## 4 Results and Discussion
### Coronal Plane versus Axial Plane  
We compared the classification accuracy with the axial plane and the coronal plane. The results are shown in Table 5. We also showed the accuracy per case (per patient) in Table 5 which is defined in Section 4.1. As shown in Table 5, both the classification accuracies per slice and per case with the coronal plane are higher than those with the axial plane as well as our previous results using the conventional loss function [16]. The results suggest that the coronal plane contains more useful information for the classification of BAPL scores which are extracted by deep learning. In the future, we will develop a method to visualize the useful features of the trained deep convolutional neural network for BAPL score classification.  
Detailed Comparison between the Proposed and Conventional Loss Functions on Three Network Architectures  
For a detailed performance comparison, we compared the precisions, recalls, F1 scores, and accuracies of three state-of-the-art networks (i.e., VGG19, ResNet50, and DenseNet121) with the proposed and conventional loss functions. The precision and recall define the ratios of correctly classified positive samples to all classified positive samples and all true positive samples, respectively. The F1 score is the harmonic mean (average) of the precision and recall. The results are summarized in Table 6. As shown in Table 6, the proposed joint loss function improved the classification performance of all three networks, confirming its usefulness in AD classification by any state-of-the-art network. Also, notably, the joint loss function greatly improved the classification performance of BAPL2 (MCI). In the networks with the conventional loss function (\(\lambda\) = 0.00), the classification performances were high for BAPL1 and BAPL3 but very low for BAPL2. The poor result for BAPL2 can be explained as follows. As the preliminary stage of AD (BAPL3), BAPL2 (MCI) shares some similar visual features with AD (BAPL3) and possibly with NC (BAPL1). Therefore, BAPL2 images were easily misclassified as AD (BAPL3) or NC (BAPL1). After adding the intra-loss, the performance indices (precision, recall, F1) of BAPL2 (MCI) were improved from 40.25%, 17.50%, and 24.39% to 100%, 51.50%, and 67.99% in VGG19, from 73.7%, 40.20%, and 52.02% to 67.80%, 62.10%, and 64.82% in ResNet50, and from 67.79%, 53.00%, 59.49% to 85.32%, 72.70%, and 78.51% in DenseNet121. Moreover DenseNet121 achieved the highest performance among the three network architectures. The classification accuracy of DenseNet121 was approximately 94.86%. In a statistical analysis (i.e., \(t\)-test) of the accuracy comparisons of the proposed joint loss (\(\lambda\) = 0.0025) and conventional loss (\(\lambda\) = 0.00), the \(p\)-values for VGG19, ResNet50, and DenseNet121 were 0.0033, 0.0042, and 0.00001, respectively. As all \(p\)-values were less than 0.05, we statistically confirmed the superior performance of the proposed joint loss over the conventional loss in all three networks.  
\begin{table}
\begin{tabular}{c c c} \hline \hline
**Plane for Analysis** & **Accuracy** & **Accuracy per Case** \\ \hline
**Axial Plane** & 85.18\% & 87.50\% \\ \hline
**Coronal Plane** & 91.03\% & 93.75\% \\ \hline \hline \end{tabular}
\end{table}
Table 5: Comparison of classification accuracy with the axial plane and the coronal plane (CNN: ResNet50).