## 3 Results  
In this section, we tested the proposed methods and the existing methods with various synthetic and real point sets. When using SVD to compute the normal vector, the iteration will be stopped when the absolute error or the relative error reaches a value that is less than a given tolerance \(1\times 10^{-6}\). Since the length of the cube has an important impact on the algorithm, unless otherwise specified, the length of the cube is \(1.0L\). In order to test the robustness of the proposed algorithm, we chose a set of typical point sets including free-form surfaces, planes, and other irregular point cloud models. Since the normal vector is very sensitive to light, we use the shading model to observe the small changes in the normal vector. All the experiments were run on a computer with Intel Core i5, 2.67 GHz CPU and 4 GB memory, running windows 7 (64 bit). Figure 4 shows the normal vector result of a free-form surface model. The model contains 0.712 million points. Figure 4a is a shading model obtained using our method. In the calculation process, 376 \(\times\) 314 \(\times\) 93 cubes were constructed, of which only 134,832 cubes contained points data (about 1.23%). It takes 1014 ms to estimate the normal vector, including 202 ms for the normal vector interpolation and 812 ms for the nearest neighbor search and the PCA algorithm. Figure 4b is the normal vector that is attached to the point cloud. Figure 4c is an enlarged image that shows that the method performs well when dealing with sharp changes of surface. Figure 4d is the shading model obtained by the ENN based algorithm, and it takes 3762 ms to estimate the normal vector, which is about 3.7 times more than our method. This is mainly due to the fact that our algorithm avoids more than 80% of the matrix decomposition. Figure 5 shows the results on several point sets with different numbers and complexity. The point cloud contains 0.121, 0.178, 0.516, 1.272, and 2.875 million points, respectively. The proposed method can deal with not only smooth transition surfaces but also the models with complex shapes and shape changes. To be objective, we estimate the normal vector for these points clouds using the EGG-based method presented by [PERSON] [6]; the '\(k\)' value is set to 8. Table 1 is the details of point number, projection point number, the cubes constructed by our method, and the comparison of the normal vector estimation with the EGG-based algorithm. The number of points and projection points in Table 1 is counted in the millions, and the unit of the time in Table 1 is millisecond. With the increase of the point cloud  
Figure 4: (**a**) Shading model of free-form surface obtained by our method; (**b**) the normal vector estimation results; (**c**) an enlarged image of local part with sharp changes; (**d**) shading model of the free-form surface according to the Euclidean nearest neighbor (ENN)-based method. Figure 5: Normal vector estimation of several point cloud with the cube’s size 1.0\(L\). (**a**) Famer Statue; (**b**) Blade; (**c**) Mother Statue; (**d**) Engine Cover; (**e**) Engine. number, the time of algorithm increases, but the time of the nearest neighbor search and normal vector estimation is obviously slower than the normal vector interpolation. We carried out experiments on the simulation of spherical point cloud data; the results are shown in Figure 6. From (a) to (e), the number of points in the simulated spherical point cloud is 0.1875, 0.375, 0.75, 1.5, and 3.0 million points, respectively. We tested the proposed algorithm and the EGG-based algorithm on the actual data and the simulated data, and plot the results in Figure 7a,b. The efficiency of the two algorithms is consistent with the data in Table 1. The average number of neighborhood points in our algorithm can reach 15, and Figure 7 shows the result that our algorithm is 3-4 times faster than the latter, and this is due to the fact that our algorithm requires less nearest neighbor searching and SVD operations.