EEG sparse source localization via Range Space Rotation

The problem of sparse Electroencephalography (EEG) source localization can be formulated as a sparse signal recovery problem. However, the dictionary matrix (Lead Field) of a realistic head model has high coherence, indicating that the sparse signal, corresponding to brain activations might not be recoverable via l₁-norm minimization techniques. In spite of the high coherence in the EEG dictionary matrix, we can still estimate the support of the underlying source signal as long as the problem satisfies the Range Space Property (RSP). In this paper, we show that one can use an initial estimate of the sparse solution to rotate the range of the sensing matrix transpose and obtain high quality source localization. We derive the conditions which the rotation matrix should meet in order to make the unique least l₁-norm solution support match the actual source support. We validate the proposed approach using simulations and a real EEG experiment, and compare the results with those obtained by other methods that have been previously proposed for EEG source localization.