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## Random Fields

The idea behind random fields is that the value at each pixel is chosen by a two-dimensional stochastic process. Imagine removing one pixel from an image. Given a type of probability distribution and all the other pixels in the image, we can determine the most likely parameters of that distribution. From the parameters we can express the probability that the missing pixel has any one particular value.

Obviously, performing this calculation would be very expensive and of little use, since the random field model for texture segmentation assumes that different regions of the image have their pixel values drawn from different distributions. Fortunately, we have the Markov property, which is stated as follows: the probability that a pixel has a certain grey level given every other pixel in the image is equal to the probability that the pixel has that same grey level given only the pixels in a neighborhood surrounding that pixel. The probability distribution is often chosen to be Gaussian, resulting in a special class of random fields called Gaussian Markov Random Fields, or GMRF's. Some parameters that can be computed from GMRFs are mean, variance, and autocovariance in different directions.

Another important relative is known as Gibbs Random Fields, or GRF's. GRFs also obey the Markov property, but they use an exponential probability distribution that is derived from an energy function. The energy function is estimated by looking at small neighborhoods called cliques (sometimes as small as pairs of pixels). Each clique contributes one parameter to the distribution.

Mark A. Ruzon
Fri Sep 5 16:40:07 PDT 1997