# Design and Analysis of High-Accuracy Telecentric Surface Reconstruction System Based on Line Laser
## 5 Experiments
### Uncertainty Analysis  
As the point cloud of ROI is obtained by the measurement procedure mentioned, it is necessary to evaluate the uncertainty of the measured results. Guide on the expression of uncertainty in measurement (GUM) is one of the international standards to evaluate the measurement uncertainty [22]. Since a measurand is usually determined from some other quantities through a functional relationship rather than measured directly, GUM requires the function model which reflects the relationship to achieve the measured uncertainty. The model for the proposed system is expressed in Equation (18), which is determined by combining Equations (16) and (17). \[P_{w}=\left[ \begin{array}{c}x_{w}\\ y_{w}\\ z_{w}\end{array} \right]=\left[ \begin{array}{c}\frac{(u-u_{0})}{m}-d\cdot n _{d1}\\ \frac{(v-v_{0})}{m}-d\cdot n_{d2}\\ -\frac{\left[n_{d1}(u-u_{0})+n_{d2}(v-v_{0})\right]}{n_{d3}m}-d\cdot n_{d3} \end{array} \right]. \tag{18}\]  
Equation (18) can be rewritten as a function model \(P_{w}=f\left(m,n_{p},n_{d},p,d\right)\), in which \(p(u,v)\) is the image coordinate of the laser line location. \(n_{p}=\left(n_{p1},n_{p2},n_{p3}\right)^{T}\) and \(n_{d}=\left(n_{d1},n_{d2},n_{d3}\right)^{T}\) are the unit normal directions of the laser plane and the unit motor direction of the displacement platform in the camera system, respectively. Nevertheless, the \begin{table}
\begin{tabular}{c c c} \hline \hline
**Parameters** & **Value** & **Uncertainty (1\(\sigma\))** \\ \hline \(M\) & 0.426920 & 0.000002 \\ \(n_{p}\) & \(\left(-0.00982,\,-0.86241,\,0.50612\right)\) & \(\left(0.00002,\,0.00035,\,0.00059\right)\) \\ \(n_{d}\) & \(\left(-0.02641,\,-0.86659,\,0.49832\right)\) & \(\left(0.00005,\,0.00005,\,0.00009\right)\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Calibration parameters in experiments. Figure 9: Experimental setup of the proposed telecentric surface reconstruction system. measurand required usually is not \(P_{w}\), but \(\Delta P_{w}\), which is the relative position for the measured points. According to Equation (18), the model of \(\Delta P_{w}\) for \(P_{w1}=f\big{(}m,n_{p},n_{d},p_{1},d_{1}\big{)}\) and \(P_{w2}=f\big{(}m,n_{p},n_{d},p_{2},d_{2}\big{)}\) is presented in Equation (19)  
\[\Delta P_{w}= \begin{bmatrix}\Delta x_{w}\\ \Delta y_{w}\end{bmatrix} = \begin{bmatrix}x_{w1}-x_{w2}\\ y_{w1}-y_{w2}\\ x_{w1}-z_{w2}\end{bmatrix} = \begin{bmatrix}\frac{(u_{1}-u_{2})}{m}-(d_{1}-d_{2} )\cdot n_{d1}\\ \frac{(v_{1}-v_{2})}{m}-(d_{1}-d_{2})\cdot n_{d2}\\ -\frac{[n_{p1}(u_{1}-u_{2})+n_{p2}(v_{1}-v_{2})]}{n_{p2}m}-(d_{1}-d_{2})\cdot n _{d1}\end{bmatrix}  \tag{19}\]  
For the function model above \(\Delta P_{w}\triangleq g(X)\), there are 3 outputs and 13 inputs considering every component of vectors. Thus, the uncertainty of \(\Delta P_{w}\) is defined by the uncertainty propagation formula as Equation (20). \[V(\Delta P_{w})=J\cdot V(X)\cdot J^{T}. \tag{20}\]  
\(J\) is the Jacobian matrix of the function model \(\Delta P_{w}=g(X)\), and \(V(X)\) is the covariance matrix which consists of the uncertainty and covariance of the inputs. The uncertainty of calibration parameters is listed in Table 1. Besides, the uncertainty of \(d\) is 0.017, which is decided by the equipment design resolution. The uncertainty of laser line location \((u,v)\) in the image is \((0,0.01)\) pixel given the laser stripe extraction algorithm is searching for the location along a single direction of the image. Note that all the inputs are independent except for the components of the unit vector, so the covariance between the components of \(n_{p}\) and \(n_{d}\) should be computed respectively while others are simply set to 0. \(V(\Delta P_{w})\) is supposed to be a \(3\times 3\) matrix as there are 3 outputs, and the diagonal elements of \(V(\Delta P_{w})\) are just the squares of the corresponding output uncertainties. Once the measure points are settled, the uncertainty of \(\Delta P_{w}\) can be achieved with \(V(\Delta P_{w})\).