The main idea here is to classify a particular image into a number of regions or classes. Thus for each pixel in the image we need to somehow decide or estimate which class it belongs to. There are a variety of approaches to do region based segmentation and to our understanding the performance does not change from one method to the other considerably. Since the emphasis of this paper lies on an integrated boundary finding approach given the raw image and the region classified image, it does not matter too much which method is being used to get the region classified image as long as the output of that method gives reasonable results.
For our purposes, we use one of the popular methods available in the literature , which models the image as a Markov Random Field (MRF) and a Maximum a posteriori (MAP) probability approach is used to do the classification. The problem is posed as an objective function optimization, which in this case is the a posteriori probability of the classified image given the raw data which constitutes the likelihood term, and the prior probability term, which due to the MRF assumption is given by the Gibb's distribution.
It can be shown that it is equivalent to,
where corresponds to the actual image data, corresponds to the region classified image and represents the classes in . The subscript represents the neighborhood of the pixel leaving out the pixel. Not only do we need to estimate the pixel classification, we also need to estimate the class properties . However, as shown in , the much simpler problem of first estimating the class properties and then doing the classification gives almost identical results. For our case we shall further assume that we know beforehand the class characteristics. Finding a global maxima to such a problem is generally computationally prohibitive. One can do it using Gibbs sampler and simulated annealing as in Geman et al.. Similar to Leahy et al, and Pappas we have used the co-ordinate wise descent method, similar to iterated conditional modes approach due to Besag . Here, at every iteration, the probability of a particular pixel being classified to different classes is computed and the pixel is classified to that class that gives the highest probability. The procedure stops when there is no change between iterations. It actually corresponds to instantaneous freezing in simulated annealing.