Image restoration with a truncated Gaussian model

The Gaussian model for image restoration has the problem of positive probability densities for pixels outside the realistic range. To solve this problem, we introduce a truncated Gaussian model (TG model). In this model, the tails of the Gaussian distribution are cut off at upper and lower bounds and are replaced by δ peaks at the cut boundaries. We analytically obtain the average performance of the TG model in a mean-field system by solving exactly the infinite-range model and using the replica method. We also compare the infinite-range model to the more realistic two-dimensional case by Monte Carlo simulations. When modeling the TG model, we introduce a generalized prior probability. This prior probability includes the Gaussian, Ising, Q-Ising spin, and TG model as special cases. Thus, we can choose an appropriate model depending on the statistical properties of the images.