This work addresses the problem of patch-based image denoising through the unsupervised learning of a probabilistic high-dimensional mixture models on the noisy patches. The model, named hereafter HDMI, proposes a full modeling of the process that is supposed to have generated the noisy patches. To overcome the potential estimation problems due to the high dimension of the patches, the HDMI model adopts a parsimonious modeling which assumes that the data live in group-specific subspaces of low dimension-alities. This parsimonious modeling allows in turn to get a numerically stable computation of the conditional expectation of the image which is applied for denoising. The use of such a model also permits to rely on model selection tools, such as BIC, to automatically determine the intrinsic dimensions of the subspaces and the variance of the noise. This yields a blind denoising algorithm that demonstrates state-of-the-art performance, both when the noise level is known and unknown.