Suppressing motion-induced phase error by using equal-step phase-shifting algorithms in fringe projection profilometry.

Phase-shifting fringe projection profilometry is a widely used and important technique for three-dimensional surface measurement, where N-step fixed-step phase-shifting algorithms are commonly used. With a pressing need to apply this technique for dynamic object/scene measurement, the motion-induced error poses a challenge in achieving high measurement accuracy. A few correction methods have been developed by involving physical markers or complicated algorithms. In this paper, the equal-step phase-shifting algorithms are proposed as a simpler yet more effective solution. By approximating the phase variations as unknown but linear phase shifts, the equal-step algorithms are naturally immune to object motion. In particular, two classical algorithms, including the four-step Carré algorithm and the five-step Stoilov algorithm, are adopted. Furthermore, a novel three-step gradient-based equal-step phase-shifting (GEPS) algorithm is proposed. These equal-step algorithms are studied through comprehensive simulations and experiments, showing that, (i) the equal-step algorithms are all effective in greatly suppressing the motion-induced errors in both ideal and noisy situations; and (ii) among the three algorithms, the Stoilov algorithm is more robust to handle the object motion and the harmonics simultaneously, while the GEPS requires a least number of frames. This study will urge the use of the equal-step algorithms for phase extraction in dynamic profilometry for immediate motion-error suppression by merely implementing a single phase-calculation equation.