On the maximum likelihood potential estimates for Gibbs random field image models

Two MLEs of Gibbs potentials in Gibbs random field image models with translation invariant pixel interactions are discussed. The unconditional MLE presents the potentials in an implicit form of a system of stochastic equations to be solved by analytic and stochastic approximation. The conditional MLE, provided a training sample holds the least upper bound (top rank) in the Gibbs energy within the parent population, results in the explicit, to scaling factors, potentials. Then only these factors have to be found using analytic and stochastic approximation. Both MLEs are consistent, in a statistical sense, but may need large training samples for determining the potentials with a tolerable accuracy. For typical in practice small samples the conditional MLE suggests how to interpolate the potentials using the available training data.