## 3 Computational Methods  
### SpatialConfiguration-Net: A Summary  
Our proposed landmark detection approach is based on the SpatialConfiguration-Net (SCN) neural network introduced in [10]. The SCN network is a fully convolutional NN and consists of two components, namely (i) local appearance and (ii) spatial configuration components. Both components generate a multidimensional heat map \(h\):  
\[h(\mathbf{x})\in\mathbb{R}^{H\times W\times N}, \tag{1}\]  
where \(\mathbf{x}\) is a location vector within the heat map, \(H\) and \(W\) are the height and width of the heat map (also the size of the input image), and \(N\) denotes the number of heat map channels (also the number of targeted landmarks). A location of the n-th landmark is predicted as the location of the global maxima in the n-th heat map channel. The local appearance component is a multi-scale pyramid style network that employs a series of convolutions and downsamplings to extract feature maps. These feature maps are then upsampled and integrated across different scales. An output of this component is the multidimensional or multichannel heat map \(h\) of dimension of \(H\times W\times N\). Every channel of \(h\) can, therefore, be treated as a separate 2D heat map (\(H\times W\)) that estimates the location of a selected landmark (i.e., \(N\) channels for \(N\) landmarks). The spatial configuration component downsamples, by a large factor, the heat map estimated by the local appearance component. It processes this heat map with another series of convolutions with larger kernels, and produces the new multichannel heat map, which is upsampled appropriately at the end. Afterwards, the heat maps from the spatial configuration component, \(h^{\text{SC}}\), and from the local appearance component, \(h^{LA}\), are merged \begin{table}
\begin{tabular}{c c c c} \hline \hline \(-\)**1a-Apex of lower incisor** & **1i–Lower incisal incison** & **–6a–Apex of lower 1 st molar** & **–6c–Cusp of lower 1 st molar** \\ \hline \(-\)6d–Distal side of lower 1 st molar & **+1a–Apex of upper incisor** & **+1a–Apex of upper incisor** & **+6a–Apex of upper 1 st molar** \\ \(+\)6c–Cusp of upper 1 st molar & **+6d–Distal side of upper 1 st molar** & **+St–Upper Stomion** & **A–Point \(\Delta\)** \\ \hline \(\mathbf{A^{\prime}}\)**–Point Soft A** & **ANS–Anterior Nasal Spine** & **APOC–Anterior point of occlusion** & **Ar–Articulare)** \\ \(\mathbf{\mathbf{\hat{B}}}\)**–Point B** & **B’–Point Soft B** & **Ba–Basion** & Ci–Clinoidale \\ Co–Condylion & Col’–Columella & Cp–Condylion posterior & Cs–Condylion superior \\ \hline D–Point D & DC—Point DC & ER—End Ramus & FMN–frontomaxillary nasal suture \\ \(\mathbf{GI^{\prime}}\)**–Glabella & Gn–Gathion & Gn’—Point Soft Gnathion & Go–Gonion) \\ Hy–Hyoid & Ir–Point Ir & L1–L1 & Li’—Lower lip) \\ L1i–Lower Lip inside & **Ls’–Upper lip** & **Me–Menton)** & **Me’—Point Soft Menton** \\ \(\mathbf{N}\)**–Nasion** & **N’–Soft Nasion** & NC—Nasal crown & Or–Orbitale) \\ \(\mathbf{P_{g}}\)**–Pogonion** & **P\(\mathbf{g}\)’–Point Soft Pogonion** & PM–Suprapogonion & Pn’—Pronasale \\ \(\mathbf{FNS}\)**–Posterior Nasal Spine** & **Fo–Portion** & **PPOC–Posterior point of occlusion** & **Pt–Pterygoid point** \\ \hline R1–R1 & R3–R3 & Rh—Rhinion & RO—Orbital roof of orbital cavity \\ RT-abo–aboRamalTangent & \(\mathbf{S}\)–Sella Turcia) & Se–Entry of Sella & SE—Sphenoethmoidal point \\ Si–Floor of Sella & Sn’–Subnasale & SOr–Supraorbitale & Sp—Dorsum of Sella \\ \(-\)**St–Lower Stomion** & Te–Temporale & **tGo–Constructed Gonion** & Th’—Throat \\ U1—U1 & U1i–Upper lip inside & W—Walker point & ZyO—Zy Orbit Ridge \\ \hline \hline \end{tabular}
\end{table} 
Table 2: A list of 72 cephalometric landmarks annotated in the AUDAX private database. All 19 landmarks from the ISBI public database are also annotated in this database (denoted encircled). The 6 most difficult to determine landmarks are underlined, while the 38 most important landmarks for the cephalometric analyses are bolded.