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aThe table shows the number of voxels labeled as "gray matter,'' "white matter,'' and "CSF," the resulting absolute volumes, the corresponding percentage of the total volume, and the total values for each data set.

aThe table shows the number of voxels labeled as "gray matter,'' "white matter,'' and "CSF," the resulting absolute volumes, the corresponding percentage of the total volume, and the total values for each data set.

Now the feature vector x is attributed to the neuron j, i.e., the codebook vector Wj with maximal activation according to Eq. (3). This is obviously equivalent with assigning the feature vector x to the codebook vector w(x) with minimal distance to w(x)

with

Thus, the fuzzy tesselation of the feature space is transformed into an exhaustive and mutually exclusive "hard" tesselation that assigns each feature vector x, i.e., each voxel of the 3D data set to the nearest-neighbor codebook vector Wj.

In a second step, each codebook vector Wj is assigned to a tissue class Xe{1,..., m} (e.g., 1 = gray matter, 2 = white matter, 3 = CSF) that is represented by the codebook vector. For this reason, for each of the N codebook vectors Wj, all the voxels of the 3D data set belonging to this codebook vector according to Eq. (47) are labeled automatically. Interactive visual inspection of the images of the 3D data set that contain the maximal number of pixels belonging to a specific codebook vector Wj usually enables a decision on which tissue class X is represented by this codebook vector. Thus, it is usually sufficient to analyze N images for assigning each codebook vector to a tissue class. If a clear decision for a codebook vector cannot be made, additional images with highlighted pixels belonging to the specific codebook vector can be viewed in order to perform a proper tissue class assignment. As a result, each of the m tissue classes X is represented by a set of codebook vectors wX.

By assigning each voxel to a codebook vector Wj according to Eq. (47) and each codebook vector to a tissue class X, all the voxels of the 3D data set can be attributed to a tissue class. This, however, is eqivalent to the segmentation of the data set with respect to the given tissue classes. Figure 7a shows a T1 weighted image, and Fig. 7b the corresponding segmented image with gray-level representation of the three tissue classes "gray matter,'' "white matter,'' and "CSF."

Besides this simple manual assignment of codebook vectors to tissue classes, segmentation can be performed by an alternative approach: Classification by supervised learning. This is explained in the following paragraph.

8.2 Supervised Classification

This approach makes use of the whole GRBF network explained in Fig. 1. If a feature vector x is presented to the input layer, the neurons of the hidden layer are activated according to Eq. (3). The hidden layer activations aj(x) are transferred via the output weights Sj to the m neurons of the output layer:

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