The Matérn class is a parametric family of autocorrelation functions that is commonly used in geostatistics. We argue that a generalized, anisotropic version of this model is suitable for capturing the correlation structure of a variety of natural images. We specify the optimal space for the MMSE reconstruction of stochastic Matérn signals from their uniformly-sampled noisy measurements (generalized sampling problem). We prove that the optimal reconstruction space is generated by the multi-integer shifts of a Matérn function which form a Riesz basis. Based on this representation, we propose a practical filter-based reconstruction method that relies on the prior identification of the Matérn parameters from the measured data. We present experimental results to justify the use of the Matérn model and to demonstrate the performance of our signal-adapted reconstruction technique.