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A mathematical procedure, called deblurring, was developed to reduce spatial blur distortion of scalp-recorded brain potentials due to transmission through the skull and other tissues. Deblurring estimates potentials at the superficial cerebral cortical surface from EEGs recorded at the scalp using a finite-element model of each subject's scalp, skull, and cortical surface constructed from their magnetic resonance images (MRIs). Simulations indicate that deblurring is numerically stable, and a comparison of deblurred data with a direct cortical recording from a neurosurgery patient suggests that the procedure is valid. Application of deblurring to somatosensory evoked potential data recorded at 124 scalp sites suggests that the method greatly improves spatial detail and merits further development.