We propose a model for denoising and deblurring. It consists of a system of linear partial differential equations with locally constant coefficients, obtained as a local linearization of the total variation (TV) models (see Rudin, L. et al., Physica D, vol.60, p.259-68, 1992). The keypoint of our model is to get the local inversion of the Laplacian operator, which is done via the fast Fourier transform (FFT). Two local schemes are developed: a pointwise one and a piecewise one. We analyze them both, and their advantages and limitations.