Correction of projector nonlinearity in multi-frequency phase-shifting fringe projection profilometry

In fringe projection profilometry, the original purpose of projecting multi-frequency fringe patterns is to determine fringe orders automatically, thus unwrapping the measured phase maps. This paper presents that using the same patterns, simultaneously, allows us to correct the effects of projector nonlinearity on the measured results. As is well known, the projector nonlinearity decreases the measurement accuracies by inducing ripple-like artifacts on the measured phase maps; and, theoretical analysis reveals that these artifacts, depending on the number of phase shifts, have multiplied frequencies higher than the fringe frequencies. Based on this fact, we deduce an error function for modeling the phase artifacts and then suggest an algorithm estimating the function coefficients from a couple of phase maps of fringe patterns having different frequencies. As a result, subtracting out the estimated phase errors yields the accurate phase maps with the effects of the projector nonlinearity on them being suppressed significantly. Experiment results demonstrated that this proposed method offers some advantages over others, such as working without a photometric calibration, being applicable when the projector nonlinearity varies over time, and having satisfied efficiency in implementation.