ML F\u27ararneter Estirnation for Markov Random Fields, with Applications to Bayesian Tomography

Markuv random fields (MRF) have proven useful for modeling the a priori information in Bayesia.n tomographic reconstruction problems. However, optimal parameter estimation of the MRF model remains a difficult problem due to the intractable nature of the partition function. In this report, we propose a fast parameter estimation scheme to obtain optimal estimates of the free parameters associated with a general MRF model formulation. In particular, for the generalized Gaussian MRF (GGMRF) case, we show that the ML estimate of the temperature T has a simple closed form solution. We present an efficient scheme for the ML estimate of the shape parameter p by an off-line numerical computatio~oif the log of the partition function. We show that this approach can be extended to compute the parameters associated with a general MRF model. In the context of tomographic rec;onstruction, the difficultly of the ML estimation problem is compounded by the fact that the parameters depend on the unknown image. The EM algorithm is used to solve this problem. We derive fast simulation techniques for efficient computation of the expectation step. We also propose a method to extrapolate the estimates when the simulations are terminated prematurely prior to convergence. Experimental results for the emission and transmission case show that the proposed methods result in substantial savings in computation and superior quality images