MEG/EEG source imaging with a non-convex penalty in the time-frequency domain

Due to the excellent temporal resolution, MEG/EEG source imaging is an important measurement modality to study dynamic processes in the brain. As the bioelectromagnetic inverse problem is ill-posed, constraints have to be imposed on the source estimates to find a unique solution. These constraints can be applied either in the standard or a transformed domain. The Time-Frequency Mixed Norm Estimate applies a composite convex regularization functional promoting structured sparsity in the time-frequency domain by combining an l(2,1)-mixed-norm and an l(1)-norm penalty on the coefficients of the Gabor TF decomposition of the source signals, to improve the reconstruction of spatially sparse neural activations with non-stationary and transient signals. Due to the l(1)-norm based constraints, the resulting source estimates are however biased in amplitude and often suboptimal in terms of source selection. In this work, we present the iterative reweighted Time-Frequency Mixed Norm Estimate, which employs a composite non-convex penalty formed by the sum of an l(2,0.5)-quasinorm and an l(0.5)-quasinorm penalty. The resulting non-convex problem is solved with a reweighted convex optimization scheme, in which each iteration is equivalent to a weighted Time-Frequency Mixed-Norm Estimate solved efficiently using a block coordinate descent scheme and an active set strategy. We compare our approach to alternative solvers using simulations and analysis of MEG data and demonstrate the benefit of the iterative reweighted Time-Frequency Mixed Norm Estimate with regard to active source identification, amplitude bias correction, and temporal unmixing of activations.