This paper studies the effect on the interaction structure of merging labels in random field models.
In image segmentation, often Markov random field models are employed in which neighbouring (blocks) of pixels have a similar texture . Generally, the number of different textures is unknown. To overcome this problem, a number of Bayesian approaches have been suggested in which the unknown number is treated as a random variable. See for example [4, 7].
A practical problem with such approaches is that most Markov random field densities are known only up to a normalising constant. When updating the number of labels in a Monte Carlo method, the normalising constants do not cancel out and have to be approximated. A more fundamental problem is that the interaction structure may change dramatically if two labels are pooled together, in other words, the class of Markov random fields is not closed under merging labels, making them unnatural models in an unsupervised image segmentation algorithm. In contrast, we show that the class of Markov connected component models  is closed under the above mentioned operation, and hence may provide more natural prior distributions for image segmentation with an unknown number of different textures.
The plan of this note is as follows. First we review some random field theory in Section 2, the main Section 3 studies the effect on the interaction structure of changes in the number of labels and compares the results to their counterparts in a continuous point process set-up [2, 9]. The paper closes with a short discussion.