## 2 Materials and Methods  
We assumed that each EEG data channel is independent of the others because brain activation patterns differ according to the motor imagery of the BCI system [32]. Based on this assumption, in the case of left and right motor imagery classification, the brain activation pattern differs according to each imagery direction; therefore, each channel can be better distinguished, and the accuracy of motor imagery classification can be increased. Figure 1 presents a flowchart describing the fundamental process of the CDC-EFA algorithm. An explanation of each step is provided. Raw EEG data need to be preprocessed to improve the signal identity. The preprocessed data points are used in the whitening  
technique to differentiate the channel data based on Gram-Schmidt orthogonalization theory [31]. Therefore, the channel whitening method is helpful for channel independence. The obtained EEG data points are changed into artificial image data because the EEG data are processed with differentiated direction, so this process further helps data discrimination. After the covariance decentering process is performed, an eigenface is constructed and then, feature extraction is processed. Centering is a method for reducing the variable correlation. The centering for random variable X is calculated using Equation (1):  
\[\text{Centering}=\overline{\text{X}}=\text{X}-\text{mean}(\text{X}) \tag{1}\]  
When extracting the classification features of motor imagery, it is necessary to precisely distinguish the classes because the left and right features are difficult to differentiate. However, when centering is performed, the unique average data of each feature are lost, making them more difficult to differentiate. The proposed decentering method reverses the centering method to enhance data characteristics. The covariance matrix calculated in the process of extracting unique facial features is a value that indicates the correlation between each data unit. The proposed covariance decentering method considers the calculated covariance matrix as a new random variable and decenters the covariance matrix by adding the average of each column to the covariance matrix. Therefore, the decentering method for an arbitrary covariance matrix is calculated using Equation (2). \[\text{C}^{\prime}=\text{C}+\text{mean}(\text{C}) \tag{2}\]  
where C and C\({}^{\prime}\) are the input and output. After adding the mean or expectation of the covariance matrix, we apply the decentering method to the data, making it possible to assign weights to data with high correlation with other data. In addition, EFA calculates a basis vector using a covariance matrix calculated from image vectors and uses the basis vector to increase the eigenvalues; thus, the data characteristics can be further emphasized. Although the original covariance matrix is symmetric, the symmetry of the covariance matrix disappears during the decentering process. Because the basis vectors are not orthogonal to each other for a non-symmetric matrix, it is necessary to find orthogonal basis vectors to determine a direction that represents the data distribution well. Therefore, in the proposed CDC-EFA method, the eigenface is constructed after the Gram-Schmidt orthogonalization-supported whitening method is performed on the eigenvector calculated from the decentered covariance matrix. Feature extraction is also performed after covariance decentralization. The application of  
Figure 1: Algorithm flowchart of the CDC-EFA method. the covariance decentering technique to the feature extraction process is calculated using Equation (3). \[\text{C}=\text{ cov}(\Phi),\text{C}^{\prime}=\text{CDC}(\text{C}),\text{ }\Omega=\text{C}^{\prime}\times\Gamma \tag{3}\]  
where the input is \(\Phi\) and \(\Gamma\), and the output is \(\Omega\). In the experiments, we evaluated the performance of CDC-EFA in the BCI system. The feature distribution and accuracy changes were analyzed to determine performance.