## 2 Materials and Methods
### Proposed Feature Extraction Method
#### 2.1.3 Feature Extraction  
Multiple features are calculated based on the extracted latent variable vectors by the CAE. The feature set includes:  
1. The latent variables.
2. The Mahalanobis distance to the normal dataset on the latent variable space.
3. Features based on errors in 2.5D image reconstruction from the latent variables.  
The Mahalanobis distance, \(DM_{latent}(\mathbf{f},\ \mathbf{S}_{nor})\), is the distance between an input 2.5D image, \(\mathbf{f}\), and a normal 2.5D patch image dataset, \(\mathbf{S}_{nor}\), used in the CAE training on the CAE latent variable space. The distribution of the normal dataset, \(\mathbf{S}_{nor}\), is considered when measuring the distance. Equation (1) is a calculation formula. Here, \(\mathbf{\mu}_{nor}\) and \(\mathbf{\Sigma}_{nor}\) are the mean vector and covariance matrix of the normal dataset, \(\mathbf{S}_{nor}\), on the CAE latent variable space. \(Enc(\mathbf{f})\) means the latent variable vector calculated by the CAE model, \(Enc(\cdot)\), from a 2.5D image, \(\mathbf{f}\).  
\[\{DM_{latent}(\mathbf{f},\ \mathbf{S}_{nor})\}^{2}=(Enc(\mathbf{f})-\mathbf{ \mu}_{nor})^{T}\mathbf{\Sigma}_{nor}^{-1}(Enc(\mathbf{f})-\mathbf{\mu}_{nor}) \tag{1}\]  
The pixel value statistics on the square difference 2.5D image, \(\mathbf{d}_{\mathbf{f}-\hat{\mathbf{r}}}\), which is between the input 2.5D image, \(\mathbf{f}\), and the reproduced 2.5D image from latent variables, \(\mathbf{\hat{f}}\), are also used as features. The formula of \(\mathbf{d}_{\mathbf{f}-\hat{\mathbf{r}}}\) is as in Equation (2). Here, \(f(x,y,c)\) represents the  
\begin{table}
\begin{tabular}{c c c c} \hline
**Layer** & **Kernel Size** & **Stride** & **Output Size** \\ \hline
2.5D Input & \(-\) & \(-\) & \(32\times 32\times 3\) ch \\ Conv + BN + ReLU & \(3\times 3\) & \(1\) & \(32\times 32\times 6\) ch \\ Max Pooling & \(2\times 2\) & \(2\) & \(16\times 16\times 6\) ch \\ Conv + BN + ReLU & \(3\times 3\) & \(1\) & \(16\times 16\times 9\) ch \\ Max Pooling & \(2\times 2\) & \(2\) & \(8\times 8\times 9\) ch \\ Conv + BN + ReLU & \(3\times 3\) & \(1\) & \(8\times 8\times 12\) ch \\ Full Connection & \(-\) & \(-\) & \(n\) \\ \hline \end{tabular} (Conv: convolution, BN: batch normalization).  
\end{table}
Table 1: The common structure of the CAEs.  
\(c\)-channel's pixel value at 2D coordinate (\(x,y\)) of image \(\mathbf{f}\). \(\mathbb{R}^{W\times H\times Ch}\) is the set of the pixel coordinates within the 2.5D images, \(\mathbf{f}\) and \(\hat{\mathbf{f}}\).  
\[\mathbf{d}_{\hat{\mathbf{f}}-\hat{\mathbf{f}}}=\left\{d(x,y,c)=\sqrt{f(x,y,c)- \hat{f}(x,y,c)}\ \left|\ (x,y,c)\in\mathbb{R}^{W\times H\times Ch}\right.\right\} \tag{2}\]  
The nine statistics for pixel intensity of \(\mathbf{d}_{\hat{\mathbf{f}}-\hat{\mathbf{f}}}\) are the maximum, minimum, mean, standard deviation, skewness, kurtosis, first quartile, second quartile, and third quartile.  
Image reproduction was performed using the convolutional decoder, which is used in the CAE training and has a pairwise structure with the CAE. Latent variable extraction from lesion images is expected to be less accurate than normal images. In addition, the reproduced images obtained from the latent variables of the lesion data are also expected to differ significantly from the input images. The above features were used to assess these differences.