1. Rey R. Ramirez & Sylvain Baillet Neurospeed Laboratory, MEG Program,
Froedtert Hospital & The Medical College of Wisconsin, Milwaukee, WI, USA
Multiresolution geodesic Bayesian algorithms for estimating the spatial extent and shape of distributed sources: Monte Carlo simulations comparing multiscale sparse Bayesian learning (mSBL), sequential mSBL (smSBL), and Matching Pursuit (mMP)
361
Introduction
The source current density distribution generating the electromagnetic signals measured in MEG/EEG is of variable spatial extent and shape. Thus, algorithms that can accurately estimate variably distributed sources of arbitrary shapes are needed. Unfortunately, most methods generate very focal or distributed estimates (depending on which priors are adopted) irrespective of the spatial extent or shape of the true source. Focal estimates can be computed with data-adaptive automatic relevance determination (ARD) priors as used in the Bayesian framework of evidence maximization, SBL, RVM, empirical, hierarchical, and variational Bayes1-5. This framework has recently been modified for multiresolution imaging (e.g., mSBL and mMAP algorithms) 6-9 by performing ARD-based inference on a dictionary generated using a multiscale set of cortically distributed geodesic basis functions (e.g., Gaussians, boxcar, etc). These methods can reconstruct variably distributed sources because the sparse point estimates computed in the larger space of the multiscale hyperparameters correspond to mixtures of variably extended basis functions. Here, we compare under different levels of noise the mSBL algorithm to a new fast sequential mSBL (smSBL) used in machine learning10, which adds, restimates, or removes one basis function at a time. We also compare results to a new fast single basis mMP algorithm that provides a lower bound on performance.
Methods
Spatially distributed sources were grown from seed vertices to desired contiguous areas ( mm2) without any shape constraints and simulated with unit amplitudes for 3 SNRs (infinite, 40, and 20 dB). We tested both single distributed sources and triplets. The noise precision hyperparameter and the basis hyperparameters were automatically learned from the data but had Gamma and Jeffreys hyperpriors respectively. The leadfield matrix was computed using a BEM forward model for a realistic cortical source-space of a subject recorded with the Elekta 306-sensor MEG system. Concatenated geodesic Gaussians and boxcar basis functions of 7 different scales were used. To quantify performance we compared several features of the estimated and simulated sources: 1) their correlation, 2) their intersection/union area ratio (IUAR); and 3) the distances between their geodesic centroids.
Results
For single distributed sources, mSBL performed better than smSBL and mMP in the noiseless and 40dB cases as measured by the performance metrics, but all algorithms performed equally well at 20dB. mMP was robust to noise and fast, but could only be used for single sources. Only mSBL accurately reconstructed multiple distributed sources. Performance did not vary as a function of true source area for any method.
Figure 1 shows true and estimated source images for an example single and a triplet simulation. Figures 2 and 3 show boxplot diagrams of the correlation/IUAR metrics for all SNRs and methods. All the median distances between centroids were around 5 mm. Figure 4 shows the correlation and IUAR boxplot diagrams for source triplets.
Linear model:
Gaussian Likelihood Model:
ARD Prior:
Multiscale Source Covariance Model:
SBL Cost Function:
Hyperpriors:
MacKay Gradient Update Rule for mSBL:
Initialize
Initialize with a single basis and
Compute
Select a candidate basis vector from all
Update noise:
Recompute/Update:
9. If converged terminate, otherwise go to 4.
Sequential mSBL:
References
[1] Tipping, M. (2001), 'Sparse Bayesian learning and the relevance vector machine.', Journal of Machine Learning Research, vol. 1, no. , pp. 211–244. [2] Wipf, D. (2004), 'Sparse Bayesian learning for Basis Selection', IEEE Trans Sig Processing, vol. 52, no. 8, pp [3] Sato, M. (2007), 'Hierarchical Bayesian estimation for MEG inverse problem.', NeuroImage, vol. 23, no. , pp. 806–826. [4] Nummenmaa, A. (2007), 'Hierarchical Bayesian estimates of distributed MEG sources: theoretical aspects and comparison of variational and MCMC methods.', NeuroImage, vol. 23, no. , pp. 669–85. [5] Nummenmaa, A. (2007), 'Automatic relevance determination based hierarchical Bayesian MEG inversion in practice.', NeuroImage, vol. 37, no. , pp. 876–889.
[6] Ramirez, R. (2006), 'Neuroelectromagnetic source imaging using multiscale geodesic neural bases and sparse Bayesian learning.', 12th Annual Meeting of the Organization for Human Brain Mapping, Florence, Italy.
[7] Wipf, D. (2007), 'Analysis of empirical Bayesian methods for neuroelectromagnetic source localization.', Advances in Neural Information Processing Systems 19. MIT Press, Cambridge, MA
[8] Ramirez, R. (2006), 'Sparse Bayesian Learning and the Relevance Vector Machine', 15th International Conference on Biomagnetism., Vancouver, BC, Canada. [9] Ramirez, R. (2007), 'Neuroelectromagnetic source imaging (NSI) toolbox and EEGLAB module.', 37th annual meeting of the Society for Neuroscience, San Diego, CA. [10] Tipping, M. (2003), 'Fast marginal likelihood maximisation for sparse Bayesian models.', Proceedings of the Ninth International Workshop on Artificial Intelligence and Statistics, Key West, FL.
Conclusion
This study suggests that mSBL should be preferred for high SNR data over smSBL and mMP, but that the latter algorithms provide fast alternatives for low SNR data. Thus, accurate estimation of distributed sources may require data denoising (e.g., SSS, ICA, etc) and accurate forward models.
Contact:
Rey R. Ramirez & Sylvain Baillet