## 3 Results and Discussion
Figure 6: 3D semicircular ducts reconstruction from the 3D high-resolution MRI scan of Volunteer 18: **(left)** the whole SCs reconstruction where we can easily locate the three ducts; and **(right)** various views the show, from top to bottom, the Horizontal SC (HSC), Posterior SC (PSC) and Anterior SC (ASC), respectively. Figure 7: Transparent 3D reconstruction of SCs ducts. The solid sphere and diamond represent the beginning and the end of the retained part (continuous thick line) of the original trajectory (thin line) for determining the fitting plane. Each transparent circle represents the fitted plane to each one of the canals. The arrows correspond to the normal vector of each plane. The color encoding is yellow for the horizontal, blue for the anterior and red for the posterior canals. Figure 8 shows the SCs Anterior-Horizontal (AH), Anterior-Posterior (AP) and Posterior-Horizontal (AH) pairs' angles as a function of the percentile chosen for rejection. The results show that in some cases there is a clear dependence of the computed SCs pair angle on the rejection threshold. PH angle presents the most regular behavior, with a plateau close to 90\({}^{\circ}\) regardless of the discarded region. Nevertheless, there are two volunteers whose angles are around 80\({}^{\circ}\). However, for the AP and AH angles, we have a large variability. The AH angle depends on the percentile threshold applied, although, for a certain value of canal discarded, the angle tends to an asymptotic angle around 90\({}^{\circ}\) too. On the other hand, the AP is totally dependent on the duct retained, and the angle has a growing trend without saturation for the range of explored lengths. Even more important, on the AP angle and for small percentile thresholds, there is a region where the angle changes more than 5\({}^{\circ}\), and it becomes evident that we must expect a large inter-observer variability for this value when an operator has to decide the position of the ampulae. A similar result can appear on the AH angle. We do not expect a large inter-observer variability on the average PH angle when this value is computed for a large set of volunteers. This could explain the high variability on the results present in the literature with different techniques (skull sectioning, imaging techniques and mathematical model). Concerning the non-planarity of the ducts, we present in Figure 9 the normal distance to the fitting plane for the trajectories that define the centerline of the SCs. We present two situations: the whole duct and a rejection threshold using the 40 th percentile. In both cases, the distance is small, comparable to the diameter of the SCs, except when we are in the ampulla regions. This result is even more evident in the PSC, where the ampulla induces a deviation of 1.5 mm. When the rejection threshold is applied removing the ampulae, these distances decrease to values smaller than the radius of the canal (0.5 mm), and we can conclude from this analysis that the central regions (that retain more than 60% of the whole length) of the SC lie in the fitting plane. Figure 8: AP, AH and PF pairs angles based on the discarded percentile of the ducts length. Gray lines correspond to different volunteers. The median angle (black thick lines) and the limits defined by the 25 th and 75 th percentiles (thick red lines) are also presented. The bottom-right panel shows the average channel length as a function of the percentile used for rejection. An important statistical indicator about the influence of the ampulla presence in the non-planarity is the standard deviation along the trajectory of the distances between the SCs ducts and the reference planes. Here, we present the average value of this statistical computed for all the volunteers (Figure 10). From this analysis, it is clear that for the PSC and HSC a rejection threshold larger than 10% does not produce a noticeable effect, as the global distance does not substantially change. However, on the ASC, this non-planarity remains important even using a rejection threshold of 40%.