# Design and Analysis of High-Accuracy Telecentric Surface Reconstruction System Based on Line Laser
## 2 System Design
### Systematic Error Analysis  
Due to the measurement principle of the intersection of the optical line and laser plane, the systematic error in this system is inevitable. The point on the object surface is located only if the point is lighted up by the laser while it is captured by camera. Nevertheless, as there must be an intersection angle between the optical axis of camera and the laser plane, a blind corner is formed for the camera when the surface of the measured object is abrupt enough. As it is shown in Figure 2, suppose the laser plane is perpendicular to the motor direction of the measured object, when the intersection angle is \(\alpha\), and the depth difference on the object surface is \(H\), the width of blind corner \(D\) is expressed as  
\[D=H\cdot\textbf{tan}\alpha. \tag{1}\]  
Figure 2 indicates that the shaded region along the movement direction is not able to be measured. The region area is characterized by the width \(D\) directly when \(H\) is constant.  
Figure 1: Telecentric surface metrology system based on laser scanning.  
Figure 2: Formation mechanism of the blind corner.  
Intuitively, the smaller \(\alpha\) reflects the narrower \(D\) within \(\alpha\in(0^{\circ},\,90^{\circ})\). Nevertheless, the resolution of the surface depth is positively associated with the intersection angle \(\alpha\) for \(\alpha\in(0^{\circ},\,90^{\circ})\). The description can be found in Figure 3, according to the telecentric camera model, which is presented in Section 3, the resolution \(r\) can be formulated as  
\[r=\frac{\Delta u}{\Delta H}=\frac{M}{\delta u}\cdot\textbf{sin}\alpha. \tag{2}\]  
\(M\) is the magnification of telecentric lens, and \(\delta u\) is the cell size of the camera sensor. The resolution \(r\) represents image pixels corresponding to unit surface depth. For instance, if the recognized depth required is \(h\), it is supposed that Equation (3) should be satisfied. Note that the \(h\) discussed here cannot be microcosmic due to the diffraction limit for optical imaging.  
\[h\cdot r\geq\Delta p, \tag{3}\]  
\(\Delta p\) is the minimum of distinguishable pixel difference, and it can be set to 0.5 pixel, while it has to be larger considering the noise aspect and other random errors. Compare the derivative of the trigonometric part of Equations (1) and (2), which are \(\tan\alpha=1/\cos^{2}\alpha\) and \(\sin\alpha=\cos\alpha/\)The variation tendency is presented in Figure 4.  
Figure 4: The variation tendency for the trigonometric part.  
Figure 3: The resolution analysis in depth direction.  
It is indicated that both \(\tan\alpha\) and \(\sin\alpha\) increase within \(\alpha\in(0^{\circ}\), \(90^{\circ})\), but \(\tan\alpha\) climbs much faster than \(\sin\alpha\), especially for \(\alpha>60^{\circ}\). Hence, the intersection angle is generally no more than \(60^{\circ}\) in surface metrology. As there is a contradiction for the intersection angle \(\alpha\) chosen, the system design must combine equipment condition with concrete demand of the applications in practice. Hence, an actual system is established for circuit boards tests as an instance.