This paper deals with numerical schemes for image restoration. These schemes rely on a duality-based algorithm proposed in 1979 by Bermudez and Moreno, and on general minimization schemes recently developped by Y. Nesterov. Total variation regularization and smoothed total variation regularization are investigated. Algorithms are presented for such regularizations in image restoration. We prove the convergence of all the proposed schemes. We illustrate our study with numerous numerical examples, and we make comparisons between the different schemes presented in the paper. Our experiments are in favor of Bermudez-Moreno approach to get a fast approximation for smoothed total variation regularization, whereas Y. Nesterov.