# Wall Structure Geometry Verification Using TLS Data and BIM Model
## 3 Case Study
\begin{table}
\begin{tabular}{c c c c c c c} \hline n & \(\alpha\) [\({}^{\circ}\)] & \(\delta\) [mm] & max [mm] & min [mm] & avg [mm] & absmax [mm] \\ \hline
1 & 0.1346 & \(-\)44 & 24 & 6 & 16 & 24 \\
2 & 0.1279 & \(-\)17 & 18 & \(-\)9 & 11 & 18 \\
3 & 0.0793 & \(-\)28 & 20 & 4 & 11 & 20 \\
4 & 0.0865 & 9 & 21 & 5 & 10 & 21 \\
5 & 0.1338 & 23 & 23 & 6 & 14 & 23 \\
6 & 0.2045 & 51 & 22 & 7 & 14 & 22 \\
7 & 0.0815 & 16 & 20 & 2 & 12 & 20 \\
8 & 0.2042 & 30 & 23 & 9 & 16 & 23 \\
9 & 0.1192 & 18 & 21 & 7 & 13 & 21 \\
10 & 0.1045 & \(-\)1 & 26 & 7 & 16 & 26 \\
11 & 0.5026 & 166 & 22 & \(-\)4 & 9 & 22 \\
12 & 0.0761 & 2 & 19 & \(-\)1 & 9 & 19 \\
13 & 0.1316 & 1 & 41 & \(-\)3 & 14 & 41 \\ \hline \end{tabular} Where n is the number of the inspected wall, \(\alpha\) is the difference between the orientation of normal vectors of IFC and regression plane, \(\delta\) is the deviation between the wall’s coefficient d, max and min are maximum and minimum deviations between the IFC plane and the segmented point cloud, avg is the average deviation between the IFC plane and the segmented point cloud, absmax is the absolute maximum deviation between the IFC plane and the segmented point cloud.  
\end{table} 
Table 2: Deviations between the BIM model and the as-built model. Figure 17: Deviation map, the wall no. 7—comparison of the IFC plane and the segmented point cloud. In Table 3, the results of the wall flatness quantification are listed, where are shown maximal and minimal distances between the regression plane and the segmented points. The deviations reached values from a maximum of 19 to a minimum of \(-24\) mm. The standard deviations (std), calculated from the orthogonal distances of the segmented points from the regression plane, reached a maximum of 2 mm. The measurement of the flatness is an integral part of the inspections carried out. By classical measurement methods, this is measured as a deviation from a straight edge (usually 2 m long) using a measuring wedge. The inspected parts are chosen randomly, therefore, the result does not give a complex picture of the entire wall surface. The advantage of laser scanning is that the flatness is inspected on the whole surface of the wall. The tolerance defined by the standard EN 13914-2 Design, preparation and application of external rendering and internal plastering Internal plastering for the inspected walls is 5 mm along 2 m of the inspected structure. The deviation maps show the parts where the criterion was exceeded. In the case of flatness inspection, the accuracy of the results is affected by the dispersion of the points around the regression plane reflecting the random error (noise) of the measurement by TLS (coordinate determination) mainly. It is expressed by the standard deviation calculated from the orthogonal distances and listed in Table 3. In practice, it means that the values under the value of standard deviation are not significant, which in our case study means 2 mm. \begin{table}
\begin{tabular}{c c c c c} \hline n & max [mm] & min [mm] & abs\({}_{\text{max}}\) [mm] & std [mm] \\ \hline
1 & 5 & \(-9\) & 9 & 2 \\
2 & 4 & \(-16\) & 16 & 2 \\
3 & 7 & \(-9\) & 9 & 1 \\
4 & 10 & \(-15\) & 15 & 2 \\
5 & 7 & \(-6\) & 7 & 2 \\
6 & 7 & \(-5\) & 7 & 2 \\
7 & 10 & \(-9\) & 10 & 2 \\
8 & 11 & \(-5\) & 11 & 2 \\
9 & 5 & \(-6\) & 6 & 1 \\
10 & 11 & \(-9\) & 11 & 1 \\
11 & 10 & \(-4\) & 10 & 1 \\
12 & 14 & \(-5\) & 14 & 2 \\
13 & 19 & \(-24\) & 24 & 2 \\ \hline \end{tabular} Where n is the number of the inspected wall, max and min are maximum and minimum deviations between the regression plane and the segmented point cloud, avg is the average deviation between the regression plane and the segmented point cloud, abs\({}_{\text{max}}\) is the absolute maximum deviation between the regression plane and the segmented point cloud, std is the standard deviation calculated from the orthogonal distances of the segmented points from the regression plane.  
\end{table} 
Table 3: Quantification of flatness of the walls. Figure 18: Deviation map, the wall no.