Roughness penalties on finite domains

A class of penalty functions for use in estimation and image regularization is proposed. These penalty functions are defined for vectors whose indexes are locations in a finite lattice as the discrepancy between the vector and a shifted version of itself. After motivating this class of penalty functions, their relationship to Markov random field priors is explored. One of the penalty functions proposed, a divergence roughness penalty, is shown to be a discretization of a penalty proposed by Good and Gaskins (1971) for use in density estimation. One potential use in estimation problems is explored. An iterative algorithm that takes advantage of induced neighborhood structures is proposed and convergence of the algorithm is proven under specified conditions. Examples in emission tomographic imaging and radar imaging are given.