## 5 Experimental Findings
### Spatial Distance-Based Metrics Segmentation Results  
The need for using both area- and distance-based metrics comes out in considering that area-based metrics do not take in account pixels' spatial distribution. This leads to the need to quantify the distance between the boundaries computed by the semi-automatic methods and the ground-truth. The boxplots in Figure 11 summarize the obtained results. In terms of HD, all the presented methods share similar characteristics with a mean value of \(\approx\)2.2 and a standard deviation of \(\approx\)0.43. This means that the boundaries of the semi-automatic masks are quite close to the manually traced ones. The MAD metric is quite similar across the examined techniques, except for the SMRG, which shows the higher mean value and standard deviation. Clustering-based approaches, on the other hand, result more stable and precise, offering a better performance on the whole dataset. Table 5 shows distance-based metrics obtained using the considered unsupervised segmentation approaches: lower distance values indicate better segmentation results. Observing the general trend, just a small deviation between the segmentations of the proposed methods and those of the experienced radiologist can be denoted. Furthermore, the achieved spatial distance-based indices are consistent with area-based metrics, confirming that clustering-based segmentation approaches allow us to reach better results with respect to SMRG. Figure 11: Boxplots of the spatial distance-based metrics achieved by the considered unsupervised segmentation approaches. From left to right: Mean Absolute Distance, Maximum Distance and Hausdorff Distance. The lower and upper bounds of each box represent the first and the third quartiles of the metric distribution, respectively. The median is represented by a red line, while outliers are displayed as red crosses. As shown at the top of Figure 12, FCM (with \(p=6.268\times 10^{-6}\) and \(p=2.670\times 10^{-7}\)) and sFCM (with \(p=8.566\times 10^{-5}\) and \(p=1.855\times 10^{-7}\)) clustering methods achieved significantly higher DI values compared to SMRG and k-means, respectively. To statistically validate the obtained results, the two-sided Wilcoxon signed rank test [50] on paired DI results was performed with the null hypothesis that the samples come from continuous distributions with equal medians (considering a significance level of 0.05). Obtained \(p\)-values are shown in Table 6. With more details, this test on paired results was used to statistically compare the distributions of the DI values achieved by two competing methods and identify significant differences. The \(p\)-values were corrected by the Bonferroni-Holm method [51] for multiple comparisons. The values reported in Tables 3-5 represent the average value \(\pm\) standard deviation over all 50 breast masses. All the values obtained on each breast mass are reported in the Supplementary Materials. \begin{table}
\begin{tabular}{c c c c} \hline \hline  & **SMRG** & **k-Means** & **FCM** \\ \hline
**k-means** & 0.973 & & \\
**FCM** & \(\mathbf{6.268\times 10^{-6}}\) & \(\mathbf{2.670\times 10^{-7}}\) & \\
**sFCM** & \(\mathbf{8.566\times 10^{-5}}\) & \(\mathbf{1.855\times 10^{-7}}\) & 0.423 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: \(p\)-Values obtained from the statistical validation procedure. The Wilcoxon rank-sum test for pairwise result comparison was used with the alternative hypothesis that the samples do not have equal medians of DI. A significance level of \(\alpha=0.05\) with the Bonferroni-Holm correction for multiple comparisons was used. **Bolface** indicates that the null hypothesis can be rejected. Figure 12: DI values achieved by the investigated traditional classic approaches. The bar graph and error bars denote the average value and the standard deviation DI values, respectively. The \(p\)-values, obtained from the statistical validation procedure, are shown at the top of the bars as brackets.