# Three-Dimensional Surface Reconstruction from Point Clouds Using [PERSON]'s Elastica Regularization
## 5 Results
### Experiments for 3D Case
Additionally, the ADMM method introduces four new parameters \(\gamma_{1}\), \(\gamma_{2}\), \(\gamma_{3}\), and \(\gamma_{4}\), which need to be carefully adjusted during the solving process. In contrast, our proposed method not only exhibits higher efficiency but it also significantly reduces the application complexity. With fast dual projection algorithm, the number of iterations and runtime is reduced, resulting in a more efficient reconstruction process. Moreover, by introducing only one dual variable \(\vec{p}\), we alleviate the burden of adjusting multiple parameters, streamlining the usage of the model in practical applications. As a result, the computational cost per iteration is significantly reduced, allowing the efficient solving of complex Euler's elastic models to be on par with lower-order TV models in terms of computational efficiency. Figure 6: Bunny reconstruction results by different models. 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\({}_{g}\)-B method; the 5 th line: final results by the TV\({}_{g}\)-B method; the 7 th line: final results by the proposed method. Figure 7: Hand reconstruction results by different models. 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. Based on the results of Figure 7, we can specifically check the effect of the balloon force by changing the value of \(\beta\). As shown in Figure 9, it can be found that it is impossible to cross a local extremum when \(\beta\) is too small. With the increase in \(\beta\), the result is more accurate, and the number of iterations required is correspondingly reduced. In addition, the number of iterations required by the proposed variational method is always less than that of the TV\({}_{g}\)-B method. However, \(\beta\) cannot grow indefinitely; once the \(\beta\) is too large, the reconstruction will fail. Table 3 shows the normalized error corresponding to the different iterations of bunny and hand reconstruction that show the relationship between the iteration and the Delaunay error. It proves that the greater the iteration, the closer the reconstructed hand point clouds are to the original model. The errors for the bunny and the hand model are different. The bunny reconstruction is significantly better than the hand, owing to the fingers' details and concave regions, which greatly increase the Delaunay error of the hand model. \begin{table}
\begin{tabular}{c c c c c c c c c c c c c} \hline  & \multicolumn{2}{c}{**Chan–Vese**} & \multicolumn{2}{c}{**GAC**} & \multicolumn{2}{c}{**TV**} & \multicolumn{2}{c}{**TV\({}_{g}\)-L1**} & \multicolumn{2}{c}{**TV\({}_{g}\)-B**} & \multicolumn{2}{c}{**Ours**} & \multicolumn{2}{c}{**ADMM**} \\  & **Iter** & **Times** & **Iter** & **Times** & **Iter** & **Times** & **Iter** & **Times** & **Iter** & **Times** & **Iter** & **Times** \\ \hline Bunny & 43 rd & 2.45 s & 3 rd & 14.14 s & 46 th & 22.51 s & 28 th & 5.23 s & 28 th & 9.19 s & 7 th & 1.97 s & 29 th & 24.42 s \\ Hand & 43 rd & 12.04 s & 3 rd & 77.46 s & 30 th & 56.67 s & 23 rd & 22.11 s & 13 th & 22.59 s & 17 th & 24.16 s & 20 th & 103.71 s \\ \hline \end{tabular}
\end{table} 
Table 2: Comparison of the number of iterations and running time required for the energy functions of different methods to reach the convergence condition (15). Figure 8: Function convergence analysis for bunny and hand. (**left**) Convergence curve of the different energy function for bunny; (**right**) convergence curve of the different energy function for hand. Figure 9: Comparison with different \(\beta\). From left to right are the results of proposed method with \(\beta=0.5,1,1.5,2\).