# Three-Dimensional Surface Reconstruction from Point Clouds Using [PERSON]'s Elastica Regularization
## 5 Results
### Experiments for 3D Case  
In order to demonstrate the superiority of the proposed [PERSON] elastica energy reconstruction model in 3D point cloud reconstruction, we conducted numerical experiments using both synthetic and real data sets. First, in the synthetic experiments, as shown in Figure 5, we aimed to reconstruct a spherical object. We compared proposed [PERSON]'s elastica model with the classical TV reconstruction model. From the experimental results, it is evident that the curves reconstructed using the TV regularization term are affected by the staircase effect, while the surfaces reconstructed using [PERSON]'s elastica as regularization exhibit greater smoothness. This indicates that [PERSON]'s elastica model performs exceptionally well in surface reconstruction. The results of the 3D surface reconstruction of the bunny point cloud data are shown in Figure 6 using six different methods. The six models all have fitting terms so they all performed fairly well. The contour generated by the Chan-Vese model is not as smooth as that generated by the GAC model, but the latter loses more contour. The TV model and the TV\({}_{g}\)-L1 model also lose lots of features and exhibit shrinking behavior. The results of the TV\({}_{g}\)-B method and the proposed model are better than the above four models, but the results of the TV\({}_{g}\)-B model in reconstruction often suffer from excessive smoothness, leading to significant contrast loss, loss of texture details, and noticeable staircase effects, particularly in regions such as the legs. The proposed method, on the other hand, outperforms the TV\({}_{g}\)-B model in preserving fine leg details and texture while minimizing the presence of staircase effects. This shows that the proposed model is the best at preserving details. \begin{table}
\begin{tabular}{c c c c c c c} \hline  & **Chan–Vese** & **GAC** & **TV** & **TV\({}_{g}\)-L1** & **TV\({}_{g}\)-B** & **Ours** \\ \hline Point Cloud 1 & 1.5337 & 1.5926 & 1.4644 & 1.3292 & 1.1669 & 1.1278 \\ Point Cloud 2 & 1.4936 & 1.7140 & 2.1126 & 1.6273 & 1.2777 & 1.2256 \\ \hline \end{tabular}
\end{table} 
Table 1: Uniform sampling error in different methods. Figure 4: The 2D reconstruction results by different models; the first row shows the reconstruction result of point clouds 1, and the second line shows the reconstruction result of point clouds 2. The 1 st line: initial surface; the 2 nd line: final result by the Chan–Vese model; the 3 rd line: final result by the GAC model; the 4 th line: final result by the TV method; the 5 th line: final results by the TV\({}_{g}\)-L1 model; the 6 th line: final results by the TV\({}_{g}\)-B method; the 7 th line: final results by the proposed method. Figure 5: Detailed comparison of the reconstruction results between the TV model and proposed model. From left to right: ground truth; initial surface; reconstruction of the “ball” using the TV\({}_{g}\)-B model; reconstruction of the “ball” using the proposed model. Figure 8 shows the convergence curve of the bunny and hand by the Chan-Vese model, the GAC model, the TV model, the TV\({}_{g}\)L-1 model, the TV\({}_{g}\)-B method, and the proposed model. We plot the normalized energy for clarity. The overall results show that all methods are convergent. In Table 2, we list the iteration count and runtime required for different models to reach the convergence criterion (15). From the table, it is evident that proposed model performs comparably to the TV\({}_{g}\)-B model in both convergence speed and runtime. This can be mainly attributed to the utilization of the fast dual projection algorithm presented in Equation (11). In contrast to the ADMM algorithm shown in (10), which requires computations of five variables and four Laplacian operators at each iteration, our proposed fast method only necessitates the calculation of two variables. Using the ADMM algorithm for reconstructing the bunny and hand requires more running times.