Here, fi is the set of voxel locations in the image domain, q is a parameter that satisfies q>1, Ujk is the membership value at voxel location j for class k such that ^k=1 Ujk = 1, y;- is the observed (multispectral) image intensity at location j, vk is the centroid of class k, and gj is the gain field at voxel j. Di is a (known) finite difference operator along the ith dimension of the image. The notation (D * g) refers to the convolution of g with the kernel D and taking the resulting value at the jth voxel. The total number of classes C is assumed to be known. The parameter q is a weighting exponent on each fuzzy membership and determines the amount of "fuzziness" of the resulting classification and is typically set to a value of 2.

The AFCM objective function (Eq. (2)) is minimized when high membership values are assigned to voxels whose intensities are close to the centroid for its particular class, and low membership values are assigned when the voxel intensity is far from the centroid. The brightness variation of the inhomo-geneities is modeled by multiplying the centroids by the gain field gj. The last two terms are first- and second-order regularization terms used to ensure that gj is spatially smooth and slowly varying. The finite difference operators act like derivatives, except they are performed on a discrete domain. The objective function is minimized using an iterative process where the membership functions are computed using a current guess of the centroids and gain field, the centroids are then updated using current estimates of the membership functions and gain field, and finally, the gain field is updated.

Because the fuzzy membership functions provide a soft segmentation and a model for partial volume effects, they have been previously used in the literature for computing tissue volumes [43]. In the comparison provided in this chapter, however, a hard segmentation, obtained by classifying each voxel in the tissue class of highest membership, was used. Figure 2 shows slices from a segmentation obtained by applying AFCM to a double-echo MR data set. Figure 2c is a hard segmentation computed by assigning each pixel to the class with the highest membership value. The membership functions are shown in Figs 2d-2f.

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