Objective  Phase Measurement Profilometry (PMP) has become the mainstream optical three-dimensional measurement method due to its advantages of high imaging accuracy and fast reconstruction speed. Phase Measurement Profilometry performs surface reconstruction through phase calculation of projected images. The accuracy of phase calculation largely determines the final three-dimensional reconstruction accuracy. Therefore, obtaining high-precision unwrapped phase is the key step of measurement. Multi-frequency heterodyne is a commonly used unwrapping phase algorithm in phase measurement profilometry. When unwrapping the wrapped phase, due to the influence of the measurement environment, the characteristics of the object surface, the nonlinear error of the camera and projector, and other factors, the fringe series will be incorrectly rounded, and the unwrapped phase will produce a jumping phase error. This jumping error will lead to the wrong concave and convex areas or burrs on the surface of the reconstructed object, which greatly affects the accuracy of 3D reconstruction. Therefore, it is necessary to study a correction method that can eliminate or compensate the jumping error. Thus, a phase correction method based on multi-frequency heterodyne principle is proposed.

  Methods  The specific flow of the phase correction method based on the principle of multi-frequency heterodyne is drawn (Fig.2). The initial wrapped phase and the unwrapped phase are calculated by using the traditional phase shift method and multi-frequency heterodyne. Because there are two different jumping errors in the unwrapped phase, the cause of the error and the error location are preliminarily determined by the gradient and amplitude of the unwrapped phase. By comparing the phase amplitude of multiple wrapped phases with different frequencies at the problem point, whether the error at the problem point really exists is determined. According to the principle of multi-frequency heterodyne, the error amplitude introduced by the error item is analyzed, and the pixel with the error is corrected to obtain a new unwrapped phase.

  Results and Discussions   Phase Measurement Profilometry is used to reconstruct the standard sphere in three dimensions, and the proposed phase correction method is used to correct the error of the unwrapped phase. By comparing the original unwrapped phase (Fig.4(a)) and the corrected unwrapped phase (Fig.4(c)), it can be observed that the phase mutation region in the original phase distribution map has been successfully eliminated and the phase error has been successfully repaired. The 470th line of the unwrapped phase before and after correction is observed (Fig.4(b) and Fig.4(d)). The abrupt phase area of the original unwrapped phase map at the corresponding pixel position has been successfully repaired. The corrected unwrapped phase transition is smooth without phase abrupt change. The point cloud reconstruction before and after correction is compared (Fig.6), the surface defect area caused by unwrapping phase error has been well repaired.

  Conclusions  In view of the jumping error in the existing multi-frequency heterodyne phase demodulation process, a phase error correction algorithm is proposed based on the full analysis of the reasons for the jumping error in the principle. The principle of this method is simple and easy to implement, the error correction is accurate and the correction speed is fast. The source terms that cause the error are analyzed, the error type is determined and phase correction is carried out through multiple discussions. After correcting the standard sphere unwrapped phase of the traditional multi-frequency heterodyne measurement, all the jumping errors in the original unwrapped phase are corrected. The experimental results show that this method can accurately locate the cause and region of the jumping error and effectively correct the jumping error in the unwrapping phase. The unwrapped phase corrected by this method is smooth without jumping, and the 3D point cloud reconstruction surface has no abnormal concave and convex areas, which realizes the elimination of the jumping error in the unwrapping phase, and verifies the feasibility and effectiveness of this correction method.