Pranav Bhagirath

52 Chapter 3 neighborhood. Since labels of neighboring pixels are typically correlated, neighborhood information is incorporated by building a graphical model G (V, E), where voxels are represented by a set of nodes (V) and the relationships among them are represented by edges (E). In the generative Markov random field (MRF) (see Boykov et al., 2001), the Bayes’ relationship is used to determine the posterior distribution: (3) where X is the unseen image to be segmented and Y is the labeling into healthy and scar. The likelihood p (X | Y) of the unseen image is estimated by assuming that the voxel intensities in X are independent given the labels. Also, a uni-modal Gaussian is often used. However, in the context of medical image segmentation, regions are not random collections of independent pixels. Instead, structures usually form coherent and continuous shapes. In this work, a conditional Markov random field (CRF) (Lafferty et al., 2001) is used which is a discriminative framework and the posterior p (Y | X) is estimated by learning a direct map from observations to the class labels (i.e. in training images). This is how it differs from other MRF approaches used in binary classification, where the posterior is estimated using Gaussian distributions. Implementation: The CRF implemented in this work used a hierarchical approach and is described in Karimaghaloo et al. (2012). There are two levels of CRF: in the first level image intensity information was used, and in the second level, a so called spin image feature vector derived from intensity information was used. In the first level CRF, the posterior distribution p (Y | X) was estimated as in a conventional CRF (Lafferty et al., 2001): where Z is a normalization termand φ, Φ and ψ are unary, pair-wise and triplet potentials respectively. Pairwise and triplet potentials measure the interaction between pixels that are immediate neighbors (pairwise) and neighbor’s neighbors (triplet). As regions in MRI images are not random collections of independent pixels but part of coherent and continuous shapes, the pairwise and triplet potentials reinforce this notion. The unary