## 3 Materials and Methods
### Feature Extraction  
Each volume of interest, VOI, must be characterized by a set of characteristics or features. These features (as a real number vector) will represent each one of the brain images. A powerful transform based on the windowing technique with variable size is the Wavelet Transform (WT). The main feature of WT is that it maintains both the time and frequency information of the analyzed signal, adopting \"scale\" instead of traditional \"frequency\"; namely, it does not generate a time-frequency interpretation but a time-scale understanding of the signal. The discrete wavelet transform (DWT) is a helpful implementation of the wavelet transform, frequently used in numerical analysis and functional analysis, for which the wavelets are discretely sampled. The procedure to perform the \begin{table}
\begin{tabular}{c c c c c c c c} \hline \hline
**Measures** & \multicolumn{4}{c}{**Prodromal (38)**} & \multicolumn{4}{c}{**PD (450)**} \\ \hline
**Sex** & \multirow{2}{*}{30} & \multirow{2}{*}{**8**} & \multirow{2}{*}{**288**} & \multirow{2}{*}{**162**} \\ **(Male/Female)** & & & & & & & \\ \hline Age (years) & 67.1 & 3.9 & 67.8 & 3.1 & 62.4 & 9.8 & 60.2 & 8.7 \\ \hline Weight (kg) & 88.8 & 14.5 & 68.5 & 9.7 & 90.8 & 16.1 & 66.3 & 11.9 \\ \hline  & Mean & Std & Mean & Std & Mean & Std & Mean & Std \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the cohort of Prodromal and PD images used in this contribution. \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Measures** & \multicolumn{4}{c}{**GeneCohort (17)**} \\ \hline
**Sex (Male/Female)** & \multicolumn{4}{c}{**14**} & \multicolumn{4}{c}{**3**} \\ \hline Age & 62.1 & 10.6 & 72.2 & 9.0 \\ \hline Weight (kg) & 87.2 & 11.3 & 64.7 & 5.7 \\ \hline  & Mean & Std & Mean & Std \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Characteristics of the cohort of GeneCohort images used in this contribution. wavelet transform is that a fully scalable modulated window is shifted along the signal for every position. This procedure is repeated modifying the amplitude of the window, which leads to obtaining a multi-resolution analysis. In the case of its application in images, the decomposing of the original image in its wavelet coefficients will generate a series of images with different scales. The computation of these coefficients is calculated as follows. First, the mother or kernel wavelets define a Hilbert basis, which is constructed as the family of functions \(\{\psi_{jk};j,k\in\mathbb{Z}\}\) by means of translations and dilations of the mother wavelet \(\psi\):  
\[\psi_{j,k}(x)=2^{\frac{j}{2}}\psi(2^{j}x-k) \tag{1}\]  
The integral wavelet transform of \(f(x)\) relative to an specific mother wavelet is defined by the following equation:  
\[[W_{\psi}f](a,b)=\frac{1}{\sqrt{|a|}}\int\limits_{-\infty}^{\infty}\psi\left( \frac{x-b}{a}\right)f(x)dx \tag{2}\]  
In this way, the wavelet coefficients \(c_{jk}\) are given by:  
\[c_{jk}=[W_{\psi}f](2^{-j},k2^{-j}) \tag{3}\]  
These operations are discretized to implement the Discrete Wavelet Transform, where translations and dilations convert to dyadic scales and positions by limiting \(a\) and \(b\) to a discrete lattice \(\{a=2^{b};a>0;a,b\in\mathbb{R}\}\) so that the DWT coefficients are now defined as [42]:  
\[c_{j,k}^{app}(n)=DS\left[\sum_{n}x(n)g_{j}^{*}(n-2^{j}k)\right] \tag{4}\]  
\[c_{j,k}^{det}(n)=DS\left[\sum_{n}x(n)h_{j}^{*}(n-2^{j}k)\right] \tag{5}\]  
where \"\(app\)\" refers to \"approximation\", hence approximation coefficients; \"\(det\)\" refers to \"detail\", hence detail coefficients; \"\(DS\)\" refers to down sampling; and \(g(n)\) and \(h(n)\) denote a low-pass filter and a high-pass filter, respectively. The parameters \(j,k\) presented in the previous equations are interpreted as the wavelet scale and translation factors. In the case of having a one-dimensional signal, 1D-DWT can be applied. Each filter yields one sub-band, low-filter outputs the low-frequency sub-band L and high-filter outputs the high-frequency sub-band.