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Finite Mixture Model

For every and ,

is independent of the individual sites and called a mixing parameter. We take as the model parameter set with

Consider two configurations and . From (1) and (2), we can compute the joint probability distribution of x and y dependent on the model parameters ( is treated as a set of random variables), namely

 (3)

We can compute the marginal distribution of Yi=y, dependent on the parameter set :
 = = (4)

This is the so-called finite mixture (FM) model. The FM model is the most frequently employed statistical model, due to its simple mathematical form [21,29]. But it is clear that the FM model only describes the data statistically. No spatial information about the data is utilized. In other words, the FM model is spatially independent and can therefore be specified fully by the histogram of the data. However, images with the same intensity distribution may have totally different structural properties. For this reason, FM model is not complete. To overcome this drawback, certain spatial properties, or constraints, have to be incorporated into the model. Here, spatial properties means quantifiable structural characteristics of an object, such as size and shape. Under certain intensity distributions, we want the model to be adaptive'' to structural information or spatially dependent in order to fit the actual image better. This leads to the consideration of MRF theory and our HMRF model.

Next: Markov Random Field Theory Up: Hidden Markov Random Field Previous: Hidden Markov Random Field
Yongyue Zhang
2000-05-11