# Fast Implicit Surface Reconstruction for the Radial Basis Functions Interpolant
## 6 Numerical Results
### Case Studies  
We sampled sets of unorganized points with normals from a number of real objects to test the implicit surface reconstruction method, as shown in Figure 8. The unorganized points are used to construct the on-surface constraints (\(f(\mathbf{x}_{i})=0\)) and the normals are used to construct off-surface constraints (\(f(\mathbf{x}_{i})>0\) or \(f(\mathbf{x}_{i})<0\)) by projecting the points along the normal direction. The implicit function (the RBF interpolant) is obtained by interpolating the constraints and solving the corresponding linear system. Then the implicit surface can be reconstructed using the improved surface reconstruction method.  
In the process of surface reconstruction, all of the on-surface constraints are selected as the seed points. Figure 9 demonstrates comparison result of the voxel growing method with one and multiple seed points based on the on-surface constraints. The on-surface constraints are constructed by the sampling points. As mentioned, for the surface following method, the main problem is the incomplete reconstruction of sub-objects. Compared with the original voxel growing method, the improved method tends to recover the complete model as long as the sampling is sufficient.  
Figure 8: Implicit surface reconstruction from a set of unorganized points with normals. (**a**) The interpolation constraints, including the on-surface constraints and the off-surface constraints. (**b**) The reconstructed implicit surface. (**c**) The evaluated voxels in voxel growing. (**d**) The evaluated voxels in cutaway view.  
Figure 7: Comparison of the evaluation time with 10000 source points using the original FMM (blue) and the single point FMM (red) with three steps. Compared with (**a**), the single point FMM in (**b**) evaluates the points leaf by leaf.  
In comparison with the marching cubes method, we test the performance of the improved voxel growing method and the leaf growing method on a variety of geological data sets, as shown in Figures 10 and 11. The surface following method does not modify the triangulation of voxels. Though the evaluation process of the three methods is different, the reconstruction results are the same when using the same size of reconstruction cell.  
Figure 11: Reconstruction results of sets of unorganized points sampled from several real objects (**a**) orebody #1, (**b**) orebody #2 and (**c**) orebody #3.  
Figure 9: Comparison result of the voxel growing method with one (**b**) and multiple (**c**) seed points based on the interpolation constraints (**a**). The black boxes point to the complete reconstruction of multiple isosurfaces.  
The performance of the reconstruction method mainly depends upon the number of interpolation constraints, the expansion order and the size of the reconstruction cell. The number of interpolation constraints determines the number of source points in the FMM evaluation. The expansion order determines the precision of evaluation, and the size of the reconstruction cell determines the model precision.  
Table 1 reports the timings of the evaluation of the marching cubes method, the voxel growing method and the leaf growing method on experimented examples. In the process of space division, the minimum bounding boxes of the sampling points are scaled by 0.01 times, respectively. Overall, because of the reduction of evaluation points, the voxel growing method has a better efficiency. The leaf growing method further improves the reconstruction efficiency. When the size of the reconstruction cell is large, the improvement in efficiency is not obvious. However, as the size of the reconstruction cell decreases, the performance advantage of the leaf growing method is more obvious. A series of experimental results show that the reduction of evaluation points can balance out the performance degradation for single point evaluation.