## 2 Principles  
### Line-Structured Light 3D Measurement Model  
Figure 1 shows a schematic diagram of the line-structured light 3D measurement process. The utilized setup includes a camera and a laser projector. A line laser is cast onto an object by the laser projector, after which the camera captures the light stripe images distorted by the object surface geometry. Finally, the 3D coordinates of the points on the light stripe are determined. By performing a precise line motion, a 3D profile of the object can be obtained. In Figure 1, the notations (\(O_{w}\); \(X_{w}\), \(Y_{w}\), \(Z_{w}\)) and (\(O_{c}\); \(X_{c}\), \(Y_{c}\), \(Z_{c}\)) represent the world coordinate system (WCS) and camera coordinate system (CCS), respectively. For simplicity of analysis, WCS and CCS are set to be identical. (\(O_{w0}\); \(u\), \(v\)) is the camera pixel coordinate system (CPCS), while (\(O_{n}\); \(x_{n}\), \(y_{n}\)) represents the normalized image coordinate system (NICS). Presumably, there is an arbitrary intersection point \(P\) between the light plane and the measured object, where \(\mathbf{P}_{c}\) = (\(X_{pc}\), \(Y_{pc}\), \(Z_{pc}\))\({}^{T}\) represents a set of coordinates in the CCS. \(\mathbf{P}_{w0}\) = (\(u_{p}\), \(v_{p}\))\({}^{T}\) denotes the coordinates of an undistorted image point. In addition, \(\mathbf{P}_{n}\) = (\(x_{pn}\), \(y_{pn}\))\({}^{T}\) and \(\mathbf{P}_{d}\) = (\(x_{pd}\), \(y_{pd}\))\({}^{T}\) represent the NICS coordinates of undistorted and distorted physical images, respectively. Generally, the camera produces a lens distortion on \(\mathbf{P}_{u}\), especially in the radial direction. For this reason, a distortion model is described in Ref. [19], which includes the term \(f_{d}(\cdot)\) of the distorted projection \(\mathbf{P}_{d}\) on the normalized image plane in the following form:  
\[\mathbf{P}_{d}=f_{d}(\mathbf{P}_{d},\mathbf{K},\mathbf{G})=\mathbf{P}_{n}\Big{(} 1+k_{1}r^{2}+k_{2}r^{4}+k_{3}r^{6}\Big{)}+\left[ \begin{array}{c}2g_{1}x_{n}y _{n}+g_{2}\big{(}r^{2}+2x_{n}^{2}\big{)}\\ g_{1}\big{(}r^{2}+2y_{n}^{2}+2g_{2}x_{n}y_{n}\big{)}\end{array} \right], \tag{1}\]  
where \(r^{2}=x_{n}^{2}+y_{n}^{2}\); \(\mathbf{K}=[k_{1},\,k_{2},\,k_{3}]\) and \(\mathbf{G}=[g_{1},\,g_{2}]\) represent the radial and tangential distortion coefficient vectors, respectively. Due to the advantages of homogeneous coordinates, the utilized coordinates are converted into homogeneous coordinates by adding 1 as the last element. By taking into account the projection and coordinate transformation, the actual projection on the image plane \(\mathbf{P}_{uv}\) can be expressed as follows:  
\[\left[ \begin{array}{c}\mathbf{P}_{uv}\\ 1\end{array} \right]=\mathbf{A}\cdot\left[ \begin{array}{c}\mathbf{P}_{d}\\ 1\end{array} \right]=\mathbf{A}\cdot\left[ \begin{array}{c}f_{d}(\mathbf{P}_{u },\mathbf{K},\mathbf{G})\\ 1\end{array} \right], \tag{2}\]  
where \(\mathbf{A}\) denotes the camera intrinsic matrix:  
\[\mathbf{A}=\left[ \begin{array}{ccc}f_{u}&0&u_{0}\\ 0&f_{v}&v_{0}\\ 0&0&1\end{array} \right]. \tag{3}\]  
Typically, \(\mathbf{A}\) consists of five parameters: \(f_{u},f_{v}\), \(u_{0}\), \(v_{0}\), and \(\gamma\). The skew factor \(\gamma\) is usually set to zero. \(f_{u}\) and \(f_{v}\) are the horizontal and vertical focal lengths, respectively, and \(u_{0}\) and \(v_{0}\) are the coordinates of the camera principal point. As shown in Equation (2), the nonlinear camera model describes the imaging process from CCS to the CPCS. From the geometric viewpoint, Equation (2) characterizes a ray passing through \(\mathbf{P}_{uv}\) and \(\mathbf{P}_{c}\). To uniquely determine \(\mathbf{P}_{c}\) in the CCS, the light plane \(\Pi\) serves as another constraint--i.e., \(\mathbf{P}_{c}\) should satisfy the equation of \(\Pi\):  
\[\left[ \begin{array}{cc}\mathbf{P}_{c}^{T}&1\end{array} \right]\mathbf{\Pi}_{ c}=0 \tag{4}\]  
where \(\mathbf{\Pi}_{c}=(a,b,c,d)^{T}\) includes the four coefficients of Equation (4). Figure 1: Schematic diagram of the line-structured light 3D measurement process.