## 2 Material and Methods
### Methodology of the Proposed Prediction Model
#### 2.2.1 Feature Extraction Based on 3D-ZM  
Moments are scalar properties with real or complex values that are used to describe a certain function. Moments are projections of function f onto a polynomial basis from a mathematical perspective. Moments are used in statistics to estimate population parameters. Moments are specific quantitative measures of the shape of a set of points. They can also be used to extract features from images by gathering relevant information. In this regard, moments as shape descriptors present powerful techniques for extracting shape-based features in AD. The ZM [28] method, one of many moment methods defined in the literature [29], is used in a variety of applications ranging from image enhancement to image classification [30; 31; 32] due to its scale, translation, and rotational invariance. Furthermore, the orthogonal basis of the function's properties results in minimal information redundancy.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Category** & **Gender (f/M)** & **Age** & **CDR** & **MMSE** \\ \hline pMCI & 39/66 & 74.95 \(\pm\) 6.6 & 0.5 \(\pm\) 0.0 & 26.6 \(\pm\) 1.7 \\ \hline sMCI & 41/80 & 74.45 \(\pm\) 7.58 & 0.5 \(\pm\) 0.0 & 27.3 \(\pm\) 1.87 \\ \hline \hline \end{tabular} Note: all data are presented in mean \(\pm\) standard deviation mode. CDR: Clinical Dementia Rating; MMSE: Mini-Mental State Examination; F: Female; M: Male.  
\end{table}
Table 1: Characteristics of the subjects used in this study.  
\[M_{p,q,r}=\int\int\int P_{p}(x)P_{q}(y)P_{r}(z)f(x,y,z)dxdydz \tag{1}\]  
where \(p\), \(q\), and \(r\) are integer coefficients indicating the various members of the basis set, and \(n\) is the order defined as \(n=p+q+r\). \(P_{p}\), \(P_{q}\), and \(P_{r}\) are polynomial basis functions (PBF). With different PBF functions, various types of moments could be derived. The simplest 3D  
Figure 1: Block diagram of the proposed MCI conversion prediction framework.  
PBF are geometric moments with the form of \(x^{p}\ y^{q}\ r^{z}\). However, deploying these moments cause a poor recognition rate since their PBFs are not orthogonal. To overcome this problem, the orthogonality condition for the basis function must be satisfied in which:  
\[\frac{3}{4\pi}\int\limits_{0}^{1}\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}Z_{ nl}^{m}(r,\theta,\phi)Z_{n^{p}t^{p}}^{m}(r,\theta,\phi)r^{2}\sin\theta d \rho d\theta d\phi=\delta_{nn^{\prime}}\delta_{ll^{\prime}}\delta_{mm^{\prime}} \tag{2}\]  
In 3D, the area of orthogonality can be defined cylindrically as:  
\[M_{p,q,r}=\int\limits_{0}^{2\pi}\int\limits_{0}^{1}\int\limits_{0}^{1}G_{r}(Z )R_{p,q}(\rho)e^{-iq\phi}f(x,y,z)\rho dzd\rho d\phi \tag{3}\]  
where \(p\), \(r\) = 0, 1, 2,... and \(q\) = \(-p\),..., \(p\). \(R_{p,q}(\rho)e^{-iq\phi}\) is the kernel function on the disk and \(G_{r}(Z)\) refers to 1D orthogonal polynomials. The 3D Zernike polynomial spherical coordinate is defined as:  
\[M_{nl}^{m}(r,\theta,\phi)=R_{nl}(r)Y_{l}^{m}(\theta,\phi) \tag{4}\]  
where \(Y_{l}^{m}\) is the spherical harmonics of degree \(l\) and order m and \(r\), \(\theta\), \(\phi\) are the spherical coordinates.  
\[\begin{array}{l}x=r\sin\theta\cos\phi\\ y=r\sin\theta\sin\phi\\ z=r\cos\theta\end{array} \tag{5}\]  
The 3D ZM is:  
\[A_{nl}^{m}=\int\limits_{0}^{2\pi}\int\limits_{0}^{\pi}\int\limits_{0}^{1}M_{ nl}^{m}(r,\theta,\phi))*f(r,\theta,\phi)rdrd\theta d\phi \tag{6}\]