Three-frequency temporal phase unwrapping using a four-fringe-based 1H + 1M + 2L algorithm

High-precision and robust absolute phase recovery is essential for fringe projection profilometry, but conventional combinations of phase-shifting algorithms and three-frequency temporal phase unwrapping usually require multiple projected fringes, limiting their efficiency in rapid three-dimensional reconstruction. This paper investigates the key factors affecting the accuracy of the fringe Hilbert transform. Uneven surface reflectivity or non-stationary ambient light introduces non-zero background components, while discontinuous surfaces generate multiple high-frequency components, causing mode aliasing and false Hilbert-transform components. To address these problems, we propose a four-fringe-based 1H + 1M + 2L algorithm and analyze its noise resistance performance. By carefully designing three-frequency four fringe patterns, the proposed method suppresses Hilbert-transform errors caused by uneven backgrounds and discontinuous surfaces, thereby improving the accuracy of three wrapped phases. The final absolute phase is then obtained through robust three-frequency temporal phase unwrapping. Experimental results demonstrate that the proposed algorithm can robustly recover the absolute phase using only four fringe patterns, providing an efficient solution for rapid and robust three-dimensional reconstruction in fringe projection profilometry.