# Enhancing Feature Selection for Imbalanced Alzheimer's Disease Brain MRI Images by Random Forest
## 3 Design of Imbalanced AD Medical Data Analysis Approach
### The Feature Extraction  
#### 3.1.1 Morphological Feature Extraction  
The VBM method is a voxel-based morphological measurement method that represents the morphological changes in brain tissue. Since it can estimate the density change of a certain voxel unit volume, it was employed to represent local brain features and brain tissue components. [PERSON] et al. [54] used the VBM method to find that the total brain gray matter of people with AD was significantly reduced when compared with the normal elderly, which was consistent with the results obtained by the manual drawing method of [55] and the pathological results. Therefore, in this paper, ROI analysis of the MRI data was carried out using SPM8 software ([[http://www.fil.ion.ucl.ac.uk/spm](http://www.fil.ion.ucl.ac.uk/spm)]([http://www.fil.ion.ucl.ac.uk/spm](http://www.fil.ion.ucl.ac.uk/spm)), accessed on 10 September 2021) and the VBM8 toolbox ([[http://dbm.neuro.uni-jena.de/vbm](http://dbm.neuro.uni-jena.de/vbm)]([http://dbm.neuro.uni-jena.de/vbm](http://dbm.neuro.uni-jena.de/vbm)), accessed on 10 September 2021). Furthermore, the gray matter volume of the lesion area was calculated as a morphological feature. Figure 2 illustrates the processing framework of morphological feature extraction. Step 1: Space standardization. Images are processed by using the VBM-DARTEL algorithm, which not only improves the segmentation accuracy however additionally preserves the original volume information and ensures the accuracy of spatial standardization. The VBM algorithm pre-segments the brain tissue, according to the prior probability information. In addition, it saves the volume of the medical image through modulation and uses the Jacobian determinant to store the volume change information of each pixel in the image, as shown in Formula (1):  
\[J= \begin{bmatrix}j_{11}&j_{12}&j_{13}\\ j_{21}&j_{22}&j_{23}\\ j_{31}&j_{32}&j_{33}\end{bmatrix} = \begin{bmatrix}\frac{\partial x^{\prime}}{ \partial x}&\frac{\partial x^{\prime}}{\partial y}&\frac{\partial x^{\prime}}{ \partial z}\\ \frac{\partial y^{\prime}}{\partial x}&\frac{\partial y^{\prime}}{\partial y}& \frac{\partial y^{\prime}}{\partial z}\\ \frac{\partial x^{\prime}}{\partial x}&\frac{\partial z^{\prime}}{\partial y}& \frac{\partial z^{\prime}}{\partial z}\end{bmatrix}  \tag{1}\]  
In Formula (1), \((x,y,z)\) and \((x^{\prime},y^{\prime},z^{\prime})\) represent the pixels before and after image registration, respectively. During the modulation process of the image, the volume information of the pixel is restored by multiplying the grayscale value of the pixel after image segmentation by the Jacobian matrix determinant of that pixel. The DARTEL algorithm introduces a Lie algebra and flow field theory into the process of VBM space standardization and uses them to carry out nonlinear image registration, which compensates for the defects in the VBM algorithm registration process and improves the accuracy of registration. The principle of the DARTEL algorithm is as follows:  
Firstly, assuming a constant flow field \(u\); a differential equation describing the deformation field changing with time can be obtained through \(u\), as shown in Formula (2):  
\[\frac{d\Phi(x)}{dt}=u\Big{(}\Phi^{(t)}(x)\Big{)} \tag{2}\]  
A deformation field can be obtained by using Formula (2), where \(x\) is the position coordinate, representing the unit transformation, and the initial value of the time point \(\Phi(x)=x\). Formula (3) can be obtained by integrating over \(\Phi^{(t)}(x)\):  
\[\Phi^{(1)}(x)=\int_{t=0}^{1}u\Big{(}\Phi^{(t)}(x)\Big{)}dt \tag{3}\]  
Further, use the Euler equation to transform Formula (2) and obtain Formula (4):  
\[\Phi^{(t+h)}=\Phi^{(t)}+hu\big{(}\Phi^{t}\big{)} \tag{4}\]  
In Formula (4), \(h\) represents the time step, and different deformation fields \(\Phi\) can be obtained by setting different time steps.