Enhanced Lissajous-based algorithm for phase-measuring in two-step phase-shifting fringe projection profilometry

Two-step phase-shifting profilometry is widely used in many applications due to its good trade-off between measurement resolution and efficiency. In this technique, however, it is not easy to accurately calculate fringe phases from only two available fringe patterns, especially when these patterns have nonuniform or even discontinuous background intensities, modulations, or fringe phases. To solve this problem, we propose in this paper an enhanced Lissajous-based algorithm. With it, intensity pairs within the neighborhood of a point form a Lissajous trajectory if their backgrounds and modulations are assumed to be constant. Fitting this Lissajous curve enables us to roughly estimate fringe phase at this point, and a linearized algorithm is suggested here for alleviating the computational complexity. After that, we provide a least-squares method to correct the errors caused by the uneven background intensities by modeling them as linearly varying values within the neighborhood. Further, from the residuals of the above step, the edges and discontinuities in the fringe patterns, where a piecewise algorithm generally have large errors, can be detected. The fringe phases at these edges are recalculated by using forward or backward extrapolations. Simulation and experimental results demonstrate this proposed algorithm to have a relatively high measurement accuracy in two-step phase-shifting fringe projection profilometry.