A comparative study on temporal phase unwrapping methods in high-speed fringe projection profilometry

Phase unwrapping, either spatial or temporal, plays an important role in fringe projection profilometry (FPP) to recover a continuous phase map. Compared with the spatial phase unwrapping (SPU), the temporal phase unwrapping (TPU), including multi/two-frequency, phase coding, and gray code methods, is more widely used due to its ability to measure discontinuous objects. However, multiple patterns are required in TPU, which limits the measurement speed and the application of dynamic object measurement. To overcome this problem, different techniques, such as binary defocusing projection, two-plus-one phase-shifting algorithm, geometry/continuity constrained phase unwrapping, and ternary/quaternary gray code phase unwrapping have been proposed and actively studied recently, which either enhance the projection speed or reduce the number of projected patterns. However, there are very few studies on how these techniques affect the accuracy of TPU and which TPU method is most accurate under the scenario of high-speed measurement. This paper compares the accuracy of the two-frequency (TF), phase coding (PC), and gray code (GC) methods with different situations, including the traditional 8-bit focused FPP (aFPP), the high-speed binary defocused FPP (bFPP), and the geometry/continuity constrained binary defocused FPP (cFPP). We classify the phase unwrapping errors caused by system noises into uniformly and non-uniformly distributed errors, and analyze their distributions and rates in different TPU methods and different FPP systems. By comparative simulations and experiments, we find that, for low-frequency phase unwrapping, all the three TPU methods have a good result, while for high-frequency phase unwrapping which is desired in high-quality measurement, GC in aFPP and TF in cFPP provide higher accuracy. Thus, for measurement where accuracy is more concerned than the speed GC in aFPP is preferred, for dynamic measurement where extreme high speed is required, TF in cFPP is suggested.