## 2 Global Registration Method Based on the OBB Regional Area-Based Descriptor
### Principle Concept and Flow Chart Diagram of the Global Registration Method
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 (as shown in Figure 5). The orientations of the OBB can be determined by using the covariance-based method, which is proposed by [PERSON] (1996) [14]. The algorithm is started by obtaining the centroid \(\overline{p}\) (\(\overline{x},\overline{y},\overline{z}\)) of the object point cloud:

Figure 3:4

Figure 3:5 Improvement of translation: (**a**) The influence of weight loss from the missing points to the centroid point position; (**b**) Two point sets are separated; (**c**) Detection of the boundary points located near the missing part. where \(n\) is the number of points in the object point cloud. The covariance matrix of the input point cloud \(COV\) is then computed as follows:

\[ \begin{array}{l}COV\\ \\ =\frac{1}{n}\left[\begin{array}{cc}\sum\limits_{i=1}^{n}\left(x_{i}-\overline{x} \right)^{2}&\sum\limits_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left(y_{i}- \overline{y}\right)&\sum\limits_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left( z_{i}-\overline{z}\right)\\ \sum\limits_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left(y_{i}-\overline{y} \right)&\sum\limits_{i=1}^{n}\left(y_{i}-\overline{y}\right)^{2}&\sum \limits_{i=1}^{n}\left(y_{i}-\overline{y}\right)\left(z_{i}-\overline{z} \right)\\ \sum\limits_{i=1}^{n}\left(x_{i}-\overline{x}\right)\left(z_{i}-\overline{z} \right)&\sum\limits_{i=1}^{n}\left(y_{i}-\overline{y}\right)\left(z_{i}- \overline{z}\right)&\sum\limits_{i=1}^{n}\left(z_{i}-\overline{z} \right)^{2}\end{array}\right]\end{array}  \tag{12}\]

The above matrix has three real eigenvalues, \(\lambda_{1}\geq\lambda_{2}\geq\lambda_{3}\), and three normalized eigenvectors, \(\mathbf{v}_{1}(v_{11},v_{12},v_{13})\), \(\mathbf{v}_{2}(v_{21},v_{22},v_{23})\), and \(\mathbf{v}_{3}(v_{31},v_{32},v_{33})\), respectively. These eigenvectors are considered as the three directions of the OBB. By projecting all points in the object point cloud onto the eigenvectors, three dimensions of the object can be determined as the distance between the nearest and farthest projected points in each eigenvector. Considering the coordinate system with the origin \(\overline{p}\) and three unit vectors \(\mathbf{v}_{1}\), \(\mathbf{v}_{2}\), and \(\mathbf{v}_{3}\) (as shown in Figure 6), the coordinates of the point \(p_{i}(x_{i}\), \(y_{i}\), \(z_{i})\) in this coordinate system are determined by following equation:

\[\left[ \begin{array}{c}x^{\prime}{}_{i}\\ y^{\prime}{}_{i}\\ z^{\prime}{}_{i}\end{array} \right]=\left[ \begin{array}{ccc}v_{1}e_{1}&v_{1 }e_{2}&v_{1}e_{3}\\ v_{2}e_{1}&v_{2}e_{2}&v_{2}e_{3}\\ v_{3}e_{1}&v_{3}e_{3}&v_{3}e_{3}\end{array} \right]\left[ \begin{array}{c}x_{ i}-\overline{x}\\ y_{i}-\overline{y}\\ z_{i}-\overline{z}\end{array} \right] \tag{13}\]

\[\left[ \begin{array}{c}x^{\prime}{}_{i}\\ y^{\prime}{}_{i}\\ z^{\prime}{}_{i}\end{array} \right]=\left[ \begin{array}{ccc}v_{11}&v_{12}&v_{ 13}\\ v_{21}&v_{22}&v_{23}\\ v_{31}&v_{32}&v_{33}\end{array} \right]\left[ \begin{array}{c}x_{i}-\overline{x} \\ y_{i}-\overline{y}\\ z_{i}-\overline{z}\end{array} \right] \tag{14}\]

\[\left[ \begin{array}{c}x^{\prime}{}_{i}\\ y^{\prime}{}_{i}\\ z^{\prime}{}_{i}\end{array} \right]=T\left[ \begin{array}{c}x_{i}-\overline{x} \\ y_{i}-\overline{y}\\ z_{i}-\overline{z}\end{array} \right] \tag{15}\]

where \(\mathbf{e}_{1},\mathbf{e}_{2}\), and \(\mathbf{e}_{3}\) are the three unit vectors of the original coordinate system. Figure 5: The oriented bounding box of a point cloud. Note that matrix \(T\) is the orthogonal matrix. It means that \(T^{-1}\) = \(T^{\rm T}\). Let:

\[