# Multi-Wavelength Digital-Phase-Shifting Moire Based on Moire Wavelength
## 3 Experiments and Results
### Measurement
Figure 3: Wrapped phase map computation: (**a**) Original captured image of the mask with binary grid of 10-pixel pitch, (**b**) generated image with moiré pattern and high-frequency grid lines, (**c**) moiré pattern after grid removal, and (**d**) wrapped phase map \(\Phi_{1}\). phase map, by projection of another pattern with greater grid pitch. Thus, three binary grid patterns with different grid pitches were projected onto the objects and a single image was captured for each pattern (10-, 12-, and 14-pixel grid pitch) (Figure 6a, Figure 7a, and Figure 8a). For each grid pitch, a moire pattern with high-frequency grid lines was generated by digital phase-shifting moire (Figure 6b, Figure 7b, and Figure 8b) and a pure moire pattern was extracted by grid removal (Figure 6c, Figure 7c, and Figure 8c). Three wrapped phase maps \(\Phi_{1},\Phi_{2}\), and \(\Phi_{3}\) (Figure 9a-c) were computed with different moire wavelengths \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\) (Section 3.2). The extended phase map \(\Phi_{12}\) (Figure 10a) with extended beat wavelength \(\lambda_{12}=\frac{\lambda_{13}\lambda_{2}}{|\lambda_{1}-|\lambda_{2}|}=135.567\) mm was computed from phase maps \(\Phi_{1}\) and \(\Phi_{2}\). To minimize the phase error caused by the noise in\(\Phi_{12}\), \(\Phi_{12}\) was used as a reference to unwrap the phase map \(\Phi_{2}\) with smaller moire wavelength \(\lambda_{2}\), to obtain the phase map \(\Phi_{12}^{\prime}\) (Figure 10b) with higher SNR. A second extended phase map \(\Phi_{23}\) (Figure 11a) with extended beat wavelength \(\lambda_{23}=\frac{\lambda_{2}\lambda_{3}}{|\lambda_{2}-\lambda_{3}|}\) = 175.986 mm was computed from phase maps \(\Phi_{2}\) and \(\Phi_{3}\). To minimize the phase error caused by the noise in \(\Phi_{23}\), \(\Phi_{23}\) was used as a reference to unwrap the phase map \(\Phi_{2}\) with smaller moire wavelength \(\lambda_{2}\), to obtain the phase map \(\Phi_{23}^{\prime}\) (Figure 11b) with higher SNR. Note that when the beat moire wavelength does not cover the full range of depth, phase jumps occur, as seen in the white regions around the manikin head in Figure 10a,b, and in the central black region of the manikin head in Figure 11a,b. When such phase jumps occur, the height cannot be correctly determined. The extended continuous phase map (Figure 12a), which covers the entire range of depth, was then computed from phase maps \(\Phi_{12}\) (Figure 10a) and \(\Phi_{23}\) (Figure 11a), resulting in a noisy phase map (Figure 12a). To improve the measurement, the extended continuous phase map with high SNR \(\Phi_{123}\) (Figure 12b) with extended beat wavelength \(\lambda_{123}=\frac{\lambda_{12}\lambda_{23}}{|\lambda_{12}-\lambda_{23}|}=590.269\) mm was calculated from high quality phase maps \(\Phi_{12}^{\prime}\) (Figure 10b) and \(\Phi_{23}^{\prime}\) (Figure 11b). Finally, to calculate a more accurate unwrapped phase map with high SNR, \(\Phi_{123}\) was used as a reference to unwrap \(\Phi_{2}\) with smaller moire wavelength \(\lambda_{2}\), to obtain the unwrapped phase map \(\varphi\) (Figure 12c), which has high SNR. This high-SNR unwrapped phase map enables accurate 3D measurement of objects with discontinuities and multiple spatially-isolated objects (Figure 13) using only three captured images. Figure 8: Moiré pattern generation: (**a**) Original captured image of objects with binary grid of 14-pixel pitch, (**b**) generated image with moiré pattern and high-frequency grid lines, and (**c**) moiré pattern after grid removal. Figure 10: Extended phase maps: (**a**) \(\Phi_{12}\) with extended beat wavelength \(\lambda_{12}\), and (**b**) \(\Phi^{{}^{\prime}}_{12}\) with higher SNR. Figure 9: Wrapped phase maps: (**a**) \(\Phi_{1}\), (**b**) \(\Phi_{2}\), and (**c**) \(\Phi_{3}\).