| Summary: | M/EEG mechanisms allow determining changes in the brain activity, which is useful in diagnosing brain disorders such as epilepsy. They consist of measuring the electric potential at the scalp and the magnetic field around the head. The measurements are related to the underlying brain activity by a linear model that depends on the lead-field matrix. Localizing the sources, or dipoles, of M/EEG measurements consists of inverting this linear model. However, the non-uniqueness of the solution (due to the fundamental law of physics) and the low number of dipoles make the inverse problem ill-posed. Solving such problem requires some sort of regularization to reduce the search space. The literature abounds of methods and techniques to solve this problem, especially with variational approaches. This thesis develops Bayesian methods to solve ill-posed inverse problems, with application to M/EEG. The main idea underlying this work is to constrain sources to be sparse. This hypothesis is valid in many applications such as certain types of epilepsy. We develop different hierarchical models to account for the sparsity of the sources. Theoretically, enforcing sparsity is equivalent to minimizing a cost function penalized by an l0 pseudo norm of the solution. However, since the l0 regularization leads to NP-hard problems, the l1 approximation is usually preferred. Our first contribution consists of combining the two norms in a Bayesian framework, using a Bernoulli-Laplace prior. A Markov chain Monte Carlo (MCMC) algorithm is used to estimate the parameters of the model jointly with the source location and intensity. Comparing the results, in several scenarios, with those obtained with sLoreta and the weighted l1 norm regularization shows interesting performance, at the price of a higher computational complexity. Our Bernoulli-Laplace model solves the source localization problem at one instant of time. However, it is biophysically well-known that the brain activity follows spatiotemporal patterns. Exploiting the temporal dimension is therefore interesting to further constrain the problem. Our second contribution consists of formulating a structured sparsity model to exploit this biophysical phenomenon. Precisely, a multivariate Bernoulli-Laplacian distribution is proposed as an a priori distribution for the dipole locations. A latent variable is introduced to handle the resulting complex posterior and an original Metropolis-Hastings sampling algorithm is developed. The results show that the proposed sampling technique improves significantly the convergence. A comparative analysis of the results is performed between the proposed model, an l21 mixed norm regularization and the Multiple Sparse Priors (MSP) algorithm. Various experiments are conducted with synthetic and real data. Results show that our model has several advantages including a better recovery of the dipole locations. The previous two algorithms consider a fully known leadfield matrix. However, this is seldom the case in practical applications. Instead, this matrix is the result of approximation methods that lead to significant uncertainties. Our third contribution consists of handling the uncertainty of the lead-field matrix. The proposed method consists in expressing this matrix as a function of the skull conductivity using a polynomial matrix interpolation technique. The conductivity is considered as the main source of uncertainty of the lead-field matrix. Our multivariate Bernoulli-Laplacian model is then extended to estimate the skull conductivity jointly with the brain activity. The resulting model is compared to other methods including the techniques of Vallaghé et al and Guttierez et al. Our method provides results of better quality without requiring knowledge of the active dipole positions and is not limited to a single dipole activation.
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