## 2 The Light Calibration Method for Line Structure based on Multiple Geometric Constraints
### Preliminary Solution of Light Plane Equation Parameters  
In this paper, the calibration plane target is shown in Figure 1; a checkerboard calibration plate and a black target plane were clamped by the fixture. The limit plane was used to ensure the coplanar relationship between the checkerboard calibration plate and the black target. The camera's internal parameters, distortion coefficients, and external parameters of the checkerboard surface could be calibrated under different poses by [PERSON]'s camera method [6]. According to the characteristics of the target, the parameters of the checkerboard surface were the same as the external parameters of the target plane.  
When the positions of the camera and the line laser were fixed, the light was projected onto the target plane by the laser, the positions of the light stripe center points in the pixel plane were obtained using [PERSON]'s algorithm [21], and the camera coordinates of these points were obtained by the camera's internal parameters. The initial value solution model for laser plane equation parameters is shown in Figure 2; \(P\) is the data point used for the calibration of the light plane, and the point was obtained by the center point detection algorithm of the light stripe. \(P^{\prime}\) is a projection of a point on the imaging plane, and \(I\) is the intersection of the laser plane and the target. During the calibration process, images of the calibration target in different positions were obtained, and the spatial coordinates of the data points in each position were obtained using the external parameters of the target surface. The spatial coordinates of the \(i\)-th data point on the \(j\)-th image were set as \(P^{j}_{Ci}=\left(X^{j}_{Ci},Y^{j}_{Ci},Z^{j}_{Ci}\right)\).  
\(O_{W}\)-\(X_{W}Y_{W}Z_{W}\) is the world coordinate system, and \(O_{C}\)-\(X_{C}Y_{C}Z_{C}\) is the camera coordinate system.  
\(O\)-\(xy\) is the image coordinate system, and \(Op\)-\(uv\) is the pixel coordinate system.  
The plane equation [22] of the light plane is:  
\[AX_{C}+BY_{C}+CZ_{C}-1000=0 \tag{1}\]  
The camera coordinates of the data points were replaced by the plane equation, and the new equation is shown in Equation (2):  
\[\begin{bmatrix}X^{1}_{Ci}&Y^{1}_{Ci}&Z^{1}_{Ci}\\ \vdots&\vdots&\vdots\\ X^{j}_{Ci}&Y^{j}_{Ci}&Z^{j}_{Ci}\\ \vdots&\vdots&\vdots\\ X^{N}_{CK}&Y^{N}_{CK}&Z^{N}_{CK}\end{bmatrix}\begin{bmatrix}A\\ B\\ \vdots\\ \vdots\\ \vdots\\ 1000\end{bmatrix}=\begin{bmatrix}1000\\ \vdots\\ 1000\\ \vdots\\ 1000\end{bmatrix} \tag{2}\]  
Figure 1: The coplanar target schematic diagram: (**a**) axonometric diagram; (**b**) side view.  
Figure 2: The initial value solution model of the light plane equation.  
In Equation (2), the two variables \(N\) and \(K\) are the target rotation times and the number of data points at each position. According to Equation (2), the initial parameters can be obtained by the least square method. According to mathematical knowledge, the value of the constant term in the light plane equation does not affect the position of the light plane. In the experiment described in this paper, the parameters of the light plane equation were very small when the constant of the light plane equation was \(-1\). In this case, the light plane equation was inconvenient for expression, and the constant of the light plane equation was set to \(-1000\) in the paper.