# Multi-Wavelength Digital-Phase-Shifting Moire Based on Moire Wavelength
## 3 Experiments and Results
### Measurement  
Object surface measurements were performed by projecting onto the object a single binary (black-and-white stripe) grid pattern for each camera-pixel grid pitch used in the calibration, and capturing a single image for each pattern. Phase-shifted moire patterns (images) were generated by digitally phase shifting a computer-generated grid (of the same pitch as the captured image) overlaid on the captured images, followed by grid removal. Phase-shift analysis was applied to extract the wrapped phase map \(\Phi\)(_x,y_) for each grid pitch. Multi-wavelength digital-phase-shifting moire based on moire wavelength was performed on a mask with surface discontinuities, two spatially isolated objects (a mask with surface discontinuities and a manikin head) and a double-hemisphere object (Figure 2). The maximum number of phase shifts possible (corresponding to the camera-pixel grid pitches employed) was used for all object measurements: 10 and 12 shifts for the 10- and 12-pixel pitch grids, respectively, used for the mask; 10, 12, and 14 shifts for the 10-, 12-, and 14-pixel pitch grids, respectively, used for the mask with manikin; and 8 and 10 shifts for the 8- and 10-pixel pitch grids, respectively, used for the double hemisphere object. The computation of object height from unwrapped phase for all 3D surface-shape reconstruction used pixelwise moire-wavelength refinement and pixelwise surface height refinement, described in Section 2.4, using the moire-wavelength-height relationship determined in Section 2.2. Two-wavelength temporal phase unwrapping was applied to the mask with surface discontinuities by projecting two binary grid patterns with the grid pitches used in the calibration, and capturing a single image for each pattern (10- and 12-pixel grid pitch) (Figures 3a and 4a). For each grid pitch, a moire pattern with high-frequency grid lines was generated by digital phase-shifting moire (Figures 3b and 4b), and a pure moire pattern was extracted by grid removal (Figures 3c and 4c). Two wrapped phase maps \(\Phi_{1}\) and \(\Phi_{2}\) (Figures 3d and 4d) were computed with moire wavelengths \(\lambda_{1}\) and \(\lambda_{2}\) (Section 3.2). The extended continuous phase map \(\Phi_{12}\) (Figure 5a) with extended beat wavelength \(\lambda_{12}=\frac{\lambda_{1}\lambda_{2}}{|\lambda_{1}-\lambda_{2}|}=\) 135.567 mm, which covers the entire range of depth, was computed from wrapped phase maps \(\Phi_{1}\) and \(\Phi_{2}\) (Figures 3d and 4d). To minimize the phase error caused by the noise in \(\Phi_{12}\), \(\Phi_{12}\) with no phase ambiguity was used as a reference to unwrap the phase map \(\Phi_{1}\) with smaller moire wavelength \(\lambda_{1}\), resulting in the unwrapped phase map \(\varphi\) (Figure 5b). The result demonstrates the ability to obtain an unwrapped phase map with high SNR to perform accurate 3D measurement (Figure 5c) of objects with discontinuities, using only two captured images. Figure 2: Objects used in surface measurement experiments: (**a**) Mask, (**b**) manikin head, and (**c**) double-hemisphere. Three-wavelength temporal phase unwrapping was applied to two spatially isolated objects: a mask with surface discontinuities and a manikin head. To handle the greater depth of the manikin compared to the mask, the beat wavelength was increased without sacrificing SNR in the extended  
Figure 4: Wrapped phase map computation: (**a**) Original captured image of the mask with binary grid of 12-pixel pitch, (**b**) generated image with moiré pattern and high-frequency grid lines, (**c**) moiré pattern after grid removal, and (**d**) wrapped phase map \(\Phi_{2}\). Figure 5: (**a**) Extended continuous phase map \(\Phi_{12}\) with extended beat wavelength \(\lambda_{12}\), (**b**) unwrapped phase map \(\varphi\), and (**c**) point cloud representation of measured points of mask.