# Multi-Wavelength Digital-Phase-Shifting Moire Based on Moire Wavelength
## 2 Method
### Digital-Moire Temporal Phase Unwrapping
#### 2.3.2 Three-Wavelength Phase-Unwrapping  
A wrapped phase map with large wavelength has few phase jumps but tends to be noisy, while a phase map with smaller wavelength has more phase jumps, but higher SNR. In two-wavelength phase unwrapping, the beat wavelength may not be sufficient to cover the entire object depth, leaving phase ambiguity. Increasing the pitch of projected patterns to enlarge the wavelengths \(\lambda_{1}\) and \(\lambda_{2}\), may help the beat wavelength to cover the entire object depth; however, this sacrifices SNR in the extended unambiguous phase map, which makes the phase unwrapping process unreliable. Three-wavelength or multi-wavelength heterodyne phase-unwrapping can further increase the beat wavelength without sacrificing SNR in the extended phase map. In three-wavelength digital-moire temporal phase-unwrapping, three binary grid patterns of different pitch are projected onto the object surface and a single image is captured for each projected pattern. The three captured patterns areused to generate three sets of moire patterns by digital phase shifting [17], each set with a different moire wavelength. Three wrapped phase maps \(\Phi_{1},\Phi_{2},\Phi_{3}\) with different moire wavelengths \(\lambda_{1}\), \(\lambda_{2}\), \(\lambda_{3}\), (\(\lambda_{3}>\lambda_{2}>\lambda_{1}\)) are computed from the three sets of moire patterns, using only three captured images. A moire wavelength is computed for each grid pattern of different pitch, using the method described in Section 2.2. Extraction of the unwrapped phase map \(\varphi(x,y)\) using the moire wavelengths \(\lambda_{1}\), \(\lambda_{2}\), and \(\lambda_{3}\), is now explained. Three scale factors \((\lambda_{12}/\lambda_{2})\), \((\lambda_{23}/\lambda_{2})\), and \((\lambda_{123}/\lambda_{2})\) from three beat wavelengths, \(\lambda_{12}=\frac{\lambda_{1}\lambda_{2}}{|\lambda_{1}-\lambda_{2}|}\), \(\lambda_{23}=\frac{\lambda_{2}\lambda_{3}}{|\lambda_{2}-\lambda_{3}|}\), and \(\lambda_{123}=\frac{\lambda_{12}\lambda_{23}}{|\lambda_{12}-\lambda_{23}|}\) are used in the phase unwrapping process to produce an extended continuous phase map. Note that the three different pitches of the three projected grid patterns must be selected such that the extended wavelength \(\lambda_{123}\) covers the entire range of depth of the objects to be measured. An extended phase map \(\Phi_{12}\) with extended wavelength \(\lambda_{12}\), is computed from phase maps \(\Phi_{1}\) and \(\Phi_{2}\) as follows:  
\[\Phi_{12}(x,y)=\left\{ \begin{array}{ll}\Phi_{1}(x,y)-\Phi_{2}(x,y),&\Phi_{1} >\Phi_{2}\\ \Phi_{1}(x,y)-\Phi_{2}(x,y)+2\pi,&\Phi_{1}\leq\Phi_{2}.\end{array} \right. \tag{10}\]  
To minimize the noise in \(\Phi_{12}\), \(\Phi_{12}\) is used as a reference to unwrap the phase map \(\Phi_{2}\) with wavelength \(\lambda_{2}\), which results in \(\Phi_{12}^{\prime}(x,y)\) with higher SNR than \(\Phi_{12}\):  
\[\Phi_{12}^{\prime}(x,y)=\Phi_{2}(x,y)+(2\pi)Round\left[\frac{\left(\frac{ \lambda_{12}}{\lambda_{2}}\right)\Phi_{12}(x,y)-\Phi_{2}(x,y)}{2\pi}\right], \tag{11}\]  
An extended phase map \(\Phi_{23}\) with extended wavelength \(\lambda_{23}\), is computed from phase maps \(\Phi_{2}\) and \(\Phi_{3}\) in a similar manner to \(\Phi_{12}\):  
\[\Phi_{23}(x,y)=\left\{ \begin{array}{ll}\Phi_{2}(x,y)-\Phi_{3}(x,y),&\Phi_{2 }>\Phi_{3}\\ \Phi_{2}(x,y)-\Phi_{3}(x,y)+2\pi,&\Phi_{2}\leq\Phi_{3}.\end{array} \right. \tag{12}\]  
Again to minimize noise, \(\Phi_{23}^{\prime}(x,y)\) with higher SNR than \(\Phi_{23}\) is calculated as follows:  
\[\Phi_{23}^{\prime}(x,y)=\Phi_{2}(x,y)+(2\pi)Round\left[\frac{\left(\frac{ \lambda_{23}}{\lambda_{2}}\right)\Phi_{23}(x,y)-\Phi_{2}(x,y)}{2\pi}\right], \tag{13}\]  
The extended continuous phase map \(\[PERSON]_{123}\) with extended wavelength \(\lambda_{123}\) that covers the entire object range of depth \(h(x,y)\), is computed from phase maps \(\Phi_{12}^{\prime}(x,y)\) and \(\Phi_{23}^{\prime}(x,y)\):  
\[\Phi_{123}(x,y)=\left\{