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   $\ell _2$   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   $\ell _2$   norm is known to overestimate the activated spatial areas. One solution to this problem is to promote sparse solutions for instance based on the   $\ell _1$   norm that are easy to handle with optimization techniques. In this paper, we consider the use of an   $\ell _0 + \ell _1$    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   $\ell _0$    pseudonorm handles the position of the nonzero elements while the    $\ell _1$   norm constrains the values of their amplitudes. We use a Bernoulli–Laplace prior to introduce this combined   $\ell _0 + \ell _1$    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    $\ell _2$   and   $\ell _1$    norms regularizations in the presence of pointwise sources. A comparison with a recent method based on multiple sparse priors is also conducted.
['Facundo Costa', 'Hadj Batatia', 'Lotfi Chaari', 'Jean-Yves Tourneret']
Sparse EEG Source Localization Using Bernoulli Laplacian Priors