## 3 Results  
### Generating a Virtual Point Cloud  
Laboratory tests were carried out on the basis of ideal data distributed over the sphere. There were six points distributed on the sphere, and an additional seventh point was the center of the sphere. Then, on their basis, a set of 98 points was generated, and it was assumed that this set would be the input system for the transformation. Then, the sphere with the set of seven base points was shifted by any (random) vector and turned by any (random) angle. Similarly to the first case, a second virtual point cloud was created on the basis of the base points. This set was called the output system. Points were generated according to the algorithm described in Section 2.2. Table 1 presents the thus-obtained coordinates of selected homologous points of virtual point clouds in the input and output systems. The distribution of the points generated in this way is shown in Figure 5. As can be seen from the data presented in Table 1 and in Figure 5, the points that are artificially generated are located in the central part of the sphere where their concentration is greatest. The graph shows the relationships in each dataset. The diagonal plots show the marginal distribution of the data in each column. The border points, which were the basis for generating a virtual point cloud, also come to the fore, where the concentration of virtual points is the lowest. After generating virtual point clouds, the distances between the individual points in each of the input and output systems were verified. The calculated  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline  & \multicolumn{3}{c}{**The Input Layout [m]**} & \multicolumn{3}{c}{**The Output System [m]**} \\ \hline
**Nr** & **X** & **Y** & **Z** & **X** & **Y** & **Z** \\ \hline
1 & \(-7.0000\) & \(4.0000\) & \(35.0000\) & \(-1.8000\) & \  
distances were identical, which confirmed the preservation of symmetry with respect to translation and rotation, as well as the invariability of the scale, and therefore the homology of points in the input and output systems. The dataset included the coordinates of 105 points in the input and output frames. The networks were learned on a set of 73 points, which included 7 basis points. The test set included 16 points, and a validation set 16 points. The points were randomly selected in addition to the 7 base points. In the STATISTICA Neural Networks (SANN) package, the top 5 were selected out of 100 learned and analyzed networks, which turned out to be MLP  
Figure 5: Virtual point cloud for laboratory research. Input coordinate system—(**a**) points distribution, (**b**) coordinates correlation diagram; output coordinate system—(**c**) points distribution, (**d**) coordinates correlation diagram. By statistically comparing the differences in coordinates between the output system and the results obtained from the analytical model and neural networks, statistically, significant differences were found. These are particularly visible in the case of the absolute values of differences. The test results confirm the better accuracy of the network; for the \(x\) and \(y\) variables, there is a significant statistical difference, and for the \(z\) variable, although the results are slightly worse, it is not a statistically significant difference. The conducted laboratory tests confirmed the correctness of the proposed method. In each case, the ANNs achieved higher accuracies than the similarity transformation algorithm. Therefore, field research was initiated. The field research was carried out on two datasets derived directly from the measurements; the set marked as Z1 contained 87 points, and the set marked as Z2 contained 141 points. Datasets based on measurement data were obtained from measurements on the Navigator XXI vessel. Two measuring stations were used. In the case of the Z1 set, three points were assumed, which were measured in one measurement series. Then, an artificial point cloud was generated from the points, and the final set consisted of 87 points.