Markov Random Field Models

Abstract

For decades, Markov random fields (MRF) have been used by statistical physicists to explain various phenomena occurring among neighboring particles because of their ability to describe local interactions between them. In Winkler (1995) and Bremaud (1999), an MRF model is used to explain why neighboring particles are more likely to rotate in the same direction (clockwise or counterclockwise) or why intensity values of adjacent pixels of an image are more likely to be the same than different values. This model is called the Ising model. There are a large number of problems that can be modeled using the Ising model and where an MRF model can be used. Basically, an MRF model is a spatial-domain extension of a temporal Markov chain where an event at the current time instant depends only on events of a few previous time instants. In MRF, the statistical dependence is defined over the neighborhood system, a collection of neighbors, rather than past events as in the Markov chain model. It is obvious that this type of spatial dependence is a common phenomenon in various signal types including images. In general, images are smooth and, therefore, the intensity values of neighboring pixels are highly dependent on each other. Because of its highly theoretical and complex nature, and intensive computational requirements, practical uses of MRF were extremely limited until recently. Due to the dramatic improvements in computer technologies, MRF modeling has become more feasible for numerous applications, for example, image analysis because many image properties, such as texture, seem to fit an MRF model, i.e., intensity values of neighboring pixels of images are known to be highly correlated with each other. The Markovian nature of these textural properties has long been recognized in the image processing community, and has widely been used in a variety of applications (e.g. image compression and image noise removal). However, these applications were limited to empirical studies and were not based on a statistical model such as the MRF model.