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OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. This is an author-deposited version published in : http://oatao.univ-toulouse.fr/ Eprints ID : 14691 Abstract-Source localization in electroencephalography has received an increasing amount of interest in the last decade. Solving the underlying ill-posed inverse problem usually requires choosing an appropriate regularization. The usual 2 norm has been considered and provides solutions with low computational complexity. However, in several situations, realistic brain activity is believed to be focused in a few focal areas. In these cases, the 2 norm is known to overestimate the activated spatial areas. One solution to this problem is to promote sparse solutions for instance based on the 1 norm that are easy to handle with optimization techniques. In this paper, we consider the use of an 0 + 1 norm to enforce sparse source activity (by ensuring the solution has few nonzero elements) while regularizing the nonzero amplitudes of the solution. More precisely, the 0 pseudonorm handles the position of the nonzero elements while the 1 norm constrains the values of their amplitudes. We use a Bernoulli-Laplace prior to introduce this combined 0 + 1 norm in a Bayesian framework. The proposed Bayesian model is shown to favor sparsity while jointly estimating the model hyperparameters using a Markov chain Monte Carlo sampling technique. We apply the model to both simulated and real EEG data, showing that the proposed method provides better results than the 2 and 1 norms regularizations in the presence of pointwise sources. A comparison with a recent method based on multiple sparse priors is also conducted.Index Terms-Electroencephalography (EEG), inverse problem, 0 + 1 norm regularization, Markov chain monte carlo (MCMC), source localization, sparse Bayesian restoration.

OATAO is an open access repository that collects the work of Toulouse researchers and makes it freely available over the web where possible. Abstract-A reliable leadfield matrix is needed to solve the magnetoencephalography/electroencephalography (M/EEG) source localization problem. The computation of this matrix requires several physical parameters, including the conductivity of the tissues that compose the subject's head. Since it is not precisely known, we modify a recent Bayesian algorithm to estimate the skull conductivity jointly with the brain activity directly from the M/EEG measurements. Synthetic and real data are used to compare our technique with two optimization algorithms, showing that the proposed method is able to provide results of similar or better quality with the advantage of being applicable in a more general case.

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In this paper, we propose a hierarchical Bayesian model approximating the ℓ20 mixed-norm regularization by a multivariate Bernoulli Laplace prior to solve the EEG inverse problem by promoting spatial structured sparsity. The posterior distribution of this model is too complex to derive closed-form expressions of the standard Bayesian estimators. An MCMC method is proposed to sample this posterior and estimate the model parameters from the generated samples. The algorithm is based on a partially collapsed Gibbs sampler and a dual dipole random shift proposal for the non-zero positions. The brain activity and all other model parameters are jointly estimated in a completely unsupervised framework. The results obtained on synthetic data with controlled ground truth show the good performance of the proposed method when compared to the ℓ21 approach in different scenarios, and its capacity to estimate pointlike source activity.