## 6 Discussion
The latter makes the AUDAX database much more demanding than the ISBI database. Figure 4 depicts the qualitative result of cephalometric landmarks' detection by using our proposed SCN-EXT method on the AUDAX private database. Seventy-two estimated (denoted by a red x) and ground-truth (blue circle) cephalometric landmarks are superimposed on the skull X-ray image. The predicted and correct location of the landmarks are connected by the green line, where the following applies: the shorter the line, the lower the radial error. It can be noticed that, with the exception of the point on the throat, all the remaining cephalometric landmarks were localized extremely accurately. The rater's annotations were also analyzed on the AUDAX database. We wanted to find out the positions of which landmarks varied the most on the skull, and, whether the results obtained with our SCN-EXT method were consistent with these findings; accordingly, if the position of the landmark varied slightly on the skull and whether this made our method more accurate, and vice versa. Just a few findings are presented in the sequel, as this analysis is not the main goal of our research. We thus conducted a statistical analysis of skull shapes on the AUDAX database. Seventy-two annotated cephalometric landmarks from all 4695 images were utilized as an input. The aim of this analysis was to determine how the locations of cephalometric landmarks differ (vary, deviate) in the population (i.e., among patients), and how this influenced landmark detection effectiveness. We carried out a so-called generalized Procrustes analysis [16; 17]. In each image, the locations of cephalometric landmarks were compensated by translation, scaling, and rotation (i.e., by a rigid transformation), resulting in a mean skull shape (and corresponding mean landmarks' locations) in the Procrustes space. Subsequently, we fitted the Procrustes mean model to the annotated cephalometric landmarks in each image by using an approach from [18], followed by the calculation of the radial error between the fitted model landmarks and the ground-truth landmarks. This error was summarized for each cephalometric landmark  
Figure 4: Sample detection result, superimposed on the X-ray image from the AUDAX private database. Cephalometric landmarks were determined by the proposed SCN-EXT method. Estimated landmarks are denoted by a red x, while ground-truth locations are superimposed as blue circles. over all images with various statistics (i.e., mean, standard deviation, median, and the 75 th percentile). It was discovered that the following 10 cephalometric landmarks have the lowest variability, namely, the landmarks PNS, APocc, W, S, Se, Ci, LLi, +St', -St', and PPocc (see Table 2 for denotations). The ten landmarks with the higher deviation from the Procrustes mean model are the landmarks Go, B, N', Gn', Go, Ba, Gl', Rh, Hy, and Th', which is the overall highest variability landmark. Both lists remained the same regardless of any statistics (e.g., mean, median, etc.) used in the comparison. Finally, we evaluated the influence of variability on the cephalometric landmark detection. We calculated the correlation between the landmark variability and detection effectiveness by using the SCN-EXT method. For both quantities, we used data regarding the points order, once in respect to the variability, the second in respect to the detection effectiveness. There was a positive correlation between the two quantities (the correlation coefficient equaled 0.505 with a \(p\) value \(5.99\times 10^{-6}\)). To sum up, the less the landmark varied, the more accurately it was detected, and vice versa. These findings are also consistent with the importance of landmarks for cephalometric analyses as defined by the AUDAX company (see Table 2). With the exception of the Gl' landmark, all the remaining nine poorly localized landmarks (see Table 9) are less important for the cephalometric analyses. Similarly, all 10 accurately localized landmarks (see Table 8) are more important for the cephalometric analyses.