Particle Filtering (Sequential Monte Carlo) Ercan Engin Kuruoğlu, ISTI-CNR, Pisa kuruoglu@isti.cnr.it

outline Review of particle filtering Case study: Source separation using Particle Filtering Application: separation of independent components in astrophysical images

non-stationary processes Special cases linear observations (h) Gaussian observation noise (n) linear state process (f) Gaussian process noise (v) non-stationary processes stationary processes Wiener filter Kalman filter

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. 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

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 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

Particle Filtering-Summary Sequential Monte Carlo technique Generalisation of the Kalman filtering to nonlinear/non-Gaussian systems/signals. Handles nonstationary signals/systems

Basic Particle Filter - Schematic Initialisation measurement Resampling step Importance sampling step Extract estimate,

Importance Sampling step For sample and set For evaluate the importance weights Normalise the importance weights,

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)

Model for the mixing In general, time-varying mixing matrix In the lack of prior knowledge, we assume

Source model Gaussian mixtures Hidden rv/state

Evolution of hyperparameters-1

Evolution of hyperparameters-2

Particle filtering Need to evaluate: Can be estimated by Kalman filter We are left with:

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.

Resampling strategy Sampling importance resampling (SIR) Residual resampling Stratified 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

Noise the noise variance is known for each pixel

Source Model: Mixture of Gaussians Each source distribution is modelled by a finite mixture of Gaussians:

A-priori distribution as “importance function”

Hierarchical structure

Rao-Blackwellisation It 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