  The Kullback-Leibler divergence or relative entropy is an
information-theoretic measure between statistical models that play an important
role in measuring a distance between random variables. In the study of complex
systems, random fields are mathematical structures that models the interaction
between these variables by means of an inverse temperature parameter,
responsible for controlling the spatial dependence structure along the field.
In this paper, we derive closed-form expressions for the Kullback-Leibler
divergence between two pairwise isotropic Gaussian-Markov random fields in both
univariate and multivariate cases. The proposed equation allows the development
of novel similarity measures in image processing and machine learning
applications, such as image denoising and unsupervised metric learning.
