## 2 Materials and Methods
### Feature Extraction  
Moment theory had been effectively employed in several engineering applications including medical imaging and pattern recognition problems [25; 29; 38]. The orthogonal moments are similar to statistical moments such as mean, variance, kurtoses are equal to first, second, and third order of the PHT, and so on. PHT moments can extract an infinite number of features from the images including texture, shape, mean, variance and this list may extend to infinity [39]. The distinct orders of PHT moments signify distinct spatial distribution of image intensity variations. In this context, a collection of these moments can be molded to be a global shape descriptor of a particular image. The features (including AD, MCI, and HC features) are then extracted using the proposed PHT algorithm (disused in the next section) from the segmented (GM) tissue. The regions for feature extraction are \begin{table}
\begin{tabular}{c c c c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**AD**} & \multicolumn{3}{c}{**MCI**} & \multicolumn{3}{c}{**HC**} \\  & \multicolumn{3}{c}{**(\(\mathbf{r}=\mathbf{258}\), \(\mathbf{1087}\)/\(\mathbf{150M}\))**} & \multicolumn{3}{c}{**(\(\mathbf{r}=\mathbf{304}\), \(\mathbf{124F}\)/\(\mathbf{180M}\))**} & \multicolumn{3}{c}{**(\(\mathbf{r}=\mathbf{330}\), \(\mathbf{150F}\)/\(\mathbf{150M}\))**} \\ \cline{2-10}  & **Mean** & **SD** & **Range** & **Mean** & **SD** & **Range** & **Mean** & **SD** & **Range** \\ \hline Age [years] & 74.8 & 7.6 & 68–77 & 74.7 & 7.1 & 61–88 & 75.3 & 5.2 & 58–88 \\ Education [years] & 14.9 & 3.4 & 8–20 & 15.9 & 2.9 & 12–20 & 15.8 & 3.2 & 12–20 \\ MMSE & 23.6 & 1.9 & 20–26 & 27.0 & 1.7 & 24–30 & 28.6 & 1.3 & 25–30 \\ CDR & 0.7 & 0.3 & 0.5-1.0 & 0.5 & 0.0 & 0.5–0.5 & 0.0 & 0.0 & 0.5–1.0 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Summary of clinical subject demographics and dementia status. Figure 1: Systematic flow diagram of the proposed approach. decided using Automated anatomical labeling (AAL) atlas. The details about the feature extraction and selection are as follows:  
Feature extraction using PHT. PHTs [28] of order \(n\) and repetition \(\ell\) of a piecewise continuous real function, \(f(x,y)\) are defined over the unit disk \(D=\left\{(x,y):x^{2}+y^{2}\leq 1\right\}\) as follows:  
\[M_{n\ell}=\lambda_{N}\int_{0}^{2\pi}\int_{0}^{1}f(r,\theta)[H_{nt}(r,\theta) ]^{*}rdrd\theta \tag{1}\]  
where \([. ]^{*}\) denotes complex conjugate, \(r\) and \(\theta\) be the polar co-ordinates radius and angle. \(H_{nt}(r,\theta)\) can be decomposed into radial and angular components:  
\[H_{nt}(r,\theta)=R_{n}(r)\exp(i\ell\theta) \tag{2}\]  
The radial kernel can be written explicitly as \(\exp(i2\pi nr^{2})\), \(\cos(\pi nr^{2})\), and \(\sin(\pi nr^{2})\) for PCET, PCT, and PST respectively. The normalization constant \(\lambda_{N}\) is given as:  
\[\lambda_{N}=1/\pi\ for\ PCET \tag{3}\]  
\[\lambda_{N}=\left\{ \begin{array}{cc}1\pi&n=0\\ 2\pi&n\
eq 0\end{array} \ for\ PCT\ \&\ PCT\ \&\ PCT\ \&\ PCT\  \tag{4}\]  
Now, let the input image rotates by an angle of \(\alpha\). PHTs of the original and its rotated sample can be related as:  
\[M_{nt}^{rotated}=\lambda_{N}\int_{0}^{2\pi}\int_{0}^{1}R_{nt}\exp(j\ell, \theta)f(r,\theta-\alpha)rdrd\theta \tag{5}\]  
By changing the variable \(\theta^{\prime}=\theta-\alpha\), we write:  
\[M_{nt}^{rotated} = \lambda_{N}\int_{0}^{2\pi}\int_{0}^{1}R_{nt}\exp(j\ell(\theta^{ \prime}+\alpha))f(r,\theta^{\prime})rdrd\theta \tag{6}\] \[M_{nt\ell}^{rotated}=M_{nt}\exp(j\ell\theta)\]  
Equation (6) expresses the rotational property of PHTs. If we nullify the exponential term of (6), the completely rotational invariant system is obtained. The straightforward way to get the absolute invariant is to take modulo on both sides of (6) which results in:  
\[\left|M_{nt}^{rotated}\right|=\left|M_{nt}\right| \tag{7}\]  
where, \(M_{nt}^{rotated}\) and \(M_{nt}\) are the PHTs of the rotated and the original images, respectively. Thus, the proposed features are the magnitude of the PHTs as texture descriptors.