In this step, we classified each pixel. Usually, the classification algorithm expects one to know how many classes (roughly) the image would have. The number of classes in the image would be the same as the number of tissue types. A pixel could belong to more than one class, and therefore we used the fuzzy membership function to associate with each pixel in the image. There are several algorithms for computing membership functions, and one of the most efficient ones is Fuzzy C means (FCM) based on the clustering technique. Because of its ease of implementation for spectral data, it is preferred over other pixel classification techniques. Mathematically, we expressed the FCM algorithm below but for complete details, readers are advised to see Bezdek and Hall [180] and Hall and Bensaid [181]. The FCM algorithm computed the measure of membership termed as the fuzzy membership function. Suppose the observed pixel intensities in a multispectral image at a pixel location j is given as yj = [yj1 yj2,..., yjN]T, (9.9)

where j takes the pixel location, and N is the total number of pixels in the data set4 in FCM (see Figs. 9.12 and 9.13) the algorithm iterates between computing the fuzzy membership function and the centroid of each class. This membership function is the pixel location for each class (tissue type), and the value of the membership function lies between the range of 0 and 1. This membership function actually represents the degree of similarity between the pixel vector at a pixel location and the centroid of the class (tissue type); for example, if the membership function has a value close to 1, then the pixel at the pixel location is close to the centroid of the pixel vector for that particular class. The algorithm

can be presented in the following four steps. If ujk is the membership value at location j for class k at iteration p, then Y*f=1ujk = 1. As defined before, yj

is the observed pixel vector at location j and vk is the centroid of class k at iteration p. Thus, the FCM steps for computing the fuzzy membership values are as follows:

1. Choose the number of classes (K) and the error threshold eth, and set the initial guess for the centroids v® where the iteration number p = 0.

2. Compute the fuzzy membership function, given by the equation iiy. v(p)ii-2 u(p = J* - Vk 11--(9.10)

3. Compute the new centroids, using the equation j (u^)2

4 This is not the N used in derivation in section 9.4.1.

Figure 9.12: Fuzzy C mean (FCM) algorithm. Input is an image volume. An observation vector is built. Initially, the current centroid is given by the initial input centroid and K the number of classes. With the observation vector the membership function is computed, and with it a new centroid is computed. This new centroid is compared to the current centroid, and if the error is too large, the new centroid is copied into the current centroid and the process repeats. Otherwise, if the error is below the threshold, the membership function is saved, and the result is a classified image.

Figure 9.12: Fuzzy C mean (FCM) algorithm. Input is an image volume. An observation vector is built. Initially, the current centroid is given by the initial input centroid and K the number of classes. With the observation vector the membership function is computed, and with it a new centroid is computed. This new centroid is compared to the current centroid, and if the error is too large, the new centroid is copied into the current centroid and the process repeats. Otherwise, if the error is below the threshold, the membership function is saved, and the result is a classified image.

4. Convergence was checked by computing the error between the previous and current centroids (yv(i'+1) — v^H). If the algorithm had converged, an exit would be required; otherwise, one would increment p and go to step 2 for computing the fuzzy membership function again. The output of the FCM algorithm was K sets of fuzzy membership functions. We were interested in the membership value at each pixel for each class. Thus, if there were K classes, then we threw out K number of images and K number of matrices for the membership functions to be used in computing the final speed terms.

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