In this paper, the minimization of a weighted total variation regularization term with L1 norm as the data fidelity term is addressed using Uzawa block relaxation methods. The unconstrained minimization problem is transformed into a saddle-point problem by introducing a suitable auxiliary unknown. Applying a Uzawa block relaxation method to the corresponding augmented Lagrangian functional, we obtain a new numerical algorithm in which the main unknown is computed using Chambolle projection algorithm. The auxiliary unknown is computed explicitly. Numerical experiments show the availability of our algorithm for salt and pepper noise removal or shape retrieval and also its robustness against the choice of the penalty parameter. This last property allows us to attain the convergence in a reduced number of iterations leading to efficient numerical schemes. Moreover, we highlight the fact that an appropriate weighted total variation term, chosen according to the properties of the initial image, may provide not only a significant improvement of the results but also a geometric filtering of the image components.