Using fuzzy rules for segmentation

The fuzzy rules for partially segmenting MR images of the brain are built to operate on the T1, T2 and proton density weighted intensity feature images. The first step in developing a set of fuzzy rules to segment an image is determining the antecedent fuzzy sets. Hence, it is necessary to find thresholds that separate tissue types in each of the 3 feature images.

In order to build fuzzy rules that apply to a large number of images, the tissue thresholds, which determine the antecedent fuzzy sets of the rule, are found via histogram analysis applied to each image slice to which the rules will be applied. Figures 3, 4, 5 show a typical set of intensity histograms with ``turning points'' which can be used to approximately separate tissue types. For example, all voxels below b1 in the PD histogram are air with those between b2 and b4 generally white matter (Figure 4), and voxels between a1 and a2 in the T1 histogram (Figure 3) are a mixture of gray and white matter. The histogram shape remains approximately the same across normal subjects and as will be seen will have an expected set of changes for patients with brain pathology. All patients with pathology have been injected with gadolinium whose magnetic properties cause enhancement in regions where the blood, brain barrier have been breached (i.e. regions where tumor exists).

The existence of turning points was discovered by examining the histograms for a set of training images. This research used 6 normal and 4 abnormal slices, which were segmented or ground truthed by expert radiologists into tissues of interest, as a training set. Projections of voxels, known to be of a given tissue type, onto one or more of the histograms shown in Figures 3, 4, 5 allowed us to choose the turning points. The turning points in the histograms are essentially the approximate boundaries between tissue types. The turning points are automatically chosen on each test slice. From the turning points in the histogram, fuzzy rules to identify 4 tissue classes (white matter, gray matter, air/bone or background, and other or skull tissues such as fat, viscous fluid in the eyes, etc.) can be generated. The rules and antecedent fuzzy sets were generated by examining the intersection of tissue types in the three intensity histograms.

**Figure 3:** T1 histogram with turning points
$\begin{figure} \centerline{ \psfig {figure=r67s17.t1.tp.ps,width=2.5in,height=1.5in} }\end{figure}$

**Figure 4:** PD histogram with turning points
$\begin{figure} \centerline{ \psfig {figure=r67s17.pd.tp.ps,width=2.5in,height=1.5in} }\end{figure}$

**Figure 5:** T2 histogram with turning points
$\begin{figure} \centerline{ \psfig {figure=r67s17.t2.tp.ps,width=2.5in,height=1.5in} }\end{figure}$

The rules adapt to each slice processed because they are generated from the turning points found on each slice. So, technically the rules' membership functions are automatically generated for each slice. The turning points are either peaks, valleys or the beginning of a hill in a histogram. The peaks and valleys can be found by searching for a maximum/minimum histogram value. The hill beginning is found by first creating intensity bins of width 30 (they contain 30 intensity levels). Next the approximate hill starting point is found by comparing the histogram sum in the first bin with the corresponding sum of the succeeding bin. If the ratio is greater than or equal to our ratio threshold of 1.8, then the middle intensity level of the bin was chosen as the beginning point of the hill. If the ratio is less than the threshold the next two bins are tested with the procedure continuing until a hill begin point is found. If a peak is found before a hill begin point, the ratio is reduced by 0.1 and the process is restarted.

Neither csf, which is a small class, nor pathology show up as a clear peak in any of the histograms. It was found that pathology and csf could be partially distinguished by viewing the voxel intensity as a percentage of the range of intensities in either T1 or T2 weighted images. This approach enabled rules to be generated for csf and pathology. The 6 fuzzy rules generated are shown in Figure 6.

**Figure 6:** Fuzzy rules for MR image segmentation.
$\begin{figure} IF {\em voxel in T1 is Set-E} \\ AND {\em voxel in T2 is Set-F} \... ...\ MIN was used as the fuzzy 'and' in the rules and $NOT(x) = 1 - x$.\end{figure}$

The fuzzy sets used to generate the fuzzy rules are shown in Figure 7 and together with the rules indicate how the turning points based on histogram shape can be used to separate voxels of different tissue types.

**Figure 7:** Fuzzy sets created using turning points from histograms (a) and Fuzzy sets created for identifying csf (b)
$\begin{figure} \centerline{ \psfig {figure=fzsets_t1.eps,width=2.5in,height=1.5i... ...{figure=fzsets_csf_t2.eps,width=2.5in,height=1.5in} } \centerline{b}\end{figure}$

Finally, the voxel memberships in all classes are normalized to 1 using:

$\begin{displaymath} \mu_i \\ gt (x) = \frac{\mu_i \\ gt (x)}{\sum_j \mu_j \\ gt (x)}\end{displaymath}$

(1)

where $i,j \in$ (csf, GrayMatter, WhiteMatter, Pathology, Skull tissues, Background). The pathology rule applied to normal slices will incorrectly label a small number of voxels as pathology. This error will need to be corrected in later processing.

**Figure 8:** Abnormal slice: (a) Raw PD image (b) Histogram
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Patients with brain tumors are typically treated with radiation and chemotherapy. A side effect of treatment is that the MR characteristics of gray and white matter are changed and the PD histogram becomes something like that shown in Figure 8. The ``valley'' shown in Figure 4 is gone and ``turning points'' b3, b4 and b5 cannot be reliably chosen.

Our strategy is to edge detect and remove the edge voxels or sharpen the boundary between gray and white matter. The edge-value operator we used is called the DIF1 operator as described in [13]. A histogram of the voxels with low edge values will leave the peaks essentially the same and deepen the valley between the peaks. This approach can be applied to normal slices with the sole effect of deepening the already existing valley in the PD histogram.

An effective edge value threshold must be chosen to make this approach work. The initial threshold is chosen to be 5, then edge detection is done and all voxels with an edge value less than 5 are used to create a PD histogram. If 2 peaks are found in the histogram the turning points are created, otherwise the threshold is increased by 5 and the histogram re-created, peak detection done, etc. The process continues until 2 peaks are found or an edge value limit (30 here) is reached. In the case that no peaks could be found approximate peaks are chosen at 1/3 and 2/3 of the region between b1 and b2 (Figure 4). Figure 9 shows an example abnormal slice with the PD edge value image thresholded at 15 and the histogram of all voxels with edge strengths < 15.

**Figure 9:** PD edge value image: (a) Thresholded at 15 (i.e. white voxels are edge voxels with edge value >15) (b) Histogram of voxels < 15.
$\begin{figure} \centerline{ \parbox{1.5in}{ \centerline{\psfig{figure=p57s18.edg... ...s18.edge15.vhist.ps,width= 1.5in,height=1.5in}} \centerline{(b)} } }\end{figure}$

All voxels can now be classified, though imperfectly, for normal or abnormal volumes. The voxels that belong to classes with memberships greater than 0.8 are generally correctly assigned. The rest of the voxels are more problematic. Hence, we re-group them with a semi-supervised clustering algorithm, ssFCM [1]. The voxels with membership greater than 0.8 are used as training voxels for ssFCM and are weighted by a value of 100. The ssFCM algorithm works as fuzzy c-means (FCM) except that training voxels cannot change clusters and will always influence the cluster centroid to which they are assigned. When they are weighted it is the same as having w (100 here) instances of the train voxels influencing the cluster center location and hence the assignment of voxels, not in the train set, to clusters.

For a typical normal slice there will be 16,816 training voxels (memberships greater than 0.8) and 13,910 unassigned voxels. The remaining 34,810 voxels were air or skull tissue voxels and are not clustered. The clustering is done into c=10 classes to allow comparisons with FCM partitions of these same images [6,10,9,12].