## 3 Results
### Simulation Study
#### 3.1.2 Simulation Study 2  
In Simulation Study 1, multiple regions with increasing noise levels were used, which covers a low to moderate range of noise (since the detection of the regions becomes problematic when considering high levels of noise). In Simulation Study 2, consideration of robustness in the face of moderate to high levels of noise is enabled by simulating a homogenous region of the CC and a background only (as per the logical image mentioned earlier); this is because the CC is still visible in this case. Initial tensors are simulated such that: the tensors have the same determinants (sizes), FA values, and eigenvalues, and they only differ in their orientation (i.e., the eigenvectors); the diffusion directions of the tensors in the simulated CC shape are parallel to y-axis, while the diffusion directions of other tensors are parallel to x-axis. Then, the three levels of noise chosen are: \(sd((X_{j})_{ls})=0.5\times 10^{-5}\) (noise1b), \(sd((X_{j})_{ls})=0.75\times 10^{-5}\) (noise2b), and \(sd((X_{j})_{ls})=0.1\times 10^{-4}\) (noise3b). These are added to the simulated region (see Figure 4). The signal-to-noise ratio (SNR) of the three levels of noise are 13, 8, and 5. The results of the segmentation of the three noisy images are shown in Figure 5. When using Log Euclidean with noise3b (in Figure 5c), the CC is not visible. The performance measures were calculated and are shown in Figure 6. Similar to Simulation Study 1, it can be seen that root Euclidean generally provided the highest values of performance measures as compared to the other methods. The findings indicated that: (i) at noise level 1b, all of the six methods yield the same results; (ii) at noise level 2b, sFCM yields the same results using log, root, and Euclidean (and hence the same values of all performance measures), whilst sFCM with each metric almost always outperforms K-means with the same metric (with exceptions that sFCM and K-means with the Euclidean metric have the same specificity and precision, and sFCM and K-means with root Euclidean have the same sensitivity); (iii) at noise level 3b, the log Euclidean metric fails to even detect the CC, whilst sFCM with root Euclidean outperforms  
Figure 3: Bar charts of the performance measures for Simulation Study 1. Overall, sFCM yields higher values of performance measures as compared to K-means, whilst root Euclidean yields the highest performance measures. To enable better visualisation of the results, a scale of 45–100 for precision and F-measure and 80–100 for the rest of the measures is used. sFCM with Euclidean, and sFCM with either metric (root or Euclidean) outperforms K-means with the same metric. Figure 4: Original and noisy tensors before and after adding three levels of noise. Figure 5: Segmentation after adding three levels of noises. From both studies, it can be seen that the sFCM method improved the segmentation of the CC, almost always providing better performance measures than the corresponding K-means method, especially in noisy images. One can observe that root Euclidean with sFCM almost always outperformed Euclidean and log Euclidean in the segmentation of the CC. With the largest level of noise (i.e., noise3b in Study 2), log Euclidean failed to even detect the shape of CC even when the cluster size was increased. This is likely to be because it is highly affected by outliers, as shown by the following example. Take three examples of tensors \(D_{1}\), \(D_{2}\), and \(D_{3}\) from the CC region of a healthy brain, as follows:  
\[D_{1}=10^{-8}\times\left( \begin{array}{ccc}0.1461&0.0329&-0.0012\\ 0.0329&0.0098&-0.0066\\ -0.0012&-0.0066&0.0170\\ \end{array} \right)\]  
\[D_{2}=10^{-8}\times\left( \begin{array}{ccc}0.1683&0.0031&-0.0226\\ 0.0031&0.0169&-0.0025\\ -0.0226&-0.0025&0.0070\\ \end{array} \right)\]  
\[D_{3}=10^{-8}\times\left( \begin{array}{ccc}0.1152&-0.0669&0.0032\\ -0.0669&0.0542&-0.0118\\ 0.0032&-0.0118&0.0140\\ \end{array}