Implicit priors for model-based inversion

While Markov random field (MRF) models have been widely used in the solution of inverse problems, a major disadvantage of these models is the difficulty of parameter estimation. At its root, this parameter estimation problem stems from the inability to explicitly express the joint distribution of an MRF in terms of the conditional distributions of elements given their neighbors. The objective of this paper is to provide a general approach to solving maximum a posteriori (MAP) inverse problems through the implicit specification of a MRF prior. In this method, the MRF prior is implemented through a series of quadratic surrogate function approximations to the MRF´s log prior distribution. The advantage of this approach is that these surrogate functions can be explicitly computed from the conditional probabilities of the MRF, while the explicit Gibbs distribution can not. Therefore, the Gibbs distribution remains only implicitly defined. In practice, this approach allows for more accurate modeling of data through the direct estimation of the MRF´s conditional probabilities. We illustrate the application of our method with a simple experiments of image denoising and show that it produces superior results to some widely used MRF prior models.