## 2 Global Registration Method Based on the OBB Regional Area-Based Descriptor  
### Principle Concept and Flow Chart Diagram of the Global Registration Method  
To find the transformation between two series of scans that are acquired from multiple single scans, the OBB regional area-based descriptors of source point cloud are matched with the descriptors of the target point cloud. Figure 2 illustrates the concept of the proposed registration method based on the OBB regional area-based descriptor to register two series of scans which are represented by the red and green point cloud for the target and source 3D point cloud that are scanned and obtained from the optical probe. Each of the proposed descriptors includes two critical components. The first component contains the information about the OBB of the point cloud. Each OBB is represented by a datum corner, \(C(x_{C},y_{C},z_{C})\), and three vectors, \(CC_{1}(x_{max},y_{max},z_{max})\), \(CC_{2}(x_{mid},y_{mid},z_{mid})\), \(CC_{3}(x_{min},y_{min},z_{min})\), corresponding with the maximum, middle, and minimum dimensions of the OBB, respectively. The second component represents the spatial distribution of the surface area of the object in the OBB. The similarity between the regional area-based descriptor representing the source point cloud and the regional area-based descriptor representing the target one is then determined by using the normalized cross-correlation. Figure 2: The proposed registration method based on the OBB (oriented bounding box) regional area-based descriptor. To make the proposed method clear in its operation procedure, the following flow chart diagram, shown in Figure 3, as well as Algorithm 1, is used to describe the proposed method in its main five steps. In step 1, the boundary points located near the scanning point missing zone can be detected through the geometric relationship with the support plane, using a set of connected edges that satisfy two conditions, in which all the boundary edges in the point missing zone have a similar orientation and each edge lacks an assigned triangle, either on the left or on the right side [7]. Then, the oriented bounding box (OBB), which is a rectangular bounding box that covers all the object point cloud, is determined in step 2. In the next step, the generation of the descriptors of the point cloud is performed. Then, the matching process between the descriptors is implemented and achieves the initial transformation that can be used for further refined registration using ICP. ```
Input: Two series of scans \(P\) and \(Q\) Output: Transformation matrix
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1 Estimate the boundary of the missing points and enclose the missing regions ```
2 Determine individual OBBs of the two series of scans
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3 Generate the descriptors of \(P\) and \(Q\) Calculate the OBB regional area-based descriptor of \(P\) Translate the OBB of \(P\) to \(Q\) using the translation matrix defined by the centers of gravity of \(P\) with its estimated missing points and \(Q\) with its estimated missing points ```
4 Rotate \(Q\) around the three axes of the coordinate system defined by the center of gravity of \(Q\) as the origin and the three unit vectors considered as the three directions of the OBB of point cloud \(P\) with every increment \(\Delta\theta\) on each axis, generating the OBB regional area-based descriptors of \(Q\)
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5 Match the descriptors between \(P\) and every \(Q\) having an increment \(\Delta\theta\) of rotation performed in Step 3.3 ```
6 Determine the best machine and obtain the initial transformation matrix
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**Algorithm 1**

Figure 3: The flow chart diagram of the proposed method. ### Estimation of Missing Regions and Initial Translation

Assuming that two series of scans to be registered are represented as two sets of unorganized points \(P=\{p_{1},\,p_{2},\,\ldots,\,p_{k}\}\) and \(Q=\{q_{1},\,q_{2},\,\ldots,\,q_{k}\}\), the initial transformation, which represents the coarse alignment between the two series, can be estimated as follows.