Fringe projection profilometry is a representative method for optical three-dimensional measurement and is widely applied in intelligent manufacturing, virtual reality, cultural heritage protection, biomedicine, and industrial inspection. Fringe projection profilometry mainly includes Moiré profilometry, Fourier transform profilometry, and phase measurement profilometry. Fourier transform profilometry can recover the three-dimensional surface information of the measured object through phase calculation, phase unwrapping, and phase-height mapping. It has the advantages of less data processing and a fast measurement speed, thus being widely used in three-dimensional reconstruction. The phase value obtained by phase calculation will be wrapped at (-π, π]. It is necessary to convert the wrapped phase into a continuous phase through phase unwrapping, and then the height distribution of the measured object can be determined by phase-height mapping. Therefore, the quality of phase unwrapping directly influences the reconstructed accuracy of the measured object. Among many phase unwrapping algorithms, Goldstein branch-cut algorithm is widely used because of its noise-immune ability and high efficiency. After identifying all residues in the wrapped phase map, the Goldstein branch-cut algorithm generates branch cuts by connecting the residues to optimize the phase unwrapping path. The shorter the total length of the branch cuts is, the better the result of phase unwrapping will be. However, the branch cuts constructed by Goldstein branch-cut algorithm cannot ensure the shortest total length and are easy to close, which causes incorrect phase unwrapping in some regions and finally affects the reconstructed accuracy. Therefore, Fourier transform profilometry based on an improved Goldstein branch-cut algorithm is proposed to ensure the accuracy of three-dimensional measurement.

The computer-generated grating fringes are projected onto the surface of the measured object by digital light processing, and the grating fringes are modulated by the height of the measured object. The deformed fringes containing the height information of the measured object are collected by a charge-coupled device, and the wrapped phase map is obtained through the operations of Fourier transform, fundamental frequency filtering, and inverse Fourier transform. First, all positive and negative residues are identified in the wrapped phase map. Then, the problem of constructing branch cuts with the shortest total length is transformed to a maximum weighted matching problem by constructing a weighted bipartite graph. The Kuhn-Munkres algorithm is applied to solve the maximum weighted matching problem, and the branch cuts with the shortest total length are obtained. Finally, the path that avoids branch cuts is selected for phase unwrapping. Pixels on the branch cuts can be unwrapped according to the unwrapped pixels around the branch cuts. The surface information of the measured object is recovered by phase-height mapping. This paper compares the total length of the branch cuts, the root mean square error, and the execution time of generating branch cuts between the proposed method and the Goldstein branch-cut algorithm. The root mean square error of the proposed method under different noises is studied to evaluate its noise-immune ability. In addition, three-dimensional reconstruction experiments are carried out on complex objects, and the reconstruction results show that the proposed method is suitable for the three-dimensional measurement of complex objects.

The Goldstein branch-cut algorithm is a powerful anti-noise method, and the quality of phase unwrapping depends on the generated branch cuts. Shorter branch cuts result in a better phase unwrapping result. The simulation results show that the proposed method constructs branch cuts with a shorter total length and takes less time for generating branch cuts than the Goldstein branch-cut algorithm, bringing a lower root mean square error (Table 1). In addition, the research on the root mean square errors of the proposed method and the Goldstein branch-cut algorithm under different noises shows that the former has a stronger anti-noise ability (Table 2). In the reconstruction experiment of complex objects, the results reconstructed by the Goldstein branch-cut algorithm are poor in some areas, while the proposed method can ensure the reconstructed accuracy of complex objects (Fig. 13).

This paper expounds the basic principles of Fourier transform profilometry and the Goldstein branch-cut algorithm. The Goldstein branch-cut algorithm is a local nearest neighbor algorithm that may not generate the shortest branch cuts. Moreover, branch cuts are easy to close, which makes phase unwrapping incorrect in some regions and increases the reconstructed error. To ensure the reconstructed accuracy of the measured object, this paper proposes Fourier transform profilometry based on an improved Goldstein branch-cut algorithm. The simulation results show that compared with the Goldstein branch-cut algorithm, the proposed method reduces the total length of branch cuts, has a stronger noise-immune ability, and can effectively improve reconstructed accuracy. Experimental results indicate that the proposed method is suitable for the three-dimensional measurement of complex objects.