This article is a companion paper of a previous work where we have developed the numerical analysis of a variational model first introduced by L. Rudin, S. Osher and E. Fatemi and revisited by Y. Meyer for removing the noise and capturing textures in an image. The basic idea in this model is to decompose an image f into two components (u+v) and then to search for (u,v) as a minimizer of an energy functional. The first component u belongs to BV and contains geometrical informations while the second one v is sought in a space G which contains signals with large oscillations, i.e. noise and textures. Y. Meyer carried out his study in the whole R^2 and his approach is rather built on harmonic analysis tools. We place ourselves in the case of a bounded set Omega of R^2 which is the proper setting for image processing and our approach is based upon functional analysis arguments. We define in this context the space G, give some of its properties and then study in this continuous setting the energy functional which allows us to recover the components u and v. We present some numerical experiments to show the relevance of the model for image decomposition and for image denoising.