# Automatic Point-Cloud Registration for Quality Control in Building Works
## 3 Results and Discussion
### Quality Control  
The BIM models (Figure 8b,c) were used as reference for the quality control. More precisely, the registration of the BIM and point clouds was done manually using a local data established by the company, which corresponds with the definition of a local origin and the direction of \(X\) and \(Y\) axes. The \(Z\) axis was always considered along the vertical. After that, two different approaches were performed for analysing the quality: (i) the models were compared with the designed building information model (BIM) to find visual discrepancies between as-built and BIM models; (ii) the accuracy of the point clouds was checked comparing against the BIM models. In particular, the accuracy of the point clouds was checked using two different dimensional analysis: (i) dimensional control of the pillars, (ii) distance between construction elements. For the dimensional control, the dimensions obtained from the as-built model, automatically registered, were compared to the dimensions measured in the reference BIM. The reference BIM provides a theoretical model very close to the final result expected for the construction, and it is updated over the time. This contributes to making the quality control during the whole process. The BIM reflects the geometry, metrical properties, material properties and construction phases of each element, together with the final layout of the building [61]. This way, the BIM is considered as a theoretical model in this case and provides the reference measurements for the structural elements chosen to validate the methodology in each phase. However, the point clouds registered with the automatic procedure proposed contain a high amount of information, and their statistical distribution is often non-normal [62]. Thus, the statistical analysis performed was based on nonparametric estimators of the median, m, such as the normalized median absolute deviation (_NMAD_) (Equation (7)) and the bi-weight mid-variance (_BWMV_) (Equation (8)). These parameters allow us to perform a comparison and, consequently, a more precise and reliable quality control. \[NMAD=14,826\cdot MAD \tag{7}\]  
\[BWMV=\frac{n\cdot\sum_{i=1}^{n}a_{i}\cdot\left(x_{i}-m\right)^{2}\cdot\left(1 -U_{i}^{2}\right)^{4}}{\left(\sum_{i=1}^{n}a_{i}\cdot\left(1-U_{i}^{2}\right) \cdot\left(1-5U_{i}^{2}\right)\right)^{2}} \tag{8}\]  
Figure 10: Automatic registration procedure: (**a**) initial state of the point clouds (all referenced to the (0,0,0), with relative coordinates regarding the position of the TLS in each acquisition); (**b**) state of the point clouds after the initial registration based on Harris 3D Noble–PFH, with a detail of the pillars where the error in registration is noticeable; (**c**) final result of the ICP registration, with a detail of the pillar shown in (**b**), which was correctly registered during this last step. Each colour represents a different point cloud. where:  
\[a_{i}=\ \left\{ \begin{array}{l}1,\ if\ |\ U_{i}\ |<1\\ 0,\ if\ |\ U_{i}\ |\geq 1\end{array} \right. \tag{9}\]  
\[U=\ \frac{x_{i}-m}{9\cdot MAD} \tag{10}\]  
being the median absolute deviation (\(MAD\)) defined by Equation (11) and described as the median (\(m\)) of the absolute deviation of the median of the data (\(m_{x}\)) and \(x_{i}\) the measurements performed on the registered point cloud. \[MAD=m\cdot(\ |\ x_{i}-m_{x}\ |) \tag{11}\]  
Table 3 shows the results of bias and dispersion for the discrepancies of TLS vs. BIM for the two study cases. Considering that the approach developed provides values admissible for quality control tasks, a comparison of measurements on the point cloud registered with the method and on the reference BIM measurements was performed. The elements where the errors in registration are most evident are pillars. Table 4 shows the admissible tolerances within the most relevant structural elements in a building according to [26]. The validation focuses on the dimensions and the cross sections of the pillars. These elements are predefined, making them adequate as control elements.