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We have developed a new sparse source imaging (SSI) method with the use of the L1-norm in EEG inverse problems to reconstruct extended cortical current sources. The new SSI method explores the sparseness in cortical current density variation maps (the transform domain) other than in the cortical current density maps (the original domain) from previously reported SSI methods. The new SSI is assessed by a series of computer simulations. The performance of SSI is compared with the well-known L2-norm MNE using the AUC metric. Our present simulation data indicate that the new SSI has significantly improved performance in reconstructing extended cortical current sources and estimating their cortical extents. The L2-norm MNE shows relatively poor performance in the same source imaging tasks. The new SSI method is also applicable to MEG source imaging.