## 3 Verification Results
### Case Study 3  
The third case study involved a sports glasses lens (Figure 19(a)). The choice of this part, which due to ergonomic intent is of a relatively simple geometry, was motivated by the complexity of the digitized data (Figure 19(b)) which requires adequate fixture and locating. 3D digitization was performed by a contact system _Cyclon II--Renishaw_, resulting in a total of 412,111 points, of which a large number represent error-points which actually belong to the fixture and measuring table. The very first pre-processing phase was the 3D filtering of resulting point data, which included volumetric filtering, filtering by segmented line, and elimination of individual points. Parameters used for volumetric filtering are listed in Table 4. The result was 175,315 filtered points, and 236,796 remaining points (Figure 20). Filtering by segmented line was performed in _y-z_ plane. Knot points defining the segmented line are listed in Table 5 and the obtained result (filtered 12,013 and 224,783 remained points) is shown on Figure 21. In this case, the geometric form and spatial orientation of object were the cause of a larger number of left-over point-errors--a total of 367 points were eliminated manually, thus leaving 224,416 points (Figure 22). Figure 17: The result of point data reduction. Figure 18: (**a**) Generated cross-sectional curves. (**b**) Generated surface model. Figure 19: **(a)** Sports glasses lens. **(b)** The results of 3D digitization. \begin{table}
\begin{tabular}{l c} \hline
**Parameter** & **Value** \\ \hline \(x_{\text{min}}\) & \(-8.5\) \\ \(x_{\text{max}}\) & \(36\) \\ \(y_{\text{min}}\) & \(43\) \\ \(y_{\text{max}}\) & \(116\) \\ \(z_{\text{min}}\) & \(17\) \\ \(z_{\text{max}}\) & \(35\) \\ \hline \end{tabular}
\end{table} 
Table 4: Parameters used for volumetric filtering. Figure 20: The result of volumetric filtering. Cross-sectional filtering/smoothing of point data was performed in the next phase of pre-processing and comprised elimination of end points in two iterations--1 st along the \(y\) axis, and 2 nd along the \(x\) axis (due to the disposition of points it was possible to eliminate end-points along two axes). This tool was applied because of a higher probability of error-points around the object's rims, caused by the transition of the digitization sensor probe from object to fixture and measuring table. Within the 1 st  
Figure 21: The result of filtering by segmented line. \begin{table}
\begin{tabular}{c c c} \hline
**Knot point** & \(y\)**-coordinate** & \(z\)**-coordinate** \\ \hline \(P_{1}\) & 41.6 & 20.0 \\ \(P_{2}\) & 51.2 & 23.1 \\ \(P_{3}\) & 50.8 & 20.8 \\ \(P_{4}\) & 61.1 & 23.9 \\ \(P_{5}\) & 73.8 & 27.0 \\ \(P_{6}\) & 80.2 & 27.2 \\ \(P_{7}\) & 88.7 & 26.7 \\ \(P_{8}\) & 94.0 & 25.3 \\ \(P_{9}\) & 106.7 & 21.4 \\ \(P_{10}\) & 116.8 & 16.2 \\ \hline \end{tabular}
\end{table} 
Table 5: Parameters of the applied segmented line filter. Figure 22: The result of manual elimination of points. iteration, along \(y\) axis, a total of 2,586 points (3 points at each end in all of the 429 cross-sections) were eliminated (Figure 23(a)). Likewise, in the 2 nd iteration along \(x\) axis, a total of 2,748 points (2 points at each end in all of the 687 cross-sections) were eliminated, so the resulting point cloud contained 219,082 points (Figure 23(b)). Bearing in mind a relatively simple object geometry, in the next phase of pre-processing cross-sectional resolution was changed from 0.1 to 0.2 mm along \(y\) axis (\(x\)-\(z\) plane was chosen as the reference plane). In this way, the number of cross-sections was lowered from 429 to 214, which lowered the total number of points from 219,082 to 109,528. Figure 23: Elimination of end-points in cross-sections. (**a**) 1 st iteration—along \(y\) axis. (**b**) 2 nd iteration—along \(x\) axis. In the final phase of pre-processing, reduction of point data was performed. Fuzzy-chordal reduction method was chosen, with \(MAD\) = 0.03 mm.