## 3 Computational Methods
### Proposed SCN-EXT Method  
The aim of this research is to develop an effective deep-learning-based method for detecting a large number of cephalometric landmarks from skull X-ray images. State-of-the-art cephalometric landmark detection methods such as [8; 9; 10; 11] have proven very effective on a small number of landmarks. Our goal, however, is to upgrade the state-of-the-art appropriately, also for more challenging kinds of detection. A substantial increase in the number of targeted landmarks requires, typically, an increase in a (convolutional) neural network's capacity. A trivial solution of increasing the number of filters for each convolution layer proved to have two drawbacks. First, doubling the number of filters squares the number of free parameters for most layers. Consequently, the memory requirements grow quadratically. Second, increasing the number of parameters typically makes the learning of an NN with the same training set and similar hyperparameters either unstable or prone to overfitting [12]. The considerable inflation of free parameters is particularly acute for the SCN network, as we have found through experimentation that this network learning has become very unstable. It should also be noted that an exhaustive fine-tuning of the initialization constants for particular SCN layers were carried out. It is expected that an additional fine-tuning of the SCN network would be required by larger expansion of the free parameters. However, the SCN network performed with high accuracy when detecting 19 cephalometric landmarks on testing images from the ISBI public database (see Section 2.2). We wanted to take advantage of the high detection effectiveness of the SCN network, but, at the same time, we wanted to avoid re-evaluating (i.e., fine-tuning) the initialization constants if the SCN network capacity was increased significantly. Therefore, we propose the following SCN network extension, denoted as SCN-EXT, which increases the capacity of the NN by adding a series of new, but with the same hyperparameters, basic building blocks of the SCN network. Figure 2: A rough block diagram of the SpatialConfiguration-Net. Depicted are ground plan views of maps (i.e., a single 2D map/channel is shown for a selected landmark). We constructed the SCN-EXT network by introducing \(J\) repetitions of the local appearance component into the SCN network, where each of these components was connected with an input image. Figure 3 depicts the basic elements and outputs of the SCN-EXT network. An output of the local appearance component is a multichannel heat map (dimensions of \(H\times W\times N\)), which is passed on to the input of the new spatial configuration component. We must, therefore, integrate \(J\) spatial configuration components into the SCN-EXT network, i.e., one for each local appearance component. The spatial configuration component also returns as an output of the matrix of dimension \(H\times W\times N\) (i.e., spatial configuration map). Subsequently, combining the outputs of all \(J\) repetitions of a particular component follows. The \(J\) outputs of the local appearance components are summed simply into the final local appearance heat map. Similarly, the spatial configuration components' outputs are combined (see Figure 3). Finally, identical to the original SCN network, both the final local appearance and the final spatial configuration heat maps are merged, by using the Hadamard product, into a prediction map, which is then utilized for predicting landmarks' locations. The described procedure for constructing the prediction map \(h\) is written formally as:  
\[h(\mathbf{x})=\left(\sum_{j=0}^{J}h_{j}^{LA}(\mathbf{x})\right)\ \circ\ \left(\sum_{j=0}^{J}h_{j}^{SC}(\mathbf{x})\right), \tag{3}\]  
where \(h_{j}^{LA}\) and \(h_{j}^{SC}\) denote heat maps of the \(j\)-th local appearance and the \(j\)-th spatial configuration component, respectively.