## 4 Experiments and Results
### Evaluation of Results  
A concave mirror, from Micro-nano Optical Corporation (Beijing, China), was measured using the calibrated DPMD system. The center of the concave surface was obtained according to the measured 3D shape data by fitting all the points. The radius could be calculated as a true value. The difference between every point and the obtained true radius was calculated. The average value of all the differences was the calculated systematic error of 26 \(\upmu\)m. To evaluate the calibrated system parameters quantitatively, the calibrated DPMD system was also tested using a standard artificial step with specular surfaces, the photos of the objects are illustrated in Figure 10. The specular step was designed and manufactured with known distances between neighboring steps, as shown in the first column in Table 2. Twelve sinusoidal fringe patterns with fringe numbers of 81, 80, and 72 were generated and displayed sequentially on the two LCD screens. The fringe patterns reflected by the specular objects being tested were deformed by the slope and shape of the measured surface. The reflected fringe patterns captured by the CCD camera are shown in Figure 11a for the step and Figure 12a for the concave mirror. Figures 11b and 12b illustrate single absolute phase maps for the specular step and the mirror, respectively. The absolute phase map was then converted into depth data, and the calibrated parameters \(d_{1}\) and \(d_{2}\) were used to obtain the 3D shape data for the specular step and the concave mirror, as shown in Figures 11c and 12c. Figures 11d and 12d show the profile maps of the specular step and concave mirror being tested.  
Figure 10: Photos of the specular objects being measured: (**a**) manufactured artificial step and (**b**) concave mirror.  
All the points measured on one step surface were fitted onto a plane to calculate the distance between neighboring steps. The measured distance between neighboring steps was calculated using the average of the distance from all the points obtained on the other step surface to the fitted plane. The measured distance, the absolute error (absolute difference between the average distance measured and the calibration distance), and the standard deviation are listed in the second to fourth columns of Table 2.  
The experimental results demonstrate that the proposed calibration method can reconstruct the 3D shape of specular objects with high precision and reliability. The absolute error in the distance between neighboring steps was below 22 \(\upmu\)m, while the value is 52 \(\upmu\)m by using the existing calibration method [37].  
\begin{table}
\begin{tabular}{c c c c} \hline \hline
**Calibration Distance** & **Measured Distance** & **Absolute Error** & **Root Mean Square Error** \\ \hline
3.987 & 3.965 & 0.018 & 0.022 \\ \hline
7.025 & 7.044 & 0.015 & 0.019 \\ \hline
5.006 & 5.030 & 0.022 & 0.024 \\ \hline
6.099 & 6.079 & 0.021 & 0.020 \\ \hline \hline \end{tabular}
\end{table}
Table 2: Evaluation results for the artificial specular step (units: mm).  
Figure 11: Measurements on the concave mirror: (**a**) one captured fringe pattern reflected by the concave mirror; (**b**) one absolute phase map from the reflected fringe patterns; (**c**) measured 3D shape of the mirror and (**d**) profile map of the mirror.