## 4 Results and Discussion
### Performance Comparisons with the State-of-the-Art DNN-Based Approaches  
In this subsection, we show some comparisons with other state-of-the-art benchmark solutions. Specifically, in this paper, we consider some well-known feed-forward deep networks in the literature, i.e., the AlexNet [111], the GoogLeNet [112], and the ResNet18 [113]. In this regard, we note that AlexNet is composed of the cascade of five convolutional layers and three (dense) fully connected layers, while the GoogLeNet is more complicated, since it is much deeper and constructed by stacking three convolutional layers, nine _inception_ modules, and two dense layers. An inception module is a particular layer obtained by concatenating several convolution operations with different filter sizes and a max pooling operation. Finally, ResNet18 is composed of 18 layers ending with a dense layer with a softmax activation. Different from the previous networks, the central part of the ResNet18 is a deep stack of residual units: these are composed of two convolutional layers (without pooling layer), with Batch Normalization and ReLU activation, using \(3\times 3\) kernels and preserving spatial dimensions. Since these architectures are of supervised type, they have been trained by using both the CP and NCP classes in the training set. The training has been performed by using the Adam algorithm [114] with the default values (\(\beta_{1}=0.9\), \(\beta_{2}=0.999\), and \(\varepsilon=10^{-7}\)), a batch size \(N_{b}=16\), and a learning rate set to \(\mu=10^{-6}\). The training has been executed for a total of 60 epochs. The results, provided by these state-of-the-art supervised DNNs, are shown in Table 9.  
These results point out that AlexNet performs worse than our approach, since it reaches merely a degree of 71% and 93% accuracy for the CT and CXR, respectively, and, in addition, in terms of other performance metrics. The performance of GoogLeNet is the same as the proposed approach under the CT dataset but is worse than the one of the proposed approach under the CXR dataset. This lower performance in the case of CXR images is due to the limited number of training images used during the learning. However, we have to remark that the high performance of GoogLeNet on CT is obtained by a deep architecture that uses a huge number of free parameters compared to the proposed approach, as shown in Table 10, which reports, for completeness, the number of trainable parameters and the training time (in minutes) for all the considered architectures and \begin{table}
\begin{tabular}{c c c c c} \hline \hline
**Architecture** & **Accuracy** & **Precision** & **Recall** & **F-Measure** \\ \hline
**CT** & 0.5460 & 0.5902 & 0.5460 & 0.4826 \\
**CXR** & 0.9137 & 0.9266 & 0.9137 & 0.9137 \\ \hline \hline \end{tabular}
\end{table} 
Table 8: Numerical results obtained by using the target histogram evaluated as in Figure 5. The results have been obtained by using the cosine distance, \(N_{T}=500\) reference images, and a number \(N_{bin}=50\) of histogram bins. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Architecture** & **Accuracy** & **Precision** & **Recall** & **F-Measure** & **AUC** \\ \hline \multicolumn{6}{c}{**Computed Tomography**} \\ \hline
**AlexNet** & 0.7110 & 0.8601 & 0.7110 & 0.7343 & 0.9460 \\
**GoogLeNet** & 1.0000 & 1.0000 & 1.0000 & 1.0000 & 1.0000 \\
**ResNet18** & 0.9130 & 0.9281 & 0.9130 & 0.9137 & 0.9130 \\ \hline \hline \multicolumn{6}{c}{**X-Rays**} \\ \hline
**AlexNet** & 0.9340 & 0.9341 & 0.9340 & 0.9340 & 0.9340 \\
**GoogLeNet** & 0.9746 & 0.9794 & 0.9694 & 0.9744 & 0.9750 \\
**ResNet18** & 0.9695 & 0.9697 & 0.9695 & 0.9695 & 0.9700 \\ \hline \hline \end{tabular}
\end{table} 
Table 9: Performance of the tested benchmark DNNs. These architectures have been trained and tested on the whole datasets in Table 2.  
datasets. In our approach, we use only one free parameter, the adaptive threshold in Equation (2).