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tumor (pixels incorrectly classified as tumor). Acronyms used in this table and later are white matter (WM), gray matter (GM), and cerebrospinal fluid (CSF).

For each of the first four items in Table 4, the scores corresponding to each algorithm are averages over all 30 segmentations and three radiologists. The percentages of true and false positives reported for items 5 and 6 are simply averaged for each algorithm. Table 4 shows that, for the data used in this study, VGC does enhance segmentations made by FCM. For example, with 99% confidence, VGC better differentiates between WM and GM (row 1); between normal and pathological tissues (row 2); and between tumor and edema (row 3); VGC is also superior to FCM for identifying CSF (row 4).

VGC segmentations are not quite as good as those obtained with the optimized, supervised 7-nn rule. This is as expected, since supervision by labeled data usually improves (or at least should improve) algorithmic outputs. VGC shows significant promise for differentiating between tumor and other tissues compared to FCM, and moreover, the ratio between its percentages of true positive and false positive tumor pixels is higher than the corresponding ratio for the supervised 7-nn rule. With 99% confidence, VGC produces significantly less false positive tumor than 7-nn. On the other hand, even with confidence as low as 90%, 7-nn's true positive tumor percentage is not significantly better than VGC's rate. So, this model seems to be a positive step on the way to truly unsupervised designs.

Velthuizen et al. [36] discuss the use of a modification of the mountain clustering method (MCM, Yager and Filev, [37]) they call M3 for segmentation of MR images. The mountain method produces c point prototypes V = {v1,..., vc}, v{ e SpVi from unlabeled data. MCM begins at c = 1 and continues to add prototypes to V until an objective function threshold is exceeded. Thus, c is neither fixed nor explicitly chosen, but is implicitly specified by the termination threshold used to stop the algorithm.

It is a misnomer to call MCM (or M3) a clustering method because no partition U of X is produced by either of these algorithms, which are more properly viewed as prototype generator methods. However, after the terminal prototypes VMCM or VM3 are found, they can be used to generate a crisp c-partition of X by applying either the HCM necessary condition (9a) or, equivalently, the 1-np classifier at (15) based on them to each point in the (unlabeled) image. Velthuizen et al. [36] use various sets of MR pixel intensities as features. Segmentation by M3 proceeds as follows:

[M3.1] Run M3 on X to find VM3 [M3.2] Construct U, a crisp partition of X with equation (9a) or (15): the label assigned to Xj is the label (index) of the closest prototype

[M3.3] Physically label each cluster in U as a tissue class by matching the pixels in an algorithmic cluster to each of the ground truth tissue clusters. Assign the algorithmic cluster to the tissue class that enjoys maximum pixel matching

[M3.4] Artificially color the labeled image

Segmentation by FCM was done with two initializations: a "standard" initialization Vo (cf. (9) in Velthuizen et al.); and with VM3. We write FCM(V) to indicate FCM initialized with V. FCM generates a terminal fuzzy c-partition UFCM of X that is hardened column by column using Eq. (7), and finally, steps [M3.3] and [M3.4] are performed on the resultant crisp partition. Velthuizen et al. evaluated segmentations of 13 MR images using both types of ground truth (methods GT1 and GT2). Segmentations were produced by four methods: a supervised k-nn rule (used to construct the type GT2 ground truth), unsupervised M3, unsupervised FCM(Vo), and unsupervised FCM(VM3). Comparisons were made visually (method C3) and quantitatively (method PR2).

Figure 9 shows T1 (weighted) input data for a patient who has a brain tumor. Figure 9(b) is the color key: CSF = cerebrospinal fluid; WM = white matter; GM = gray matter, GM-2 = (falsely labeled) gray matter. A supervised k-nn segmentation is shown in Fig. 9c. This image results from an operator choosing labeled subsets of pixels from each tissue class, and then using the standard k-nn rule to label the remaining pixels. This is repeated until a panel of radiologists agree that the k-nn segmentation is good enough to be used as type GT2 ground truth. Ten of the 13 images discussed in this study used this method (GT2) as a basis for comparing the results of the three algorithms (unsupervised M3, unsupervised FCM(Vo), and unsupervised FCM(VM3). The other three images had manual ground truth (GT1).

Figure 9d shows a segmentation achieved by FCM(Vo). The tumor is not detected. Instead, FCM( Vo) finds two gray matter regions that do not correspond to anatomical tissues. The M3 segmentation in Fig. 9e is much better—it finds many of the tumor pixels and does not have a GM-2 tissue region. Finally, panel 9f exhibits the segmentation resulting from the initialization of FCM with the output of M3. This view should be compared to Fig. 9c. It's hard to see on a printed copy, but there is excellent correspondence between the tumor regions in these two views. Table 5, adapted from Velthuizen et al., shows the average performance on pathological tissues for segmentations of 13 images made by unsupervised M3, unsupervised FCM(Vo), and unsupervised FCM(VM3).

When FCM is initialized with Vo, segmentation is not as good as M3—it has nearly 5% more false positives and about 7% fewer true positives in tumor. In edema, the recognition rates are about the same. When FCM is initialized with VM3, there is substantial improvement in the true positive rate for both tissue classes and a slight decrease in edema false positives.

To appreciate the importance of the validity issue when segmenting images by clustering, see Fig. 10a the original digital mammogram of a patient with a breast tumor (image

FIGURE 9 MR segmentations [37].

WNB WFCHVy

FIGURE 9 MR segmentations [37].

MDB001 in the MIAS database [53]). Views (b) to (f) in Fig. 10 are segmentations of this image made by the FCM algorithm given in Table 2. Parameters of these runs are p = 1, m = 2, both norms Euclidean, and termination threshold e = 1.0.

All five segmentations are shown in shades of grey, assigned to each pixel by hardening the columns of the terminal partition with Hin Eq. (7). The variable in these five views is c,

TABLE 5 Average true and false positive pixel counts (in %) for pathological tissues
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