Statistical Mechanics of the Bayesian Image Restoration under Spatially Correlated Noise

We investigated the use of the Bayesian inference to restore noise-degraded images under conditions of spatially correlated noise. The generative statistical models used for the original image and the noise were assumed to obey multi-dimensional Gaussian distributions whose covariance matrices are translational invariant. We derived an exact description to be used as the expectation for the restored image by the Fourier transformation and restored an image distorted by spatially correlated noise by using a spatially uncorrelated noise model. We found that the resulting hyperparameter estimations for the minimum error and maximal posterior marginal criteria did not coincide when the generative probabilistic model and the model used for restoration were in different classes, while they did coincide when they were in the same class.