It is of great significance to infer activation extents under different cognitive tasks in neuroscience research as well as clinical applications. However, the EEG electrodes measure electrical potentials on the scalp instead of directly measuring activities of brain sources. To infer the activated cortex sources given the EEG data, many approaches were proposed with different neurophysiological assumptions. Traditionally, the EEG inverse problem was solved in an unsupervised way without any utilization of the brain status label information. We propose that by leveraging label information, the task related discriminative extended source patches can be much better retrieved from strong spontaneous background signals. In particular, to find task related source extents, a novel supervised EEG source imaging model called Graph regularized Variation-Based Sparse Cortical Current Density (GVB-SCCD) was proposed to explicitly extract the discriminative source extents by embedding the label information into the graph regularization term. The graph regularization was derived from the constraint that requires consistency for all the solutions on different time points within the same class. An optimization algorithm based on the alternating direction method of multipliers (ADMM) is derived to solve the GVB-SCCD model. Numerical results show the effectiveness of our proposed framework.