The surface Laplacian is known to be a theoretical reliable approximation of the cortical activity. Unfortunately, because of its high pass character and the relative low density of the EEG caps, the estimation of the Laplacian itself tends to be very sensitive to noise. We introduce a method based on vector field regularization through diffusion for denoising the Laplacian data and thus obtain robust estimation. We use a forward-backward diffusion aiming for source energy minimization while preserving contrasts between active and nonactive regions. This technique uses headcap geometry specific differential operators to counter the low sensor density. The comparison with classical denoising schemes clearly demonstrates the advantages of our method. We also propose an algorithm based on the Gauss-Ostrogradsky theorem for estimation of the Laplacian on missing (bad) electrodes, which can be combined with the regularization technique in order to provide a joint validation framework.