JPEG and Wavelet compression artifacts leading to Gibbs effects and loss of texture are well known and many restoration solutions exist in the literature. So is denoising, which has occupied the image processing community for decades. However, when a noisy image is compressed, a new kind of artifact may appear from the interaction of both degradations. This new kind of artifact is surprisingly never mentioned or studied in the image processing community, with only a few rare exceptions. Yet the importance of such artifacts in very high resolution satellite imaging has recently been recognized. Indeed, such images are mainly used for highly accurate subpixel stereo vision, an application where the presence of this kind of artifact (even if barely visible) is particularly harmful. In this work we present a thorough probabilistic analysis of the kind of degradation that results from the interaction of noise and compression called wavelet outliers , and conclude that their probabilistic nature is characterized by a single parameter q/o that can be inferred from a noise model and a compression model. This analysis provides the conditional probability for a Bayesian MAP estimator, whereas a patch-based local Gaussian prior model is learnt from the corrupted image iteratively, like in state of the art denoising algorithms (non-local Bayes), albeit with the additional difficulty of dealing with non-Gaussian noise during the learning process. The resulting joint denoising and decompression algorithm has been experimentally evaluated under realistic conditions. The results show its ability to simultaneously denoise, decompress and remove wavelet outliers better than the available alternatives, both from a quantitative and a qualitative point of view. As expected, the advantage of our method is more evident for large values of q/o, a situation that naturally occurs in satellite images containing very dark areas (shadows).