The Markov Random Field in Materials Applications: A synoptic view for signal processing and materials readers

The Markov random field (MRF) is one of the most widely used models in image processing, constituting a prior model for addressing problems such as image segmentation, object detection, and reconstruction. What is not often appreciated is that the MRF owes its origin to the physics of solids, making it an ideal prior model for processing microscopic observations of materials. While both fields know of their respective interpretations of the MRF, each knows very little about the other’s version of it. Hence, both fields have “blind spots,” where some concepts readily appreciated by one field are completely obscured from the other. With this in mind, the objectives of this article are to 1) develop a synoptic view of the MRF, the related Gibbs distribution, and the Hammersley–Clifford theorem that links them, in such a way that signal processing and materials readers will see them from the same perspective; and 2) explain physics-based regularization using the MRF and describe how it can provide insight into the performance of MRF-based segmentation methods. While the MRF has already been used in many machine learning contexts, we will use a simpler, more transparent method to illustrate the fundamental behavior of the MRF, with the understanding that this behavior should be inherent in more complex learning approaches.