# Three-Dimensional Surface Reconstruction from Point Clouds Using [PERSON]'s Elastica Regularization
## 5 Results
### Experiments for 3D Case
The number of iterations required to reach the convergence condition are \(17,31,38,\) and \(42\). To better demonstrate the effectiveness of the [PERSON] elastica model compared to the TV model in the 3D reconstruction process, we magnified local images of the reconstruction results as shown in Figure 10. In the reconstruction process of the point clouds \"bunny\" and \"hand\", this figure clearly demonstrates that the TV model exhibits noticeable staircase artifacts in the reconstructed legs of the \"bunny\" and the pinky of the \"hand\". In contrast, the incorporation of [PERSON]'s elastica as a regularization term successfully overcomes this issue. Furthermore, in Figure 10 we also zoomed in on the left ear of the bunny and the index and middle fingers of the hand. As can be seen, TV leads to a less accurate surface reconstruction from the respective point cloud. In contrast, the utilization of [PERSON]'s elastica yields more precise and faithful reconstructions. Figure 11 shows the Delaunay tetrahedron error for the bunny and hand using six different methods. It is clear that the Chan-Vese model, GAC model, TV model and TV\({}_{g}\)-L1 model all move towards stabilization, but their results are not as good as those of the TV\({}_{g}\)-B method (3) and the proposed elastica method (11). When the number of iterations reaches 50, the result of the reconstruction shows a tendency toward stabilization. In Table 4, it can be seen that different models have different reconstruction errors. In order to better compare the accuracy of the reconstruction, we normalized the results to better present them. By setting the maximum error to 1 and the  
Figure 11: The errors of the bunny and hand with different reconstruction models. (**left**) Delaunay tetrahedron errors of different methods in the point clouds data of the bunny; (**right**) Delaunay tetrahedron errors of different methods in the point clouds data of the hand. \begin{table}
\begin{tabular}{c c c c c c} \hline \hline  & **10 th Error** & **20 th Error** & **30 th Error** & **40 th Error** & **50 th Error** \\ \hline Bunny & 41.36 & 40.90 & 39.19 & 39.32 & 39.34 \\ Hand & 67.20 & 67.15 & 67.12 & 67.10 & 67.09 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Delaunay tetrahedron error in different iterations. Figure 10: Detailed comparison of the reconstruction results between the TV model and proposed model. The 1 st line: reconstruction of the “bunny” using the TV model; the 2 nd line: reconstruction of the “bunny” using the proposed model; the 3 rd line: reconstruction of the “hand” using the TV model; the 4 th line: reconstruction of the “hand” using the proposed model. minimum error to 0, we can represent the errors between different algorithms on a scale of 0 to 1. This allows for a more intuitive observation of the reconstruction results, as the errors are normalized within a standardized range. The Chan-Vese model and GAC model still perform as normal, and in some point clouds with large gradient changes, TV and TV\({}_{g}\)-L1 have a lower error value, but these four models do not perform well in the concave regions. However, the TV\({}_{g}\)-B method and the proposed elastica model are more stable in that they can achieve a relatively small error in diverse point clouds. As seen in Figure 12, the surface reconstruction results from point clouds with more intricate shapes are accurately handled by the TV\({}_{g}\)-B method and the newly suggested variational method. These two methods produce smooth and visually appealing reconstructed surfaces due to the utilization of a continuous signed distance function. In contrast, when binary-based segmentation models, such as the TV model and TV\({}_{g}\)-L1 model, are implemented, the reconstructed shapes are not as smooth as those produced by the proposed method. This difference is due to the discontinuity present in the binary representation compared to the continuous level set function used in the proposed method.