## 3 Methodology
### Hybrid Feature Extraction  
After image segmentation, the hybrid feature extraction is accomplished by using HoG, LBP, and LTP feature descriptors, where these descriptors are selected based on the feature importance calculation. In image processing applications, the HoG descriptor is often used for extracting feature values from medical images. In the HoG feature descriptor, the magnitude, and orientation of the brain scans \(I_{N}\) are initially computed. The vertical gradient \(G_{v}\) and horizontal gradient \(G_{h}\) are mathematically specified in Equation (5).  
\[G_{v}=I_{N}\times[-1,0,1]^{T},\ G_{h}=I_{N}\times[-1,0,1] \tag{5}\]  
The computed vertical gradient \(G_{v}\) and horizontal gradient \(G_{h}\) are utilized to calculate the gradient magnitude \(M(x,y)\) and angular orientation \(\theta(x,y)\) that are defined in Equations (6) and (7).  
\[M(x,y)=\sqrt{G_{v}^{2}(x,y)+G_{h}^{2}(x,y)} \tag{6}\]  
\[\theta(x,y)=tan^{-1}\bigg{(}\frac{G_{v}(x,y)}{G_{h}\ (x,y)}\bigg{)} \tag{7}\]  
The gradient magnitude \(M(x,y)\) and angular orientation \(\theta(x,y)\) partitions the 3D brain scans into different cells. Furthermore, the orientation related to the similar cells is integrated and quantized into final histogram bins and then the respective bins are combined into the final histogram [23; 24]. The total number of features \(T_{hog}\) are estimated by utilizing Equation (8).  
\[T_{hog}=B_{img}\times B_{s}\times N_{b} \tag{8}\]  
where \(N_{b}\) represents the number of bins, \(B_{img}\) specifies the number of blocks per 3D brain scan, and \(B_{s}\) denotes block size.  
In addition, the LBP and LTP encode the relation between the neighborhood pixel and the referenced pixel by calculating the gray-level difference. The LBP is an effective texture feature descriptor, which transforms the 3D brain scans into labels based on luminance value. In a 3D brain scan \(I\), the position of the pixel is represented as \((x,y)\), which is derived by utilizing the central pixel value \(x_{c}\) of the threshold to signify the neighborhood pixel \(n_{p}\). Additionally, the binary pixel value is weighted using the power of two, and further, the  
Figure 3: Sample segmented 3D brain scans.  
value is summed to generate a decimal number and it is stored in the location of \(x_{c}\). The LBP is mathematically specified in Equations (9) and (10) [25].  
\[LBP\left(x,y\right)=\sum_{i=0}^{n_{p}-1}I(x_{i}-x_{c})^{2i},\ I(x)=\left\{ \begin{matrix}1,\ x\geq 0\\ 0,\ x\leq 0\end{matrix}\right\} \tag{9}\]  
\[U(LBP(x,y))=\left|I(x_{c-1}-x_{i})-I(x_{0}-x_{i})\right|+\sum_{i=1}^{n_{p}-1} \left|I(x_{c}-x_{i})-I(x_{c-1}-x_{i})\right| \tag{10}\]  
where \(u\) denotes maximum jumping time and \(x_{i}\) specifies the gray-level value of the center pixel \(x_{c}\).  
Similarly, the LTP is an extension of LBP that uses a thresholding constant for the pixel intensify values of three. In the LTP feature descriptor, the thresholding value is defined by using Equation (11) [26].  
\[\begin{array}{c}1\\ \{\begin{array}{cc}0\\ -1\end{array}&if\ n_{p}>x_{c}+T_{v}\\ \end{array}\\ \{\begin{array}{cc}0\\ -1\end{array}&if\ n_{p}>x_{c}-T_{v}\ and\ n_{p}<x_{c}+T_{v}\\ \end{array}\\ \begin{array}{cc}-1\end{array}&if\ n_{p}<x_{c}-T_{v}\end{array} \tag{11}\]  
where \(T_{v}\) denotes thresholding constant. By contrast, the extracted 9824 feature vectors are given as the input to the binary cuckoo search algorithm for feature optimization. The graphical representation of the feature importance calculation is shown in Figure 4.