## 2 Global Registration Method Based on the OBB Regional Area-Based Descriptor
### Principle Concept and Flow Chart Diagram of the Global Registration Method
(**a**) The OBB of the object is subdivided into smaller boxes; (**b**) The object point cloud and OBB; (**c**) Triangle mesh is generated over the point cloud data; (**d**) the regional area-based descriptor of the object. ### Matching Descriptors

In the first step of the matching process, the OBB's parameters of point cloud \(P\) are compared with the OBB's parameters of point cloud \(Qi\). If the OBB matching satisfies the given condition, the regional area-based descriptor of point cloud \(P\) is then compared with the regional area-based descriptor of \(Qi\). These two OBBs should be satisfied with the following equation:

\[\left\{ \begin{array}{l}\frac{C_{P}C_{P1}.C_{Q}C_{Q1}}{11C_{P}C_{P1}1.1C_{Q}C_{ Q1}11}<\alpha_{thresh}\\ \frac{C_{P}C_{P2}C_{Q2}C_{Q2}}{11C_{P2}C_{Q2}11C_{Q2}11}<\alpha_{thresh}\\ \frac{C_{P}C_{Q2}C_{Q3}}{11C_{P}C_{P3}11.1C_{Q}C_{Q3}11}<\alpha_{thresh}\\ \end{array} \right. \tag{19}\]

where \(\alpha_{thresh}\) is the given adequate threshold. The similarity between the regional area-based descriptors that represent the object point clouds \(P\) and \(Q\) is determined by using the normalized cross-correlation. The normalized cross-correlation between \(F_{P}=\left\{f_{P_{P}},\,v=0,\,\ldots,\,n\right\}\) and \(F_{Q}=\left\{f_{Q^{p}},\,v=0,\,\ldots,\,n\right\}\) is computed as follows:

\[C\left(F_{P},F_{Q}\right)=\frac{\sum\limits_{v=0}^{n}\left(f_{P ^{p}}-\overline{f}_{P}\right)\left(f_{Q^{p}}-\overline{f}_{Q}\right)}{\sqrt{ \sum\limits_{v=0}^{n}\left(f_{P^{p}}-\overline{f}_{P}\right)^{2}\cdot\sum \limits_{v=0}^{n}\left(f_{Q^{p}}-\overline{f}_{Q}\right)^{2}}} \tag{20}\]

where \(\overline{f}_{P}=\frac{1}{n+1}\sum\limits_{v=0}^{n}f_{P^{p}}\,\overline{f}_{Q }=\frac{1}{n+1}\sum\limits_{v=0}^{n}f_{Q^{p}}\)

The matching is the best in Equation (20) in which the normalized cross-correlation \(C\left(F_{P},F_{Q}\right)\) reaches its peak. ### Transformation Estimation and Refinement

After the best matching is defined, the correspondence feature vectors \(FV_{P}\) and \(FV_{Q}\) can be determined. Based on these feature vectors, the initial transformation matrix \(\mathbf{T}_{initial}\) between two series of scans can be estimated by aligning the frame \(\left\{C_{Q},\,v_{Q1},\,v_{Q2},\,v_{Q3}\right\}\) that represents the series of scans 1 to the frame \(\left\{C_{P},\,v_{Qbest1},\,v_{Qbest2},\,v_{Qbest3}\right\}\) that represents the series of scans 2. Although the parameters of the transformation matrix can be obtained in the initial registration, the accuracy may not be satisfactory due to the error caused by the limited number of the rotating \(Q\) iterations. Fortunately, a refined registration such as the iterative closest point (ICP) algorithm can be performed to achieve precise registration between multi-view point cloud. ## 3 Results and Discussion

### Case Study on Measured Data

To verify the feasibility of the developed methodology, some experiments were performed and evaluated. In the experiments, the point cloud of a hammer head were acquired from the NTU-developed optical 3D measuring probe being developed by using a random speckle pattern projection and triangulation measurement principle [12; 15]. The dimensions of the hammer head are approximately \(110\times 35.5\times 21\) mm\({}^{3}\). Firstly, the hammer head was placed on a fixed table, allowing for viewing the object from several positions around the object. However, the whole object surface could not be detected due to optical occlusion. Therefore, after the first series of automated scans to obtain point cloud \(P\), the object was reoriented manually and the second series of automated scans was continuously carried out to acquire point cloud \(Q\). Figure 8 illustrates the coarse registration process between the two series of scans. The width of the normalized cross-correlation peak depends on the number of iterations and \(\Delta\theta\), as shown in Figure 9. Meanwhile, the height of the normalized cross-correlation depends on the number of OBB segments \(k_{1}\times k_{2}\times k_{3}\) and \(\Delta\theta\).