A fast MAP algorithm using high order Gibbs priors

The MAP method has the ability to deal with ill-posed problems, involving noisy data, by reducing the number of admissible solutions. However, this method is usually iterative and time consuming and not appropriate to work in a real time basis. In the context of image reconstruction/restoration, Gibbs priors with quadratic potential functions have been used because they allow simple mathematical formulations. However, these priors lead to smooth solutions with a poor representation of transitions. To overcome this problem, several authors have proposed other potential functions, non quadratic, that deal better with the transitions, but lead to an increase of complexity. In this paper, we come back to the Gibbs priors with quadratic potential functions to show that MAP algorithms can be formulated as a linear filtering problem. The recursive nature of the filtering methods allows us to obtain very fast and efficient restoration/reconstruction MAP algorithms. Furthermore, we show that by using quadratic potential functions involving high order differences between neighbors it is possible to significantly improve the solution at the transitions. In fact, with this strategy we are increasing the order of the filter that models the original MAP problem. The approach presented in this paper can be used as a method to design Gibbs priors based on the filter design theory.