## 2 Materials and Methods
### Multi-Atlas-Based Data Enhancement
\begin{array}{l}\sigma_{x}^{(i)}=\sqrt{\frac{1}{|\Omega_{i}|-1}\sum \limits_{\pi\in\Omega_{i}}\left(x_{\pi}-\mu_{x}^{(i)}\right)^{2}}\\ \sigma_{y}^{(i)}=\sqrt{\frac{1}{|\Omega_{i}|-1}\sum\limits_{\pi\in\Omega_{i}} \left(y_{\pi}-\mu_{y}^{(i)}\right)^{2}}\\ \sigma_{z}^{(i)}=\sqrt{\frac{1}{|\Omega_{i}|-1}\sum\limits_{\pi\in\Omega_{i}} \left(z_{\pi}-\mu_{z}^{(i)}\right)^{2}}\\ \end{array} \right.\enspace, \tag{6}\]where \(x_{\pi}\), \(y_{\pi}\), and \(z_{\pi}\) are the coordinates of pixel \(\pi\) in the brain volume. The formula of the mapping \(f_{i}\) is:  
\[f_{i}(\pi) = \left(\left\langle\frac{s\big{(}x_{\pi}-\mu_{i}^{(i)}\big{)}}{ \xi_{\pi}^{(i)}}\right\rangle,\left\langle\frac{s\big{(}y_{\pi}-y_{\pi}^{(i)} \big{)}}{\xi_{\pi}^{(i)}}\right\rangle,\left\langle\frac{s\big{(}z_{\pi}-\mu_{ i}^{(i)}\big{)}}{\xi_{\pi}^{(i)}}\right\rangle\right)\quad, \tag{7}\]  
where \(\left\langle\cdot\right\rangle\) stands for the operation of rounding a floating point variable to the closest integer. For any feature with index \(\varphi\in\{1\ldots n_{\varphi}\}\), the atlas function has the form \(A_{\varphi}:\mathcal{S}^{3}\rightarrow\mathbb{R}^{3}\), which for any atlas point \(\hat{\pi}\in\mathcal{S}^{3}\) is defined as:  
\[A_{\varphi}(\hat{\pi})=\left(\hat{\mu}_{\hat{\pi}}^{(\varphi)},\hat{\sigma}_{ \hat{\pi}}^{(\varphi)},\hat{\
u}_{\hat{\pi}}^{(\varphi)}\right)\, \tag{8}\]  
where the components \(\hat{\mu}_{\hat{\pi}}^{(\varphi)}\), \(\hat{\sigma}_{\hat{\pi}}^{(\varphi)}\), and \(\hat{\
u}_{\hat{\pi}}^{(\varphi)}\) are established with the following formulas:  
\[\left\{ \begin{array}{l}\hat{\
u}_{\hat{\pi}}^{(\varphi)}=\sum\limits_{M_{i} \in\mathcal{M}^{(\Gamma)}}|\Psi(i,\hat{\pi})|\\ \hat{\mu}_{\hat{\pi}}^{(\varphi)}=\left(\hat{\
u}_{\hat{\pi}}^{(\varphi)}\right) ^{-1}\sum\limits_{M_{i}\in\mathcal{M}^{(\Gamma)}}\left(\sum\limits_{\pi\in \mathbf{Y}_{i}(\hat{\pi})}I_{\pi}^{(\varphi)}\right)\\ \hat{\sigma}_{\hat{\pi}}^{(\varphi)}=\sqrt{\left(\hat{\
u}_{\hat{\pi}}^{( \varphi)}-1\right)^{-1}\sum\limits_{M_{i}\in\mathcal{M}^{(\Gamma)}}\left(\sum \limits_{\pi\in\mathbf{Y}_{i}(\hat{\pi})}\left(I_{\pi}^{(\varphi)}-\hat{\mu} _{\hat{\pi}}^{(\varphi)}\right)^{2}\right)}\end{array} \right. \tag{9}\]  
where:  
\[\Psi_{i}(\hat{\pi})=\left\{\pi\in\Omega_{i}\cap\Gamma^{(v)},f_{i}(\pi)\in \mathcal{C}_{\mathcal{S}}(\hat{\pi})\right\}. \tag{10}\]  
The feature values of each pixel \(\pi\in\Omega_{i}\) (\(i=1\ldots n_{\varphi}\)), no matter whether \(\pi\) belongs to a record of the training or test data, are updated with the following formula:  
\[\widetilde{I}_{\pi}^{(\varphi)}\leftarrow\min\left\{\max\left\{\alpha,\left\langle \overline{\mu}+\overline{\sigma}\frac{I_{\pi}^{(\varphi)}-\hat{\rho}_{f_{i}( \pi)}^{(\varphi)}}{\sigma_{f_{i}(\pi)}^{(\varphi)}}\right\rangle\right\},\beta \right\}\, \tag{11}\]  
where parameters \(\overline{\mu}\) and \(\overline{\sigma}\) represent the target average and standard deviation, respectively, and their recommended values are:  
\[\left\{ \begin{array}{l}\overline{\mu}=\gamma_{0.5}=(\alpha+\beta)/2\\ \overline{\sigma}=(\beta-\alpha)/10\end{array} \right.. \tag{12}\]  
Algorithm 1 summarizes the construction and usage of the atlas. The updated intensity values compose the pre-processed data denoted by \(P_{2}\) in Figure 1. Both \(P_{1}\) and \(P_{2}\) follow the very same classification and post-processing steps.