Two-frequency phase-shifting method vs. Gray-coded-based method in dynamic fringe projection profilometry: A comparative review

Two-frequency phase-shifting phase unwrapping and Gray-coded-based phase unwrapping are two typical temporal phase unwrapping methods to recover discontinuous or spatially isolated surfaces in fringe projection profilometry (FPP). But in dynamic measurement, environmental noise, required number of patterns and inter-frame motion affect the performance of these two methods on measuring anti-noise ability, efficiency and robustness in varying degrees. So, it is a challenge to quantitatively compare these methods' abilities and determine the optimum one with appropriate system parameters and under a certain condition. In this paper, a detailed comparative study is conducted for the two-frequency phase-shifting method including hierarchical, number-theoretical and heterodyne approaches and the Gray-coded-based method including traditional Gray code approach, complementary Gray code approach and Gray code approach with tripartite phase unwrapping (Tri-PU). First, the principles of six approaches are reviewed; then, their anti-noise ability was quantitatively compared based on the proposed noise model; next, theoretical comparison of measuring efficiency, comparative review of improved methods and recommendation on usage are given to boost their efficiency; finally, motion-induced error model in phase unwrapping was presented to determine the appropriate system parameters in dynamic measurement. Theoretical analysis, numerical simulation and experimental results are consistent and demonstrate that (i) Gray-coded-based method has better anti-noise ability than two-frequency phase-shifting method; (ii) two-frequency phase-shifting method is more efficient than traditional Gray-coded-based method when the number of encoding fringe periods exceeds 8; (iii) hierarchical approach and Gray-coded approach with Tri-PU have best performance to resist motion-induced error in their respective types of methods. Comparative results can give a helpful guidance to choose the "best" approach in different application scenes and quantitatively determine the projected fringe periods and imaging or projecting speed in dynamic measurement for FPP.