## 3 Results and Discussion  
Based on the proposed method, we computed the centerline for all three SCs of each volunteer's right ear. From the initial skeleton (Figure 3, top), we removed the common crus structure. The initial SCs centerlines considered in the study are the remaining sections of the ducts: the anterior and posterior ducts paths are defined from the common intersection point to their ampullae. Figure 4 shows the local radius for all the studied volunteers and the mean value for each SCs. As expected from the data available in the literature [24], all the A, P and H semicircular ducts present a central region where the radius reaches a roughly constant value \(r\sim 0.5\) mm regardless of the volunteer. A histogram of the whole data gives an accurate approximation of the SCs' radii (Figure 5). There are no data for radius smaller than 0.3 mm, being the mode within 0.5-0.6 mm for all ducts, which corresponds to the value obtained on the central regions (Figure 4). In contrast, the extreme values of the plots show an increase of the radius value that can be up to three times (1.5 mm) the central region radius. These last values correspond to the ampullae. As happens in other studies, in many volunteers, the SCs are far from simple toroids and are bent (Figure 6). This bending is more relevant close to the ampulla region, and consequently the orientation of the plane that fits to the trajectory that defines the duct (Figure 7) is largely affected by the presence of the ampulla, be it complete or partial, in the calculations. In all previous works,  
Figure 5: SCs’ radii histogram for each duct. \begin{table}
\begin{tabular}{l l} \hline
1 & **Begin** \\
2 & **Input**: centerline curves; \\
3 & Define canals radii (Distance from volume surface); \\
4 & Define ampullae threshold range _ini_ and end (Local radii thresholding); \\
5 & **for** (\(\text{Ampullae}_{ini}\)) **to** (\(\text{Ampullae}_{end}\)) \\
6 & Remove ampullae points from centerlines; \\
7 & Define plane fits best centerlines (Least-squares of normal distance to the plane); \\
8 & Define normal vectors to planes; \\
9 & Define relative orientation between canals (Angles between vectors); \\
10 & Define non-planarity (Normal distance from canal to plane); \\
11 & Define representative ellipse (Least-squares algebraic distance); \\
12 & **end** \\
13 & **Output**: Radii, Planes, Ellipses, Nonplanarities and Angles \\
14 & **end** \\ \hline \end{tabular}
\end{table} 
Table 3: Pseudo-code of morphological characterization. the determination of the ampulle position was done by an expert operator. There is an inherent inter-operator variability due to this decision, and we sought to evaluate its role on the determination of the geometric parameters. We explored the effect of progressively removing the ampulle based on a threshold of the radius that defines the section to be removed. Once the distribution of radii \(r(\vec{d})\) in decreasing order was obtained for each canal, the threshold was defined using an increasing percentile. A zero value means that the whole trajectory was retained, and 100% that everything was rejected. For any intermediate value, we found the first and last points that fulfill the requirement. Due to the roughness of the channels, the diameter is not a monotonic function of the canal position \(\vec{d}\), and it may happen that the diameter increases again in some sections (see Posterior SC behavior in Figure 4, for a position \(d/d_{Max}=0.8\)). In those situations and for a given threshold, we could retain sections with a diameter larger than the rejection threshold value. This is why (see Figure 8, bottom right) we obtained retained lengths for the posterior duct that can be up to 10% larger than the value that corresponds to the rejection threshold.