A priori par normes mixtes pour les problèmes inverses. Application à la localisation de sources en M/EEG

We are interested by under-determined inverse problems, and more specifically by source localization in magneto and electro-encephalography (M/EEG). Although there is a physical model for the diffusion (or “mixing”) of the sources, the (very) under-determined nature of the problem leads to a difficult inversion. The need for strong and physically relevant priors on the sources is one of the challenge. For M/EEG classical sparsity prior based on the l1 norm is not adapted, and gives unrealistic results. We propose to take into account a structured sparsity thanks to the use of mixed norms, especially a mixed norm with three indices. The method is then applied on MEG signals obtained during somesthetic stimulation. When stimulated, hand fingers activate separate regions of the primary somatosensory cortex. The use of a three level mixed norm allows to take this prior into account in the inverse problem in order to correctly recover the organization of associated brain regions. We also show that classical methods fail for this task. MOTS-CLÉS : normes mixtes, problème inverse, opérateurs de proximité, électroencéphalographie, magnétoencéphalographie