# Focal Dice Loss-Based V-Net for Liver Segments Classification
## 3 Experimental Results  
In order to assess the performances of the different classifiers obtained by varying the \(FDL\) formulation and the associated modulation factor \(\gamma\), we assessed the \"hard\" Dice coefficient \(D_{H_{\text{t}}}\) for each class \(t\), the average \(D_{H_{\text{avg}}}\) Dice coefficient across all classes, the mean accuracy \(A_{avg}\), and the confusion matrix, in order to see the relationships of the errors from different segments. We compared the \(FDL\) presented by [PERSON] et al. with the ones presented in this paper. We considered five values for the modulation parameter: \(\gamma\in\{1,1.25,1.5,1.75,2\}\). The case \(\gamma=1\) is equivalent to using the Dice coefficient as defined in Equation (7). For each experiment, we checked if the application of the post-processing was beneficial or not. Experimental results are reported in Tables 3-6, for models trained with \(FDL^{A}\), \(FDL^{B}\), \(FDL^{W}\), and \(DL\), respectively. Figure 5 shows the segmentation obtained for a single CT scan, adopting \(FDL^{A}\) with \(\gamma=2\). In the case of \(FDL^{W}\), the best model was the one with \(\gamma=1.25\) after post-processing. A possible justification is that larger values of \(\gamma\) result in flattening too much the loss of the volumetric predictions, so that those having a good Dice coefficient are too penalized. For the FDLs defined in this paper, the model trained \(FDL^{A}\) achieved the best performance also in the case \(\gamma=2\), after post-processing, while the one trained with \(FDL^{B}\) obtained the best performance in the configuration \(\gamma=2\). \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline
**Models** & \(D_{H_{0}}\) & \(D_{H_{1}}\) & \(D_{H_{2}}\) & \(D_{H_{3}}\) & \(D_{H_{4}}\) & \(D_{H_{5}}\) & \(D_{H_{6}}\) & \(D_{H_{7}}\) & \(D_{H_{8}}\) & \(D_{H_{avg}}\) & \(D_{H_{std}}\) & \(A_{avg}\) \\ \hline \(\gamma=1\) & 99.86 & 80.64 & 83.88 & 82.22 & 74.49 & 75.05 & 77.59 & 83.66 & 79.5 & 81.88 & 7.56 & 98.82 \\ \hline \(\gamma=1,PP\) & 99.86 & 80.75 & 84.13 & 83.20 & 74.68 & 75.65 & 77.81 & 83.66 & 79.61 & 82.15 & 7.47 & 98.84 \\ \hline \hline \end{tabular}
\end{table} 
Table 6: Results for the models trained with the Dice loss, which corresponds to the case of \(\gamma=1\) for all the previously defined focal Dice losses. Comparisons for the different liver segments. PP stands for Post-Processing. \(D_{H_{i}}\) is the “hard” Dice coefficient for class \(t\). It is reported as a percentage. \(D_{H_{avg}}\) is the arithmetic mean of \(D_{H_{i}}\). \(D_{H_{std}}\) is the standard deviation of \(D_{H_{i}}\). The average accuracy is denoted as \(A_{avg}\). \begin{table}
\begin{tabular}{l c c c c c c c c c c c c} \hline \hline
**Models** & \(D_{H_{0}}\) & \(D_{H_{1}}\)  
Concerning \(FDL^{B}\), in our experiments, we found that the model trained with \(\gamma>2\) could not generalize, and the \(FDL_{t}\) for at least a class remained one (worst case) during the training. A possible explanation could be similar to the one provided for the \(FDL^{W}\) from the paper of [PERSON] et al. [54]. When you increase the value of \(\gamma\), even the high values of \(p_{i}\) are penalized, as we can see from Figure 3.  
We tested if \(FDL^{A}\) was better than the other formulations. To do this, we performed the Wilcoxon signed-rank test, obtaining that \(FLD^{A}\) showed superior performances to \(FDL^{B}\) with \(p_{value}=0.003\), to \(FDL^{W}\) with \(p_{value}=0.034\), and to \(DL\) with \(p_{value}=0.011\). Considering, as usual, the significance threshold \(\alpha_{th}=0.05\), the obtained results were statistically significant. We also report the normalized confusion matrices in Figure 6. Elements in the main diagonal of the row-normalized and column-normalized confusion matrices are the _precision_(s) and _recall_(s) of the corresponding classes, respectively.