## 4 Experiments and Results
### Loss Analysis on the LiTS Dataset  
To verify the effectiveness of the boundary loss function, we use different loss functions to optimize our 2.5D FCN and to test the effect on the LiTS dataset. As shown in Table 2, in the segmentation of the liver, the Dice Global with the cross-entropy, dice and combined loss function is not very different. However, the Dice Global with contour and boundary loss functions achieves excellent results. The Dice Global with proposed the loss is 96.1%, which surpasses the other loss functions. In the segmentation of the tumor, the Dice Per Case with a dice loss function is approximately 3.3% higher than that with a cross-entropy loss function. Compared with the loss of cross-entropy and dice, the combined loss function can be improved by at least 2.5% for tumor segmentation, as shown in Figure 7. The above results indicate that the combined loss function is superior to the similarity coefficient loss function or cross-entropy loss function alone. In addition, further optimization with the contour loss function and the boundary loss function after using the combined loss function in the experiment can improve the accuracy to 5.6% and 6.3%, respectively. The proposed boundary loss function can optimize the boundary to a certain extent, as shown in Figure 8, to remove false-positive cases more effectively for the subsequent post-processing method. Therefore, our method can obtain higher segmentation accuracy and the Dice Per Case with the proposed loss function is 74.5%, which is the highest value among loss functions that are compared.  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Loss**} & \multicolumn{2}{c}{**Tumor**} & \multicolumn{2}{c}{**Liver**} \\ \cline{2-5}  & **Dice Per Case** & **IoU** & **Dice Per Case** & **IoU** \\ \hline
2D FCN + Combined loss & 65.2 & 48.4 & 91.3 & 84.0 \\
2D FCN + Contour loss & 72.4 & 56.8 & 93.2 & 87.3 \\
2D FCN + Boundary loss & 73.6 & 58.3 & 93.9 & 88.5 \\
2.5D FCN + Combined loss & 68.2 & 51.8 & 92.6 & 86.3 \\
2.5D FCN + Contour loss & 73.8 & 58.5 & 94.1 & 88.9 \\
2.5D FCN + Boundary loss & **74.5** & **59.4** & **94.3** & **89.3** \\ \hline \hline \end{tabular}
\end{table}
Table 1: Performance of different configurations of the proposed method on the LiTS dataset (Dice and IoU: %).  
Figure 8: The first column on the left side is the original image; the second column is the corresponding GroundTruth; the third column is the segmentation result using the combined loss function; and the fourth column is the segmentation results using the boundary loss function. In the third and fourth columns, the colors used to represent the segmentation results are as follows: liver TP (red), liver FP (yellow), liver FN (pink), tumor TP (green), tumor FP (purple), tumor FN (light green), liver and tumor TN (black).  
Figure 7: By comparing the cross-entropy loss function, dice loss function, combined loss function, contour loss function and boundary loss function, 2.5D FCNs have an improved segmentation effect when the boundary loss function is used. The boundary loss function can better optimize the segmentation of the boundary of the liver and tumor. Afterwards, the corresponding post-processing method can achieve an excellent segmentation effect.