We examine the general regularization model which is based on total-variation for the structural part and a Hilbert-space norm for the oscillatory part. This framework generalizes the Rudin-Osher-Fatemi and the Osher-Sole-Vese models and opens way for new denoising or decomposition methods with tunable norms, which are adapted to the nature of the noise or textures of the image. We give sufficient conditions and prove the convergence of an iterative numerical implementation, following Chambolle's projection algorithm. In this paper we focus on the denoising problem. In order to provide an automatic solution, a systematic method for choosing the weight between the energies is imperative. The classical method for selecting the weight parameter according to the noise variance is reformulated in a Hilbert space sense. Moreover, we generalize a recent study of Gilboa-Sochen-Zeevi where the weight parameter is selected such that the denoised result is close to optimal, in the SNR sense. A broader definition of SNR, which is frequency weighted, is formulated in the context of inner products. A necessary condition for maximal SNR is provided. Lower and upper bounds on the SNR performance of the classical and optimal strategies are established, under quite general assumptions.