Phase extraction accuracy comparison based on multi-frequency phase-shifting method in fringe projection profilometry

•The noise-induced phase error of the multi-frequency phase-shifting algorithms is analyzed and compared.•The phase unwrapping reliability of the multi-frequency phase-shifting algorithms is analyzed and compared.•The results of this paper can provide a reference for algorithm selection in 3D measurement applications. For fringe projection profilometry (FPP), phase extraction accuracy determines three-dimensional (3D) measurement precision. The combination of phase-shifting (PS) and multi-frequency temporal phase unwrapping (MTPU) can reconstruct discontinuous or isolated surfaces with high accuracy and robustness. However, due to the introduction of inevitable random noise, the phase extraction is biased. To this end, this paper firstly compares the wrapped phase errors based on various PS methods in detail. Then, the phase errors and unwrapping reliability based on various dual-frequency temporal phase unwrapping methods are compared. Finally, the effects of three-frequency temporal phase unwrapping and reasonable selection of fringe frequency on phase extraction accuracy are further discussed. The accuracy of the wrapped and unwrapped phase extraction and the phase unwrapping reliability are compared experimentally. This paper can provide a reference for algorithm selection in 3D measurement applications.