResampling strategy
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Model for observationsAssume linear, instantaneous mixing (extension to the convolutional case is possible)
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Model for the mixingIn general, time-varying mixing matrix
In the lack of prior knowledge, we assume
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Source modelGaussian mixtures
Hidden rv/state
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Evolution of hyperparameters-1
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Evolution of hyperparameters-2
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Particle filteringNeed to evaluate:
Can be estimated by Kalman filterWe are left with:
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Choice of importance functionTo be decided on, a choice can be:
Evaluation of this requires only one step of Kalman Filtering for each particle.
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Resampling strategySampling importance resampling (SIR)Residual resamplingStratified sampling
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Astrophysical source separation
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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
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Noisethe noise variance is known for each pixel
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Source Model: Mixture of GaussiansEach source distribution is modelled by afinite mixture of Gaussians:
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A-priori distribution as importance function
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Hierarchical structure
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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:
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Simulation results 2
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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
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Computer vision applicationsNow lets have a look some results obtained using particle filters in computer vision problems