## 3 Results
### Regularization Analysis  
This paper proposed a double regularization constraint on the model during training, hoping to avoid the excessive deformation and voxel folding problems that may be caused by two-stage registration. In this section, we analyzed the impact of the additional bending energy loss on the registration performance. Table 3 presents the registration accuracy and the percentage of folded voxels of MCANet when the bending energy weight coefficient \(\beta\) took different values, and the data are presented in the form of a line chart in Figure 6.  
To intuitively show the constraint of the bending energy regularization term on the deformation field, Figure 7 visualizes the deformation field under different values of \(\beta\). By observing the visualization results of the deformation field, it can be seen that, with the increase of \(\beta\), the folded gridlines (white areas) in the gridded deformation field were significantly reduced, and the distortion of the gridlines became more natural and smoother. The median of Dice score, SDlogJ, and folded voxel percentage of MCANet under different \(\beta\) values can be seen in Figure 6. It can be seen that the Dice score shows a slight decrease  
\begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\beta\) & **Dice Score** & **SDlogJ** & \(|J_{\mathbf{\rho}}(\mathbf{u})|\leq 0\) & \(\%|J_{\mathbf{\rho}}(\mathbf{u})|\leq 0\) \\ \hline
0 & \(0.649\pm 0.016\) & \(0.462\pm 0.025\) & \(36,706\pm 4812\) & \(0.689\pm 0.090\) \\
0.2 & \(0.645\pm 0.017\) & \(0.451\pm 0.084\) & \(30,019\pm 4473\) & \(0.564\pm 0.084\) \\
0.4 & \(0.647\pm 0.015\) & \(0.446\pm 0.033\) & \(24,922\pm 4420\) & \(0.468\pm 0.083\) \\
0.6 & \(0.643\pm 0.015\) & \(0.430\pm 0.052\) & \(20,965\pm 3117\) & \(0.394\pm 0.059\) \\
0.8 & \(0.641\pm 0.016\) & \(0.436\pm 0.041\) & \(16,878\pm 2758\) & \(0.317\pm 0.052\) \\
1 & \(0.627\pm 0.018\) & \(0.429\pm 0.026\) & \(14,274\pm 2776\) & \(0.268\pm 0.052\) \\
1.2 & \(0.639\pm 0.013\) & \(0.416\pm 0.065\) & \(13,560\pm 2412\) & \(0.255\pm 0.045\) \\
1.4 & \(0.637\pm 0.013\) & \(0.394\pm 0.021\) & \(11,693\pm 2418\) & \(0.219\pm 0.045\) \\
1.6 & \(0.630\pm 0.016\) & \(0.418\pm 0.042\) & \(10,585\pm 2227\) & \(0.199\pm 0.042\) \\
2 & \(0.618\pm 0.016\) & \(0.421\pm 0.023\) & \(7672\pm 1557\) & \(0.144\pm 0.029\) \\ \hline \hline \end{tabular}
\end{table}
Table 3: The effect of regularization coefficient \(\beta\) on registration performance.  
Figure 6: The Dice scores, SDlogs, and folded voxel of MCANet at different values of \(\beta\).  
with increasing \(\beta\), but this stabilizes at \(\beta=1.4\), indicating a maintained registration accuracy at higher \(\beta\) values. SDlogJ decreases significantly with the increase of \(\beta\), reflecting the increase of smoothness and the decrease of deformation in the deformation field. However, when \(\beta\) increases to 1.4 or higher, SDlog tends to decrease. The number and percentage of folded voxels also decrease markedly with higher \(\beta\), highlighting reduced folding and enhanced deformation quality. Overall, \(\beta=1.4\) emerges as an optimal choice, balancing high registration accuracy with smooth and authentic deformation.