The phase-shifting method can extract the phase with high resolution, high precision, and high robustness. However, since the number of fringe patterns is generally three or more, it is sensitive to positional movement. The 2+1 phase-shifting method can reduce the phase error caused by motion. However, the local specular reflection of the measured object causes local intensity saturation, which leads to phase error. Although the multi-exposure method can extract fringe patterns with a better signal-to-noise ratio (SNR), it is difficult to quantify the exposure time and range. In addition, the number of fringe patterns is huge, and the measurement efficiency is low. The adaptive fringe projection method adaptively adjusts the projected pixel intensity according to the pre-acquired image, but its ability to correct the phase error is limited by the following two aspects. One is the pixel-matching error between the camera and the projector, and the other is the limitation of the grayscale range of the camera and projector. The polarization method has a great suppression effect on specular reflection, but it reduces the SNR on low-reflection surfaces. In addition, the adjustment of the optical path is complicated. For the three-dimensional (3D) measurement of highly reflective objects based on the 2+1 phase-shifting method, a novel double 2+1 phase-shifting method is proposed, which can not only correct the saturation-induced wrapped phase error but also has a higher measurement efficiency.

First, the computer simulates the intensity-saturated fringe pattern. When the fringe intensity is saturated, there are high-frequency components in addition to the fundamental frequency component, and the high-frequency components increase with the increase in the intensity saturation coefficient. Therefore, the intensity-saturated fringe pattern can be expressed as a high-order Fourier series. Second, based on the analysis of the Fourier spectrum, the intensity-saturated fringe pattern can be approximately represented by a third-order Fourier series. Third, we calculate the ideal wrapped phase and the actual wrapped phase based on the 2+1 phase-shifting method. We subtract the ideal wrapped phase from the actual wrapped phase and simplify the phase difference according to the Fourier spectrum to establish a saturation-induced wrapped phase error model. From the saturation-induced wrapped phase error model, it can be seen that the 2+1 phase-shifting method contains a saturation-induced wrapped phase error of one time the fundamental frequency. Fourth, based on the above model, the opposite wrapped phase error can be obtained by doing a π phase shift to the original 2+1 phase-shifting fringes. Since the background image (the variable a in Eq. (5)) does not require a π phase shift, the additional fringe sequence is one less than the original fringe sequence. Finally, an additional fringe sequence with a phase shift of π is projected to generate the opposite wrapped phase error. The phase unwrapping adopts three-frequency hierarchical temporal phase unwrapping, and we reduce the saturation-induced wrapped phase error by fusing the unwrapped phases of the original fringe sequence and the additional fringe sequence.

The multi-exposure method can obtain a high-precision unwrapped phase with enough exposure time and a wide exposure time range. The unwrapped phase extracted by the proposed method is close to the multi-exposure method, which is better than the traditional method and the adaptive fringe projection method (Fig. 6). We use the unwrapped phase extracted by the multi-exposure method (23 exposures) as the ground truth and calculate the root mean square error (RMSE) of the traditional method, the adaptive fringe projection method, and the proposed method, respectively. The RMSE of the proposed method is 69.92% lower than the traditional method and 65.2% lower than the adaptive fringe projection method (Table 1). Since the computational cost of each algorithm is similar, the measurement efficiency mainly depends on the number of fringes (Table 2). The number of fringes using the traditional method is 7, the number of fringes using the multi-exposure method (23 exposures) is 7×23=161, and that of fringes using the adaptive projection fringe method is 7×3+15=36. In contrast, the number of fringes using the proposed method is 7+6=13. Compared with that of the multi-exposure method and the adaptive fringe projection method, the measurement efficiency of the proposed method is increased by 91.9% and 63.9%, respectively.

The 2+1 phase-shifting method has good performance in suppressing motion errors. However, highly reflective surfaces such as metals, plastics, and ceramics can cause intensity saturation of the fringe pattern, resulting in phase extraction errors. To this end, we propose a new saturation-induced wrapped phase error model of the 2+1 phase-shifting method. Based on the above saturation-induced wrapped phase error model, an efficient and high-precision double 2+1 phase-shifting method for reconstructing strongly reflective surfaces is proposed. Compared with the traditional 2+1 phase-shifting method and 2+1 phase-shifting method of adaptive fringe projection, the proposed method greatly reduces the saturation-induced phase error. Compared with the 2+1 phase-shifting method of adaptive fringe projection and multi-exposure 2+1 phase-shifting method, the proposed method requires fewer additional fringes and thus has higher measurement efficiency. Therefore, the proposed method has potential applications in the 3D reconstruction of highly reflective surfaces.