## 2 Global Registration Method Based on the OBB Regional Area-Based Descriptor
### Principle Concept and Flow Chart Diagram of the Global Registration Method
\begin{array}{l}x^{\prime}{}_{min}=\min\left\{x^{\prime}{}_{\shortmid},\ i=1,...,\ n \right\},\ x^{\prime}{}_{max}=\max\left\{x^{\prime}{}_{\shortmid},\ i=1,...,\ n\right\}\\ y^{\prime}{}_{min}=\min\left\{y^{\prime}{}_{\shortmid},\ i=1,...,\ n\right\}, \ y^{\prime}{}_{max}=\max\left\{y^{\prime}{}_{\shortmid},\ i=1,...,\ n\right\}\\ z^{\prime}{}_{min}=\min\left\{z^{\prime}{}_{\shortmid},\ i=1,...,\ n\right\}, \ z^{\prime}{}_{max}=\max\left\{z^{\prime}{}_{\shortmid},\ i=1,...,\ n\right\} \end{array} \]

The parameters of the OBB can be determined as follows:

\[\left\{ \begin{array}{l}x_{C}=\overline{x}+v_{11}x^{\prime}{}_{min}+v_{21}y^ {\prime}{}_{min}+v_{31}z^{\prime}{}_{min}\\ y_{C}=\overline{y}+v_{12}x^{\prime}{}_{min}+v_{22}y^{\prime}{}_{min}+v_{32}z^{ \prime}{}_{min}\\ z_{C}=\overline{z}+v_{13}x^{\prime}{}_{min}+v_{23}y^{\prime}{}_{min}+v_{32}z^{ \prime}{}_{min}\end{array} \right. \tag{16}\]

\[ \begin{array}{l}CC_{1}=\left(x^{\prime}{}_{\max}-x^{\prime}{}_{\min}\right) v_{1}\\ CC_{2}=\left(y^{\prime}{}_{\max}-y^{\prime}{}_{\min}\right)v_{2}\\ CC_{3}=\left(z^{\prime}{}_{\max}-z^{\prime}{}_{\min}\right)v_{3}\end{array}  \tag{17}\]

### Generating Descriptors of Point Cloud

#### 2.4.1 The OBB Regional Area-Based Descriptor

Each proposed feature descriptor includes two components [15]. The first component contains the information about the OBB of the point cloud. Each OBB is represented by a 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 distribution of the surface area of the object in the OBB. We assume that the three directions of the OBB are divided into \(k_{1}\), \(k_{2}\), and \(k_{3}\) segments (as shown in Figure 7a); the total number of subdivided boxes in the OBB is \(n=k_{1}\times k_{2}\times k_{3}\). The surface area of the object in each subdivided box \(V_{ijk}\) (\(i=0,...,k_{1}-1;j=0,...,k_{2}-1;k=0,...,k_{3}-1\)) is denoted as \(f_{v}\), where \(v=k(k_{1}k_{2})+j(k_{1})+l_{2}\)\(f_{v}\) can be determined by subdividing the considered surface into the triangle mesh [16] (as shown in Figure 7c). The distribution of \(S_{v}\) is defined as the regional area-based descriptor, shown in Figure 7d. Thus, the feature descriptor \(FV\) can be described as follows:

\[FV=\left\{C,CC_{1},CC_{2},CC_{3},f_{0},\ldots,f_{v},\ldots,f_{nv}\right\} \tag{18}\]

Figure 6: The coordinates of point \(p_{i}\) (green dot) in the absolute and relative coordinate systems. where \(n_{V}=k_{1}k_{2}k_{3}-1\). #### 2.4.2 Generating Database and Calculating Descriptors

Firstly, the regional area-based descriptor of point cloud \(P\) is computed as \(FV_{P}=\{C_{P};\)\(C_{P}C_{IP},\)\(C_{P}C_{2P},\)\(C_{P}C_{3P},\)\(r_{Pv},\)\(v=0,\ldots,n\}\) in the OBB of \(P\). Then we shift the OBB of \(P\) to \(Q\) using the translation matrix defined by the centers of gravity of \(P\) and \(Q\). We consider the coordinate system \(O_{Q}x_{q}y_{q}z_{q}\) with the origin \(O_{Q}\) is the center of gravity of \(Q\) and three unit vectors \(\mathbf{v}_{1}\), \(\mathbf{v}_{2}\), and \(\mathbf{v}_{3}\), which are considered as the three directions of the OBB of point cloud \(P\). Rotating \(Q\) around the three axes of the coordinate system \(O_{Q}x_{q}y_{q}z_{q}\) is performed with every increment \(\Delta\theta\) on each axis. For each rotation, the regional area-based descriptor \(Q\) in the OBB of \(P\) is \(FV_{Qi}=\{C_{Qi};\)\(C_{Qi}C_{1 Qi},\)\(C_{Qi}C_{2 Qi},\)\(C_{P}C_{3 Qi},\)\(r_{Qv},\)\(v=0,\ldots,n\}\) which can be then determined as described as Algorithm 2. To enhance the accuracy, the descriptors \(FV_{P}\) and \(FV_{Qi}\) can ignore the subdivided boxes in which the missing parts of point cloud \(P\) are located. Figure 7: Illustration of the regional area-based descriptor.