Particle Filtering in MEG: from single dipole filtering to Random Finite Sets
A. Sorrentino CNR-INFM LAMIA, Genova
methods for image and data analysis
www.dima.unige.it/~piana/mida/[email protected]
Co-workers
Genova group:
Cristina Campi (Math Dep.)Annalisa Pascarella (Comp. Sci. Dep.)Michele Piana (Math. Dep.)
Long-time collaboration
Lauri Parkkonen (Brain Research Unit, LTL, Helsinki)
Recent collaboration
Matti Hamalainen (MEG Core Lab, Martinos Center, Boston)
Basics of MEG modeling
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2 approaches to MEG source modeling
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Automatic current dipole estimate
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Number of sources constant Source locations fixed
Bayesian filtering allows overcoming these limitations
Common methods:
Manual dipole modeling Automatic dipole modelingEstimate the number of sourcesEstimate the source locationsLeast Squares for source strengths
Manual dipole modeling still the main reference method for comparisons (Stenbacka et al. 2002, Liljestrom et al 2005)
Bayesian filtering in MEG - assumptions
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Bayesian filtering in MEG – key equations
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ESTIMATES
Linear-Gaussian model Kalman filterNon-linear model Particle filter
Likelihood function
Transition kernel
Particle filtering of current dipoles
The key idea: sequential Monte Carlo sampling.
(single dipole space)
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Draw random samples (“particles”) from the prior
Update the particle weights
Resample and let particles evolve
A 2D example – the data
A 2D example – the particles
The full 3D case – auditory stimuliS. et al., ICS 1300 (2007)
Comparison with beamformers and RAP-MUSIC
Two quasi-correlated sources
Pascarella et al., ICS 1300 (2007); S. et al. , J. Phys. Conf. Ser. 135 (2008)
Beamformers: suppression of correlated sources
Comparison with beamformers and RAP-MUSICPascarella et al., ICS 1300 (2007); S. et al. , J. Phys. Conf. Ser. 135 (2008)
Two orthogonal sources
RAP-MUSIC: wrong source orientation, wrong source waveform
Rao-BlackwellizationCampi et al. Inverse Problems (2008); S. et al. J. Phys. Conf. Ser. (2008)
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Can we exploit the linear substructure?
Analytic solution
(Kalman filter)Sampled(particle filter)
Accurate results with much fewer particles
Statistical efficiency increased (reduced variance of importance weights)Increased computational cost
Bayesian filtering with multiple dipolesA collection of spaces (single-dipole space D, double-dipole space,...)A collection of posterior densities (one on each space)Exploring with particles all spaces (up to...)
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One particle = one dipole One particle = two dipoles One particle = three dipoles
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Random Finite Sets – why
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Non uniquess of vector representations of multi-dipole states:(dipole_1,dipole_2) and (dipole_2,dipole_1) same physical state, different points in D X D
Consequence: multi-modal posterior densitynon-unique maximumnon-representative mean ),(),( 122212 dddd
Where is the set of all finite subsets of (single dipole space) equipped with the Mathéron topology
A random finite set of dipoles is a measurable function
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Random Finite Sets - how
Probability measure of RFS: a conceptual definition
Belief measure instead of probability measure
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Probability Hypothesis Density (PHD): the RFS-analogous of the conditional mean
The integral of the PHD in a volume = number of dipoles in that volume
Peaks of the PHD = estimates of dipole parameters
Model order selection: the number of sources estimated dynamically
Multi-dipole belief measures can be derived from single-dipole probability measures
RFS-based particle filter: ResultsS. et al., Human Brain Mapping (2009)
Monte Carlo simulations:1.000 data setsRandom locations (distance >2 cm)Always same temporal waveforms 2 time-correlated sources peak-SNR between 1 and 20
Results: 75% sources recovered (<2 cm) Average error 6 mm, independent on SNR Temporal correlation affects the detectability very slightly
RFS-based particle filter: ResultsS. et al., Human Brain Mapping (2009)
Comparison with manual dipole modeling
Data: 10 sources mimicking complex visual activation
The particle filter performed on average like manual dipole modeling performed by uninformed users (on average 6 out of 10 sources correctly recovered)
In progress
Source space limited to the cortical surface
Two simulated sources
In progress
Two sources recovered with orientation constraint
Only one source recovered without orientation constraint
References
- Sorrentino A., Parkkonen L., Pascarella A., Campi C. and Piana M. Dynamical MEG source modeling with multi-target Bayesian filtering Human Brain Mapping 30: 1911:1921 (2009)
-Sorrentino A., Pascarella A., Campi C. and Piana M. A comparative analysis of algorithms for the magnetoencephalography inverse problem Journal of Physics: Conference Series 135 (2008) 012094.
-Sorrentino A., Pascarella A., Campi C. and Piana M. Particle filters for the magnetoencephalography inverse problem: increasing the efficiency through a semi-analytic approach (Rao-Blackwellization) Journal of Physics: Conference Series 124 (2008) 012046.
-Campi C., Pascarella A., Sorrentino A. and Piana M. A Rao-Blackwellized particle filter for magnetoencephalography Inverse Problems 24 (2008) 025023
- Sorrentino A., Parkkonen L. and Piana M. Particle filters: a new method for reconstructing multiple current dipoles from MEG data International Congress Series 1300 (2007) 173-176