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After FCM segmentation of the unlabeled pixels, the final rat brain segmentation obtained was reportedly satisfactory.

This technique has also been used to study HIV-positive lesions in human MR images. In agreement with the USF-KB system, this group reports that using FCM alone is not sufficient to produce satisfactory segmentations of MR images. They noted again that the computational complexity of FCM was a problem and that FCM had difficulty in directly utilizing spatial (nonfeature) information derived from the MR image. Furthermore, they reported that FCM sometimes mislabeled "easy" pixels of classes that did not have concave hulls in feature space.

3.3 Supervised Segmentation: Track Su

Our discussion of fuzzy methods for supervised medical image segmentation begins with the semisupervised FCM (ssFCM) method. Techniques of this kind in the context of c-means clustering were first discussed by Pedrycz [44]. Algorithms in this category are (i) clustering algorithms that (ii) use a finite design set XL C ffi of labeled data to (iii) help clustering algorithms partition a finite unlabeled data set XT C , and then (iv) terminate without the capability to label other points in W. The prefix "semi" is used because these schemes are not supervised in the sense that labeled training data are used to find the parameters of a classifier D that is subsequently used to complete segmentation of XT as shown in the Su track of Fig. 6.

The semisupervised approach is applicable in domains such as image segmentation, where users may have a small set of manually derived labeled data, and can use it to supervise classification of the remaining pixels in a single image. Initially, partitions of X for semisupervised clustering algorithms have the form

where

When n = Xtr i-, the n{ need not be equal, nor is it necessary that the columns of Utr be crisp. The basic idea is to use (Xtr, Ur) and Xte to find Ute. Roughly speaking, semisupervi-sion exchanges the generalization capability of a classifier trained with (Xtr, Utr) for access to the structural information possessed by the points in both Xtr and Xte while searching for clusters Ute of Xte.

The development and use of ssFCM for MRI segmentation is discussed by Bensaid et al. [45]. In this model a modified version of FCM (i.e., ssFCM) is applied to Xte. The training data set Xtr guides ssFCM toward improved clustering of the unlabeled pixels by splitting the FCM update conditions at (10) for U and V into labeled and unlabeled components. The labeled components of U are fixed as in (18) and can be weighted by class so that ssFCM effectively uses many copies of the relatively few training data in Xr. We exemplify this model in Section 3.D, where ssFCM is used to make volume estimates of tumors.

Segmentation in the USF-KB model has been augmented by the addition of fuzzy rules. Namasivayam and Hall [46] have shown that over a large set of MR images from different patients, fuzzy rules perform most reliably when they are based on relative differences in pixel intensities for different tissue types. Relative fuzzy rules and ssFCM applied to the unlabeled pixels in Xte yield more accurate and much faster (compared to FCM alone) segmentations of normal MR images. ssFCM has also been used in the reclustering stage of the USF-KB system. In this application, crisply labeled training pixels are chosen by a set of rules that identify tissue types with a high degree of confidence.

3.4 Three-Dimensional Applications

The last topic in our chapter considers research that attempts to extend 2D techniques such as segmentation into three dimensions. Typical applications include volume estimation and visualization, both of which are needed in pre- and postoperative surgical analysis. Nonfuzzy work in this area includes Xiaoping et al. [48]. Research along these lines based on either unsupervised and supervised fuzzy methods is quite sparse.

The USF-KB system has been used to estimate tumor volume [35]. Interslice variability makes direct volumetric clustering of pooled sets of MR slices problematic. From a knowledge perspective, developing a model of the entire structure of the brain, even qualitatively, is extremely complex (if possible at all; but see Hata et al. [26,27]). The USF-KB system exploits the fact that clustering performs well on individual slices. A 3D segmentation can be constructed by combining labeled tumor areas from each processed slice (as long as there are no gaps) into a single volume.

Qualitative models called templates in the USF-KB system model specific regions (in the axial plane) of the brain. These models attempt to capture expected changes in the brain's anatomical structure at different points in the volume. Also, slices are processed in contiguous order to allow knowledge gleaned from processing one slice to be propagated axially to assist in classification and labeling decisions in spatially adjacent slices. This is roughly equivalent to imposing an extra layer of supervision onto multiple copies of track USB of Fig. 6.

Figure 11, from [35], compares the total tumor volume estimated by the USF-KB system with the tumor volume based on hand labeling by radiological experts at five observation times (weeks 0, 7, 13, 16, and 20) during the course of treatment of a brain tumor patient. The patient had diagnosed glioblastoma multiforme and was undergoing both chemo and radiation therapy during the 20-week period. There were

FIGURE 11 Knowledge-based vs manual volume estimates [35].

FIGURE 11 Knowledge-based vs manual volume estimates [35].

approximately 9 slices per volume in the axial plane (each 5 mm thick with no gaps between slices) at each of the five sampling periods. The graph shows that the unsupervised knowledge-based system very closely models physician-generated ground truth.

Results from a series of experiments at USF to measure tumor volume from MR brain images using ssFCM, the k-nn rule, and a seed-growing approach called ISG are reported in Vaidyanathan et al. [34]. The objective was to determine how sensitive these three methods were to training data chosen by observers (technicians with some medical physics training). It is difficult to choose training data from the raw images so that the measured tumor volume is consistent over multiple trials. Four patient cases were used in the study, with repeat scans available for 2 patients (3 for patient 1, and 5 for patient 2).

Experiments reported by Vaidyanathan et al. are summarized in Table 7, where tumor volume variability (in percent) resulting from the choice of the training data for each trial is reported. The differences in volume estimates for multiple training sets chosen by one or more observers are given in terms of averages over the 10 tumor volumes obtained from the four patient cases. The tumors involved were either menin-gioma or glioblastoma multiforme. All patients were undergoing therapy during the course of repeat scans. This experiment indicates that, for the cases studied, ssFCM and ISG are less sensitive to the choice of training data by a single observer than the k-nn rule. When more than one observer

TABLE 7 Variability in % of tumor volumes
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