## 3 Materials and Methods
### Clustering Setting
#### 3.2.4 Post-Processing  
The result of the segmentation process is a binary image in which a label is assigned to each pixel: 0 if the pixel does not belong to the mask, and 1 otherwise. The aim of the post-processing phase is to properly choose, among the connected-components resulting from the previous step, the only one representing the lesion. The post-processing itself consists of several sub-steps allowing for removing connected-components that do not meet specific morphological criteria (Figure 5). The first post-processing phase allows us to extract all the connected-components from the raw binary mask. For each connected-component, a list of morphological features (i.e., extreme points, area, centroid, eccentricity, mean intensity and solidity) are computed that the algorithm is going to exploit in the following steps. In order to remove spurious connected-components with an area that is too small to be considered as a lesion, an area-based selection is performed: all the connected-components with an area smaller than 0.75 \(\times\) meanArea of all the connected-components. In such a way, spurious regions with a very small area are removed, reducing the number of valid candidates to be analyzed in the following steps. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Unsupervised** & **Number of** & **Maximum Number** & **Minimum** & **Exponent of Fuzzy** & \(\langle p,q\rangle\) \\
**Algorithm** & **Clusters** & **of Iterations** & **Tolerance** & **Partition Matrix** & **Parameters** \\ \hline k-means & 2 or 3 & 100 & N.A. & N.A. & N.A. \\ FCM & 2 or 3 & 100 & \(1\times 10^{-5}\) & 2.0 & \textless{}1,1\textgreater{} \\ sFCM & 2 or 3 & 100 & \(1\times 10^{-2}\) & 2.0 & \textless{}1,2\textgreater{} \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Parameters setting for k-means, FCM and sFCM clustering algorithms. Figure 5: Diagram of the post-processing steps for both central slice and other slices. Clinical evidence allows us to state that breast lesions generally grows in an isotropic way preserving a pseudo-spherical appearance [39]. As a consequence, their particular shape can be described in terms of eccentricity, which is defined as the ratio of the distance between the _foci_ of an ellipse and its major axis length. Eccentricity values lay in the range \([0,1]\): the extreme values represent degenerate cases identifying a circle and a line segment, respectively. The aim of this post-processing step is to delete from the list of the lesion candidates the regions with eccentricity lower than 0.3 and higher than 0.9 (these threshold values were determined experimentally). Eccentricity values close to 1 relate to very elongated lesions, while eccentricity values close to 0 relate to almost perfectly round shapes. In the case of breast lesions, even if overall there are rounded lesions, the lesion has a 'lobed' trend for which the final value of the eccentricity never takes values below 0.2-0.4. Using 0.3 as eccentricity lower limit in the post-processing phase, allow us to eliminate those (almost perfectly) circular connected components with pixel values similar to the lesion (and which are therefore incorrectly selected), but which are instead part of the background. As a consequence of their tendency to be round-like shaped, breast lesions exhibit an high solidity, which is defined as the ratio between the region area and the (including the region) convex polygon area. Considering this morphological information, the lesion ROI identification can be turned into an optimization problem with the ultimate goal of maximizing ratioCentral, defined in Equation (18):  
\[\text{ratioCentral}_{i}=\frac{\mu_{i}s_{i}}{\epsilon_{i}}, \tag{18}\]  
where \(\mu_{i}\) is the mean intensity, \(s_{i}\) is the solidity and \(\epsilon_{i}\) denote the eccentricity of the \(i\)-th connected-component, respectively. The post-processing in the other slices differs from the central slice.