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Sparse EEG Source Localization Using Bernoulli Laplacian Priors

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 ℓ<sub>2</sub> 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 ℓ<sub>2</sub> norm is known to overestimate the activated spatial areas. One solution to this problem is to promote sparse solutions for instance based on the ℓ<sub>1</sub> norm that are easy to handle with optimization techniques. In this paper, we consider the use of an ℓ<sub>0</sub> + ℓ<sub>1</sub> 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 ℓ<sub>0</sub> pseudonorm handles the position of the nonzero elements while the ℓ<sub>1</sub> norm constrains the values of their amplitudes. We use a Bernoulli-Laplace prior to introduce this combined ℓ<sub>0</sub> + ℓ<sub>1</sub> 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 ℓ<sub>2</sub> and ℓ<sub>1</sub> norms regularizations in the presence of pointwise sources. A comparison with a recent method based on multiple sparse priors is also conducted.