# A Two-Step Surface Reconstruction Method Using Signed Marching Cubes
## 3 Results  
We implemented the two-step surface reconstruction method for the RBF interpolant and tested some experiment examples on a Windows 64-bit PC with 2.20 GHz Intel(R) Core(TM) i7-8750H and 16 GB RAM. The Scal-FMM library was used to implement the single point FMM for fast evaluation. The open source 3D computer graphics software visualization toolkit (VTK) was used for the visualization. The improved method was validated on several geological data sets. It is certain that other types of data (e.g., medical or mechanical engineering CT scans) can be used to validate the algorithm. In our experiments, we used the biharmonic spline and the triharmonic spline as the basis functions. Different numbers of evaluation points \(N_{e}\) and source points \(N_{s}\) on the calculation time and max residual were used to compare the performance of the evaluation algorithm, as shown  
Tables 1 and 2 demonstrate the max residual of the evaluation process using different expansion orders. In general, the higher the order, the smaller the max residual. According to the statistical data, the max residual is small enough when the expansion order was greater than 8. For the two-step reconstruction method, the recommended higher expansion order is \(p=9\)-12, and the recommended lower expansion order is \(p^{\prime}=5\)-8. The accuracy of the reconstructed model is exactly the same as that of the original marching cubes method. Taking a boundary voxel as an example, the exact function values \(S(\mathbf{x})\) of a vertex \(\mathbf{a}\) of voxels were calculated using Equation (1) and the evaluated values \(S_{p}(\mathbf{x})\) were calculated using Equation (9). For point, the evaluation error can be calculated as \(\left|S(\mathbf{a})-S_{p}(\mathbf{a})\right|<\varepsilon_{p}\), which is small enough using an appropriate expansion order, and the calculated intersection point is basically identical. Therefore, it can be considered that the accuracy of the proposed method is consistent with the original marching cubes method. Six ore body models were reconstructed using the implicit modeling method based on the RBF interpolant to prove the applicability of the two-step reconstruction method, as shown in Figure 6. The model reconstruction results and corresponding constraint points are shown in the figure. It is worth noting that the branch marked in the black box is separated from the main part of the model, and the complete model can be established by using the Marching Cubes method and Signed Marching Cubes method, but the black box marked branch cannot be established using the Voxel Growing method, as seen in Figure 6f. The method proposed in this paper can recover the complete isosurface efficiently, and is useful for reconstructing the models with several separated parts. \begin{table}
\begin{tabular}{c c c c} \hline \hline \multirow{2}{*}{\(\mathbf{p}\)} & \multicolumn{3}{c}{\(\mathbf{N_{e}}\)} \\ \cline{2-4}  & **10,000** & **100,000** & **1,000,000** \\ \hline
12 & \(2.93\times 10^{-9}\) & \(3.92\times 10^{-9}\) & \(3.83\times 10^{-9}\) \\
11 & \(2.03\times 10^{-8}\) & \(2.16\times 10^{-8}\) & \(2.64\times 10^{-8}\) \\
10 & \(1.29\times 10^{-7}\) & \(1.91\times 10^{-7}\) & \(1.56\times 10^{-7}\) \\
9 & \(9.30\times 10^{-7}\) & \(1.38\times 10^{-6}\) & \(1.54\times 10^{-6}\) \\
8 & \(8.27\times 10^{-6}\) & \(8.18\times 10^{-6}\) & \(9.25\times 10^{-6}\) \\
7 & \(1.04\times 10^{-4}\) & \(8.60\times 10^{-5}\) & \(8.80\times 10^{-5}\) \\
6 & \(9.92\times 10^{-4}\) & \(8.61\times 10^{-4}\) & \(1.06\times 10^{-3}\) \\
5 & \(7.60\times 10^{-1}\) & \(9.20\times 10^{-1}\) & \(7.40\times 10^{-1}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The max residual of the RBF interpolant (biharmonic spline) using the fast multipole method with different expansion orders (the number of source points \(N_{s}\) is 10,000). \(N_{e}\) is the number of evaluation points.