## 4 Accuracy Analysis  
The following section describes the analysis of the measurement errors due to the perturbations of the degrees of freedom in the PMD system. Generally, the measurement result depends on the accuracy of the measurement tool and device. Therefore, it is necessary to analyze measurement errors caused by each coordinate value, which is called the degree of freedom (DOF). The error analysis aids in the identification of which DOF need to be tightened and which might be loosened. Thus, there are three DOF \((x_{c},y_{c},z_{c})\) for the camera coordinates. For the coordinate measurement of the screen pixels, there are six DOFs; these are translations \((x_{s},y_{s},z_{s})\) and rotations (screen tilt \(x\), screen tilt \(y\), screen tilt \(z\)) in three directions. For the coordinate measurement of the SUT, the rotation of the SUT around the \(z\) axis does not cause measurement error; only the measurement results need to be rotated. Therefore, there are five DOFs with three translations \((x_{m},y_{m},z_{m})\) and two rotations (flat tilt \(x\), flat tilt \(y\)) for the SUT. Regardless of the errors by the LCD screen, such as thermal distortion, roughness, and noise, the 14 DOFs need to be considered [9].  
As mentioned above, the DOFs are analyzed in the simulation. The schematic diagram of the PMD system is shown in Figure 5. Assuming that the pinhole \(O\) coordinates are the origin of the world coordinate system, the distance between the SUT and the camera \(O\) is 1525 mm, and the distance between the LCD screen and the SUT is 1500 mm in the \(z\) direction, while the angle between the camera and the \(z\) direction is 5\({}^{\circ}\). Korsch ray tracing [30] is used to calculate the slope of an ideal planar element in the x and y directions. For DOF, initial values are considered to be the accurate values. For the coordinate values of the camera \((x_{c},y_{c},z_{c})\), the screen pixel \((x_{s},y_{s},z_{s})\), and the SUT \((x_{m},y_{m},z_{m})\), a perturbation of 1 mm is artificially added to the accurate value of each DOF, respectively. Similarly, a perturbation of 1 mrad is artificially added to the accurate value for the five DOFs of the rotations, respectively. When error is added artificially to each DOF, the others remain unchanged.  
Based on the above analysis, the Zernike polynomial coefficients are analyzed after the surface shape is reconstructed with the Zernike polynomial [25; 26] under different DOFs with errors. The differences from the 4 th to 11 th terms of the Zernike polynomial coefficient between the results of the accurate surface figure and the results gained using different perturbations are shown in Table 1. The terms from the 4 th to 11 th Zernike polynomial coefficients are defocus, oblique astigmatism, vertical astigmatism, vertical coma, horizontal coma, vertical trefoil, oblique trefoil, and primary spherical, respectively. The root sum square (RSS) is the value for evaluating the effects on the measurement results due to the perturbation of each DOF or in every term of the Zernike polynomial coefficient.  
Figure 5: Schematic diagram of PMD.  
In general, the first eight terms of the Zernike polynomial coefficients are greatly affected when the DOF can be not obtained accurately. Therefore, according to the pre-set perturbation for all DOFs shown in the first column in Table 1, the accuracy of the alignment can be estimated in this measuring system. When the positions of the camera, the screen, and the SUT are separately changed under a given perturbation, these alignment errors for the 4 th to the 11 th Zernike polynomial coefficients can be calculated, as shown in Table 1. This table shows that the coordinate error of \(z_{e}\) and \(z_{s}\) will induce some relatively large errors in terms of defocus, while screen tilt error about the z axis will produce relatively significant errors in terms of astigmatism; however, the other terms are so small that they can be ignored.