# Multi-Wavelength Digital-Phase-Shifting Moire Based on Moire Wavelength
###### Abstract  
Multi-wavelength digital-phase-shifting moire was demonstrated using multiple moire wavelengths determined by system calibration over the full working depth. The method uses the extended noisy phase map as a reference to unwrap the phase map with a shorter wavelength, and thus achieve a less noisy and more accurate continuous phase map. The moire wavelength calibration determines a moire-wavelength to height relationship that permits pixelwise refinement of the moire wavelength and height during 3D reconstruction. Only a single pattern has to be projected and, thus, a single image captured to compute each phase map with a different wavelength to perform digital-phase-shifting moire temporal phase unwrapping. Only two captured images are required for two-wavelength phase unwrapping and three captured images for three-wavelength phase unwrapping. The method has been demonstrated in the 3D surface-shape measurement of an object with surface discontinuities and spatially isolated objects. 3D shape measurement; fringe projection profilometry; phase shift analysis; temporal phase unwrapping; moire profilometry; digital moire; moire wavelength Article  
Multi-Wavelength Digital-Phase-Shifting Moire Based on Moire Wavelength  
[PERSON] and [PERSON]  
Department of Systems Design Engineering, University of Waterloo, 200 University Avenue West, Waterloo, Ontario, N2L 3G1, Canada; f24 [EMAIL_ADDRESS]  
* Correspondence: [EMAIL_ADDRESS]  
* [2019] *  
fringe frequency requires the capture of at least three phase-shifted images. To apply temporal phase unwrapping using three different phase maps, multiple-image phase shifting PSP methods typically require the projection and capture of nine images, which would tend to cause phase error in measuring dynamic objects, either moving or deforming. It has been thus desirable to devise a method of temporal phase unwrapping that uses fewer projected patterns to enable faster 3D image capture, while still maintaining a high quality fully continuous phase map. Heterodyne temporal phase unwrapping [9] employs multiple phase maps with different fringe frequencies or wavelengths to create a new synthetic phase map with extended unambiguous phase range. Two shorter wavelengths can be used to create a longer synthetic beat wavelength (the synthetic wavelength at the beat frequency of two close frequencies) [4; 10], and the unambiguous and continuous phase value of the beat wavelength can be used as a reference phase to unwrap the phase of the shorter wavelengths [10; 11]. Thus, fewer projected and captured patterns are possible (e.g., six patterns for temporal phase unwrapping with 3-step phase-shifting) [4]. Three and multiple-wavelength temporal phase unwrapping can further extend the beat wavelength to increase the measurable height with less noise [3; 12; 13; 14; 15; 16]. Another method of reducing the number of projected and captured phase-shifted images during measurement is by digital moire. In digital moire, only a single grid image needs to be projected and captured during measurement, and phase shifting is performed digitally in a post process to generate multiple phase-shifted images [17]. This paper combines heterodyne multi-wavelength temporal phase unwrapping and digital moire in a new method of multi-wavelength digital-phase-shifting moire based on moire wavelength. Temporal phase unwrapping has the advantage over spatial phase unwrapping in permitting the measurement of discontinuous surfaces, heterodyne multi-wavelength temporal phase unwrapping has the benefit of requiring fewer sets of projected and captured phase-shifted patterns at different wavelengths, and digital moire has the advantage of requiring the projection and capture of only a single grid pattern to generate several phase-shifted images. Temporal phase unwrapping has been performed with digital-moire; however, direct physical measurement of the optical system geometry parameters was needed to calculate fringe wavelengths [18].