## 2 Materials and Methods
### Multi-Atlas-Based Data Enhancement  
In image segmentation, atlases are usually used as approximative maps of the objects that should be present in the image in the normal case. Atlases are usually established based on prior measurements. The image data before the current segmentation is registered to the atlas, and the current segmentation outcome is fused with the atlas to obtain a final, atlas assisted segmentation result. In this study, we use atlases in a different manner. We build a spatial atlas for each feature using the training records, which contains the local average and standard deviation of the intensity values taken from normal pixels only. Of coarse, the training records need to be registered to each other so that we can build consistent atlases. These atlases are then used to refine the feature values in both the training and test records, before proceeding to the ensemble learning-based classification. First of all, it is important to define the spatial resolution of atlases, which depends on a single parameter denoted by \(S\). Each atlas is a discrete spatial array defined on \(\hat{\mathcal{S}}^{3}\), where \(\hat{\mathcal{S}}=\{-S,-S+1,\ldots 0\ldots,S-1,S\}\). Within the atlas array, the neighborhood of atlas point \(\hat{\pi}\) having the coordinates \((\hat{x},\hat{y},\hat{z})\) is defined as:  
\[\mathcal{C}_{\hat{\mathcal{S}}}(\hat{\pi})=\{(\hat{\alpha},\hat{\beta},\hat{ \gamma})\in\mathcal{S}^{3},|\hat{\alpha}-\hat{x}|\leq\delta\quad\land\quad| \hat{\beta}-\hat{y}|\leq\delta\quad\land\quad|\hat{\gamma}-\hat{z}|\leq\delta \}\enspace, \tag{4}\]  
where \(\delta\) is a small positive integer, typically one ore two, which determines the size of the neighborhood. The mathematical description of the atlas building process requires the introduction of several notations. Let \(\mathcal{M}\) stand for the set of all MRI records of the chosen data set, while \(\mathcal{M}^{(T)}\) and \(\mathcal{M}^{(E)}\) represent the current set of training and evaluation (or test) records, respectively. Obviously, \(\mathcal{M}=\mathcal{M}^{(T)}\cup\mathcal{M}^{(E)}\), and \(\mathcal{M}^{(T)}\cap\mathcal{M}^{(E)}=\Phi\). Let \(M_{i}\) be the record with index \(i\), belonging to either \(\mathcal{M}^{(T)}\) or \(\mathcal{M}^{(E)}\) in a certain scenario. The set of pixels belonging to record \(M_{i}\) is denoted by \(\Omega_{i}\), for any \(i=1\ldots n_{\rho}\). The set of all pixels from all MRI records is \(\Omega=\bigcup_{i=1}^{n_{\rho}}\Omega_{i}\). Any pixel \(\pi\in\Omega\) has a feature vector with \(n_{\varphi}\) elements. For any pixel \(\pi\), the value of the feature with index \(\varphi\in\{1\ldots n_{\varphi}\}\) is denoted by \(I_{\pi}^{(\varphi)}\). Further, let \(\Gamma^{(v)}\) and \(\Gamma^{(\tau)}\) be the set of negative and positive pixels, respectively, as indicated by the ground truth. Obviously, \(\Omega=\Gamma^{(v)}\cup\Gamma^{(\tau)}\), and \(\Gamma^{(v)}\cap\Gamma^{(\tau)}=\Phi\). A rigid registration is defined in the following, so that we can map all volumes onto the atlas. For each record \(M_{i}\) (\(i=1\ldots n_{\rho}\)), a function \(f_{i}:\Omega_{i}\to\mathcal{S}^{3}\) is needed that maps the pixels onto the atlas, to find the corresponding atlas position for all brain pixels. These functions map the gravity center of each brain to the atlas origin, and the standard deviations of the \(x\), \(y\), and \(z\) coordinates are all transformed to \(S/\xi\), where \(\xi=2.5\) is a predefined constant. From the pixel coordinates, we compute averages and standard deviations as follows:  
\[\left\{ \begin{array}{l}\mu_{x}^{(i)}=\frac{1}{|\Omega_{i}|}\sum\limits_{\pi \in\Omega_{i}}x_{\pi}\\ \mu_{y}^{(i)}=\frac{1}{|\Omega_{i}|}\sum\limits_{\pi\in\Omega_{i}}y_{\pi}\\ \mu_{z}^{(i)}=\frac{1}{|\Omega_{i}|}\sum\limits_{\pi\in\Omega_{i}}z_{\pi}\\ \end{array} \right.\enspace, \tag{5}\]  
and then:  
\[\left\{