We propose a new statistical model for image restoration in which neighbourhoods of wavelet subbands are modeled by a discrete mixture of linear projected Gaussian Scale Mixtures (MPGSM). In each projection, a lower dimensional approximation of the local neighbourhood is obtained, thereby modeling the strongest correlations in that neighbourhood. The model is a generalization of the recently developed Mixture of GSM (MGSM) model, that offers a significant improvement both in PSNR and visually compared to the current state-of-the-art wavelet techniques. However the computation cost is very high which hampers its use for practical purposes. We present a fast EM algorithm that takes advantage of the projection bases to speed up the algorithm. The results show that, when projecting on a fixed data-independent basis, even computational advantages with a limited loss of PSNR can be obtained with respect to the BLS-GSM denoising method, while data-dependent bases of Principle Components offer a higher denoising performance, both visually and in PSNR compared to the current wavelet-based state-of-the-art denoising methods.