A total variation (TV) model with an L1-fidelity term and a spatially adapted regularization parameter is presented in order to reconstruct images contaminated by impulse noise. This model intends to preserve small details while homogeneous features still remain smooth. The regularization parameter is locally adapted according to a local expected absolute value estimator depending on the statistical characteristics of the noise. The numerical solution of the L1-TV minimization problem with a spatially adapted parameter is obtained by a superlinearly convergent algorithm based on Fenchel-duality and inexact semismooth Newton techniques, which is stable with respect to noise in the data. Numerical results justifying the advantage of such a regularization parameter choice rule are presented.