# Registration of Urban Aerial Image and LiDAR Based on Line Vectors
## 2 Methods
### Line Vector  
As shown in Figure 2, the unit direction vector of lines can be expressed by three direction cosines, namely \(\mathbf{o}=(l,m,n)^{\mathrm{T}}\), which satisfies \(l^{2}+m^{2}+n^{2}=1\), i.e., \(\sigma^{\mathrm{T}}\mathbf{o}=1\), but these three parameters cannot determine the position of the line. The position vector of \(H\), an arbitrary point on the line directed by the origin is \(\mathbf{r}\), according to which the position of the line can be determined and line vector \(L\) can be obtained. Line vector \(L\) is a six-vector, including the primary part and secondary part. The directional vector is the primary part of the line vector, while the moment vector is the secondary part of the line vector. The directional vector does not change as the origin changes, but the moment vector does. For this reason, we know that the line vector is a vector attached to a line in the space and six-dimensional linear space in the real number field:  
\[\mathbf{L}=\left(\begin{array}{c}\mathbf{o}\\ \mathbf{p}\end{array}\right)=\left(\begin{array}{c}\mathbf{o}\\ \mathbf{r}\times\mathbf{o}\end{array}\right)=\left(\mathbf{L},m,n,r,s,t\right)^{\mathrm{T}} \tag{1}\]  
The directional vector and moment vector satisfy: \(\mathbf{p}\cdot\mathbf{o}=\left(\mathbf{r}\times\mathbf{o}\right)\cdot\mathbf{o}=\mathbf{r}\cdot\left( \mathbf{o}\times\mathbf{o}\right)=0\). According to the two constraints that \(\sigma^{\mathrm{T}}\mathbf{o}=1\) and \(\mathbf{p}\cdot\mathbf{o}=0\), which derive from the unit line vector itself, there are four degrees of freedom in the line vector coordinate.  
Figure 1: Steps of the aerial images and LiDAR data registration based on line vectors.  
Assume there are two line vectors \(\mathbf{L}_{1}\) and \(\mathbf{L}_{2}\), which are \(\mathbf{L}_{1}=\left(\mathbf{o}_{1}\mathbf{p}_{1}\right)^{\text{T}}\), \(\mathbf{L}_{2}=\left(\mathbf{o}_{2}\mathbf{p}_{2}\right)^{\text{T}}\). Operations on line vectors are performed similar to those on Cartesian vector, also referred to as line dot- and cross-products [17], and are defined by:  
\[\mathbf{L}_{1}\cdot\mathbf{L}_{2}=\left(\begin{array}{c}\mathbf{o}_{1}\cdot\mathbf{o}_{2}\\ \mathbf{o}_{1}\cdot\mathbf{p}_{2}+\mathbf{p}_{2}\cdot\mathbf{o}_{1}\end{array}\right) \tag{2}\]  
\[\mathbf{L}_{1}\times\mathbf{L}_{2}=\left(\begin{array}{c}\mathbf{o}_{1}\times\mathbf{o}_{2} \\ \mathbf{o}_{1}\times\mathbf{p}_{2}+\mathbf{p}_{1}\times\mathbf{o}_{2}\end{array}\right) \tag{3}\]  
The advantage of the linear expression method based on line vector coordinate is in determining a line not relying on any point or surface, as well as directly representing a line direction by using a directional vector and line position by a moment vector.  
Assuming any two lines with their directional vectors set as \(\mathbf{o}_{1}\) and \(\mathbf{o}_{2}\), respectively, and their moment vectors as \(\mathbf{p}_{1}\) and \(\mathbf{p}_{2}\), respectively, if these two lines are coplanar, then they will meet the requirement of:  
\[\mathbf{o}_{1}\cdot\mathbf{p}_{2}+\mathbf{o}_{2}\cdot\mathbf{p}_{1}=0 \tag{4}\]  
Equation (4) is the coplanar formula of the two lines in a simple way. Compared with the two traditional methods of using point coordinates and the orientation vector to express the coplanar formula, the line vector may directly make use of the information of direction and position.