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FIGURE 10 Illustration of the validity problem when segmenting with clustering. See also Plate 7.

the number of tissue classes to seek. View (b) depicts the tumor as one of four tissue classes, and this is probably a fairly accurate representation of the tissue structure in this image. However, FCM reliably delivers a partition of the input data at any specified value of c, and as you can see, if c = 12 tissue classes are requested, as in view 10f, FCM happily complies. Medically, of course, this is not a good result. Moreover, computing a validity functional such as the partition coefficient in Eq. (13) will not tell an uninformed user that c = 4 or c = 6 is probably the preferred algorithmic interpretation of this image. We believe that the best approach to this problem lies with a rule-based approach to validity.

Clark et al. [28] have reported results from a completely automated system that extracts glioblastoma multiforme tumors from sequences of transaxial MR images (for the purpose of tumor volume tracking during treatment). FCM is used during preprocessing to remove extracranial areas (air, bone, skin, fat, muscles, etc.), which helps to limit processing in subsequent steps. A rule-based expert system then extracts the intracranial region and initiates the process of iteratively refining the segmentation. Multispectral histogram analysis of the intracranial region followed by region analysis provides the final tumor labeling. Results are provided for 13 unseen volume data sets, with comparisons to supervised, radiologist-labeled "ground truth'' tumor volumes, and supervised fc-nearest neighbor tumor segmentations.

There are many, many studies that perform segmentation with fuzzy models along track USA. Without ground truth images that can be used for quantitative assessment as in Table 5, none of these studies will carry much weight in terms of establishing the clinical efficacy (or lack of it) of a particular model. Human expertise is needed, and it is more efficiently used in the other two tracks shown in Fig. 6. Nonetheless, investigations in track USA are numerous and popular; we conjecture that this is because it is the easiest track to work in without the aid of medically knowledgeable clinicians.

3.2 Unsupervised Segmentation: Track USB

One of the earliest (nonfuzzy) articles discussing the use of rules in the context of MR segmentation was by Menhardt and Schmidt [39]. The use of fuzzy rule-based guidance for the segmentation of medical images, track USB of Fig. 6, apparently began with Li et al. [40]. These authors used FCM as the first step in a knowledge-based (KB) system that automatically segments and labels glioblastoma-multiforme tumors in successive MR slices of the human brain and subsequently estimates total tumor volume. We call the approach discussed in [40] the USF-KB system.

Initial clustering of image sections in the USF-KB approach is performed by FCM with overclustering, i.e., the image is deliberately segmented into more clusters than are known to exist. When FCM uses for c the number of clusters determined by an operator during manual labeling, FCM often separates tissue types incorrectly. Overclustering is based on the premise that multiple clusters containing the same tissue type are easier to merge than the separation of tissue regions in under-segmented clusters. This is in some sense the reverse of the VGC approach: Clusters are merged using mid-level rules rather than low-level pixel-based processing.

After initial FCM segmentation, the overclustered partition and the cluster center for each region are provided to a (crisp) rule-based expert system that contains knowledge gathered from a variety of sources such as radiologist expertise and empirical observations. The USF-KB system includes heuristics concerning tissue characteristics in feature space (T1, T2, p) and anatomical information about the internal structure of the brain.

Using knowledge-based and model-based recognition techniques, the system iteratively locates tissues of interest. These focus-of-attention tissues are analyzed by matching measured to expected characteristics. For tumor segmentation, focus-of-attention is used with additional stages of fuzzy clustering to separate normal brain tissue from pathology, then to separate tumor from nontumor pathology. This cycle is important, as it allows the results of applying FCM to the raw images to guide knowledge application, which in turns guides further clustering, making it possible to break down the more complex problem of tumor segmentation into smaller, more easily attainable goals.

Table 6 [35] shows results from USF-KB processed slices of a single patient with diagnosed glioblastoma multiforme scanned over five repeat sessions. Acronyms are false positive (FP), true positive (TP), false negative (FN), tumor size (TS), and correspondence ratio (CR). The slices were used to refine the rules in the knowledge-based system. Ground truth for these slices in the form of tumor pixel counts were made by a human operator. False positives are nontumor pixels that are mislabeled.

The correspondence ratio shown in Table 6 is computed with the formula CR = (TP -(1/2 * (FP + FN)))/TS. The true positives and false negatives sum to the tumor size, TP + FN = TS. This is one way to assess the overall quality of the segmentation. The value of CR maximizes when there are no FPs or FNs. On the other hand, FPs not in the tumor can cause this number to be negative. Thus, values close to 1 indicate segmentations that closely match human estimates. Table 6 shows that the processed slices for this patient were relatively accurate.

Hillman etal. [41] and Chang etal. [42,43] report success at Texas A&M with a track USB scheme that is in some sense diametrically opposite to the USF-KB model. This group introduces fuzzy rules as a preprocessing stage followed by FCM clustering to segment the remaining pixels in rat and human brain MR images. Fuzzy rules endow the system with somewhat greater reliability than crisp rules at a significant savings in CPU time and storage. Approximately 80% of the pixels are labeled by intensity thresholding with the fuzzy rule-base before clustering. The mean vector of each set of labeled pixels (after a hardening alpha-cut) provides FCM with initial prototypes.

TABLE 6 Pixel counts for a glioblastoma-multiforme patient using the USF-KB system

Patient ID

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