## 3 Results  
### Testing Results  
_Data 1_  
At the first sight, Data 1 represents a simple object consisting of simple geometrical shapes. Figure 4 depicts the original data colored according to the \(T\) values. It is important to note that the \(T\) value distinguishes areas with different levels of ruggedness--this is the very principle of the method as without these distinguishing capabilities, the progressive dilution could not be satisfactory. Blue color indicates the most rugged/curved parts, red color the flattest ones. The figure clearly shows that the edges and joints are well distinguished from flat surfaces--including details such as the round blue dots in the lower part of the figure, indicating bolt heads. On a standard present-day laptop, the \(E_{3}\) calculation for all points in this cloud took 1.2 s; the transformation of \(E_{3}\) to \(T\) was practically immediate, and the dilution step took 0.4 s.  
Figure 5 indicates the results of processing with selected parameters and confirms that the progressive dilution works as intended. Figure 5a shows data after uniform dilution using a = 0.02 m, the remaining variants show the same data filtered using the same a but higher b, thus preserving details with the same accuracy as uniform dilution but lowering the numbers of points in flat areas. \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Data** & **Uniform Dilution [m]** & **R [m]** & **a/b [m] for Progressive Dilutions** \\ \hline \multirow{3}{*}{Data 1} & \multirow{3}{*}{0.01; 0.02; 0.03; 0.05} & \multirow{3}{*}{0.050} & (a) 0.01/0.02; 0.02/0.04; 0.03/0.06; 0.05/0.10; \\  & & & (b) 0.01/0.03; 0.02/0.06; 0.03/0.09; 0.05/0.15; \\  & & & (c) 0.01/0.05; 0.02/0.10 \\ \hline \multirow{3}{*}{Data 2} & \multirow{3}{*}{0.01; 0.02; 0.03; 0.05} & \multirow{3}{*}{0.100} & (a) 0.01/0.02; 0.02/0.04; 0.03/0.06; 0.05/0.10; \\  & & & (b) 0.01/0.03; 0.02/0.06; 0.03/0.09; 0.05/0.15; \\  & & & (c) 0.01/0.10; 0.02/0.10; \\ \hline \multirow{3}{*}{Data 3} & \multirow{3}{*}{0.005; 0.01; 0.02} & \multirow{3}{*}{0.050} & (a) 0.005/0.01; 0.01/0.02; 0.02/0.04; \\  & & & (b) 0.005/0.015; 0.01/0.03; 0.02/0.06; \\ \cline{1-1}  & & & (c) 0.005/0.025; 0.005/0.050; 0.01/0.05; 0.02/0.10 \\ \hline \hline \end{tabular}
\end{table} 
Table 1: The minimum distances of neighboring points used for dilution and testing. Figure 4: Data 1—The original point cloud colored according to the \(T\) values; red—flatest parts, blue—most curved/rugged parts. Table 2 presents the results of the numerical testing of Data 1. As can be expected, increasing the a value leads to a decrease in the number of points in the diluted point cloud and the RMSD grows. Obviously, a is crucial for the capture of details as it sets the maximum possible density (maximum detail) of the resulting cloud. The quality of results should be, therefore, compared for the uniform and progressive dilutions with the same a. If, for example, the number of points after progressive dilution with settings of a = 0.01 m a b = 0.02 m is approximately 30% lower than that of uniform dilution while the RMSD remains practically unchanged (0.0005 m vs. 0.0006 m, respectively), it can be perceived as a successful application of the tested algorithm. The detail presented in Figure 6e allows a better visual evaluation of the performance of the progressive dilution algorithm. Clearly, the density of points remains much higher on the edges and in the joints between stones than on the flat sides of the stones. The edges of details, such as the bolt head in the lower left corner are well preserved, but their flat parts are successfully diluted.