# Fast Point Cloud Registration Method with Incorporation of RGB Image Information
## 2 Methods
### Initial Alignment Based on RGB Images
#### 2.1.4 Solution of Initial Transformation Matrix  
The optimized matched pairs of feature points can be solved for the homography matrix H of the projection transformation. The initial transformation matrix of the point cloud is obtained by converting the camera motion parameters decomposed in H into the relative motion parameters of the laser scanner according to the fixed coordinate transformation relationship between the camera and the laser scanner. Assuming that the positions of the laser scanning device and the visible camera are fixed, the coordinate systems of the laser scanning device and the camera are defined as \(O_{s}-x_{s}y_{s}z_{s}\) and \(O_{c}-x_{c}y_{c}z_{c}\), respectively. The rotation parameter \(R_{h}\) and translation parameter \(t_{h}\) are regarded as the rigid transformation relationship between the two coordinate systems. \(X_{s}\) and \(X_{c}\) are considered as the coordinates under the two coordinate systems, then the transformation relationship between \(X_{s}\) and \(X_{c}\) is expressed as  
\[X_{s}=R_{h}X_{c}+t_{h} \tag{2}\]  
In the position shown in Figure 4 below, a conventional laser scanner is placed coaxially with the visible camera. \(x_{s}\) and \(z_{c}\) axes, \(y_{s}\) and \(x_{c}\) axes, and \(z_{s}\) and \(y_{c}\) axes of the coordinate system of the laser scanner and the camera coordinate system are parallel to each other, the rotation parameters between these two coordinate systems are calculated as follows:  
\[R_{h}=R_{x}\Big{(}\frac{\pi}{2}\Big{)}R_{y}\Big{(}-\frac{\pi}{2}\Big{)}= \begin{bmatrix} 1&0&0\\ 0&\cos\big{(}\frac{\pi}{2}\big{)}&-\sin\big{(}\frac{\pi}{2}\big{)}\\ 0&\sin\big{(}\frac{\pi}{2}\big{)}&\cos\big{(}\frac{\pi}{2}\big{)}\end{bmatrix} \begin{bmatrix}\cos\big{(}-\frac{\pi}{2}\big{)}&0&\sin\big{(}-\frac{\pi}{2} \big{)}\\ 0&1&0\\ -\sin\big{(}-\frac{\pi}{2}\big{)}&0&\cos\big{(}-\frac{\pi}{2}\big{)}\end{bmatrix}  \tag{3}\]  
The point \(p(x,y)\) in the RGB image captured by the camera can be transformed by projection to obtain the corresponding point \(p^{\prime}(x^{\prime},y^{\prime})\) in the other image, then the mapping relationship between the corresponding points of the two images can be written as  
\[p^{\prime}=\mathrm{H}\times p \tag{4}\]  
Figure 4: Coordinate conversion relationship between laser scanning equipment and camera. H is denoted as a \(3\times 3\) homography matrix. In H, \(h_{33}\in\) H is a scaled free degree and constant 1, so there are 8 effective degrees of freedom. Extending the above homography matrix transformation relationship, a mapping relationship for feature point matching can be established. The matrix H with 8 degrees of freedom can be found by establishing 8 equations through 4 sets of matching points. The relative rotation parameter \(R_{c}\) and the translation parameter \(t_{c}\) for the camera at two positions are obtained from the SVD [29] decomposition of the matrix H.  
The corresponding points in the two point clouds obtained by the laser scanning equipment are marked as \(X_{s1}\) and \(X_{s2}\), and the coordinates of the two corresponding points under the camera are \(X_{c1}\) and \(X_{c2}\), then the transformation of these two points under the two coordinate systems is expressed as  
\[ \begin{array}{c}X_{s1}=R_{h}X_{c1}+t_{h}\\ X_{s2}=R_{h}X_{c2}+t_{h}\end{array}  \tag{5}\]  
From the relative rotation parameter \(R_{c}\) and translation parameter \(t_{c}\) of the camera at the two positions, we have  
\[X_{c2}=R_{c}X_{c1}+\lambda t_{c} \tag{6}\]  
Substituting into the above two equations, we can get  
\[X_{s2}=R_{h}R_{c}{R_{h}}^{-1}X_{s1}-R_{h}R_{c}{R_{h}}^{-1}t_{h}+\lambda R_{h} t_{c}+t_{h} \tag{7}\]  
The relative rotation parameter \(R\) and translation parameter \(t\) of the laser scanning device at two positions can be obtained as follows:  
\[ \begin{array}{c}R=R_{h}R_{c}{R_{h}}^{-1}\\ t=\lambda R_{h}t_{c}+t_{h}-R_{h}R_{c}{R_{h}}^{-1}t_{h}\end{array}