The purpose of this thesis is presenting the Rudin-Osher-Fatemi filter for image analysis. We will consider mostly the application for denoising, however there are other relevant application, such as restoring damaged inscriptions. This filter involves solving a minimization problem with the objective function built summing an $L^2$-norm squared, which measure the distance from the reference image, and the total variation functional. Therefore, one of the main concerns of this work is to understand the behaviour of this problem under a variational point of view, this involves studying the total variation functional and the main properties of the bounded variation functions, on which it is defined.