CHAPTER 6
Markov Random Field Models
Teerasit Kasetkasem
6.1
Introduction
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 ofMRF 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.
The pioneering work by Geman and Geman (1984) introduced a statistical methodology based on an MRF model. Their work inspired a continuous
stream of researchers to employ MRF models for a variety of image analysis
tasks such as image classification and segmentation. In their paper, a noiseless
image was assumed to have MRF properties. The noiseless image was disturbed
P. K. Varshney et al., Advanced Image Processing Techniques for Remotely Sensed Hyperspectral Data
© Springer-Verlag Berlin Heidelberg 2004
Markov Random Field Models
Teerasit Kasetkasem
6.1
Introduction
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 ofMRF 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.
The pioneering work by Geman and Geman (1984) introduced a statistical methodology based on an MRF model. Their work inspired a continuous
stream of researchers to employ MRF models for a variety of image analysis
tasks such as image classification and segmentation. In their paper, a noiseless
image was assumed to have MRF properties. The noiseless image was disturbed
P. K. Varshney et al., Advanced Image Processing Techniques for Remotely Sensed Hyperspectral Data
© Springer-Verlag Berlin Heidelberg 2004