  This paper is concerned with variational and Bayesian approaches to
neuro-electromagnetic inverse problems (EEG and MEG). The strong indeterminacy
of these problems is tackled by introducing sparsity inducing
regularization/priors in a transformed domain, namely a spatial wavelet domain.
Sparsity in the wavelet domain allows to reach ''data compression'' in the
cortical sources domain. Spatial wavelets defined on the mesh graph of the
triangulated cortical surface are used, in combination with sparse regression
techniques, namely LASSO regression or sparse Bayesian learning, to provide
localized and compressed estimates for brain activity from sensor data.
Numerical results on simulated and real MEG data are provided, which outline
the performances of the proposed approach in terms of localization.
