We propose a new class of models for image restoration and decomposition by functional minimization. Following ideas of Y. Meyer in a total variation minimization framework of L. Rudin, S. Osher, and E. Fatemi, our model decomposes a given (degraded or textured) image u(0) into a sum u + v. Here u is an element of BV is a function of bounded variation (a cartoon component), while the noisy (or textured) component v is modeled by tempered distributions belonging to the negative Hilbert-Sobolev space H-s. The proposed models can be seen as generalizations of a model proposed by S. Osher, A. Sole, L. Vese and have been also motivated by D. Mumford and B. Gidas. We present existence, uniqueness and two characterizations of minimizers using duality and the notion of convex functions of measures with linear growth, following I. Ekeland and R. Temam, F. Demengel and R. Temam. We also give a numerical algorithm for solving the minimization problem, and we present numerical results of denoising, deblurring, and decompositions of both synthetic and real images.