# Automated Bone Age Assessment with Image Registration Using Hand X-ray Images
## 5 Xception-41 Regressor  
In this section, the bone age will be predicted by a deep learning regressor while using a set of X-ray input images \(\mathbf{\hat{X}}\) that has been pre-processed into a standardized form. The chosen network will be based on the Xception architecture [18] of the 41 CNN layers version. The primary goal of this work is to design a network that can generalize the bone age prediction by deploying a large number of training data. Because the total number of training, validation, and testing data is large in number, which is 12,811 images, a deep network is suitable to be implemented without worrying much about the overfitting problem. Therefore, a residual network scheme is crucial in overcoming the challenge of diminishing gradient while training a deep network. Furthermore, a deep network will use a large number of parameters if a standard convolution is used and, hence, an alternative of separable convolution scheme will be implemented in order to reduce memory consumption. Given the two criteria, Xception-41 architecture fits these needs perfectly, where it extensively applies residual separable convolution scheme through eight repeating modules. The separable convolution is a factorized version of the standard convolution using a composite function of depthwise and pointwise convolutions. The bone age will be regressed by using the RSNA dataset that is divided according to the ratio of 8:1:1 for training, validation, and testing, respectively. The network will be trained using Adam optimizer with variable learning rates, \(LR\) using the following piecewise function that depends on the epoch value (\(ep\)), where its value is fixed for the first 100 epochs, and it will then linearly decrease for another 50 epochs. \[LR= \begin{cases}0.001&\text{if }\ 0\ <ep\leq 50\\ 0.0005&\text{if }\ 50<ep\leq 100\\ LR-\delta_{LR}&\text{if }\ ep>100\end{cases}  \tag{10}\]  
where \(\delta_{LR}\in\mathbb{R}^{1}\) is the reduction gradient of the learning rate. The full Xception architecture consists of three modules, which are entry (\(\psi_{1}\)), middle (\(\psi_{2}\)), and exit modules (\(\psi_{3}\)) as shown in Figure 7. Let the regressor output, \(\mathbf{R}_{BA}\in\mathbb{R}^{1}\) be formulated, as follows  
Figure 6: Some output samples of the X-ray image that have passed through the image registration process. The first-row samples illustrate the original input images, while the second-row samples illustrate the image that has been transformed. \[\mathbf{R}_{BA}(\mathbf{\hat{X}})=\psi_{3}(\psi_{2}(\psi_{1}(\mathbf{\hat{X}}))) \tag{11}\]  
The entry module \(\psi_{1}\) takes an input image of 299 \(\times\) 299 pixels, where it comprises two layers of standard convolution and three layers of separable convolution. Carrying over from the previous subsection definitions, a single-layer residual convolution can be formulated as  
\[\hat{\mathbf{V}}(m,n,o_{out})=\mathbf{\hat{V}}(m,n,o_{in})+\mathbf{X}(m,n,o_{ in}) \tag{12}\]  
Therefore, a two-layer residual convolution can be written as  
\[\mathbf{\hat{V}}(m,n,o_{out})=\mathbf{\hat{V}}(\mathbf{\hat{V}}(m,n,o_{in}))+ \mathbf{X}(m,n,o_{in}) \tag{13}\]  
Using the combination of \(\mathbf{V}\) and \(\mathbf{\hat{V}}\), the output feature maps \(\mathbf{\hat{X}}\) of \(\psi_{1}\) can be represented by  
\[\mathbf{\hat{X}}_{\psi_{1}}=(\prod_{i=1}^{2}\mathbf{V}_{i}\cdot\prod_{j=1}^{3 }\mathcal{M}(\mathbf{V}_{j}))\mathbf{\hat{X}} \tag{14}\]  
where \(\mathcal{M}\) is a maximum pooling operator. The output feature maps \(\mathbf{\hat{X}}_{\psi_{1}}\) will be reduced to \(\frac{1}{4}\) of the original \(\mathbf{\hat{X}}\) size. Taking over from the entry module, the middle module consists of eight repetitive three-layer residual convolution networks.