Due to its low resolution, any EEG inverse solution provides a source estimate at each voxel that is a mixture of the true source values over all the voxels of the brain. This mixing effect usually causes notable distortion in estimates of source connectivity based on inverse solutions. To lessen this shortcoming, an unmixing approach is introduced for EEG inverse solutions based on piecewise approximation of the unknown source by means of a brain segmentation formed by specified Regions of Interests (ROIs). The approach is general and flexible enough to be applied to any inverse solution with any specified family of ROIs, including point, surface and 3D brain regions. Two of its variants are elaborated in detail: arbitrary piecewise constant sources over arbitrary regions and sources with piecewise constant intensity of known direction over cortex surface regions. Numerically, the approach requires just solving a system of linear equations. Bounds for the error of unmixed estimates are also given. Furthermore, insights on the advantages and of variants of this approach for connectivity analysis are discussed through a variety of designed simulated examples.