## 1 Introduction  
As a representative structured light three-dimensional (3D) measurement technique, fringe projection profilometry [1; 2; 3] is widely applied in medical imaging, the industrial field, and online measurement due to its non-contact measurement, high precision, and low cost. Phase-shifting projected fringe profilometry mainly consists of four steps: projecting grating stripes with a certain phase difference onto the target, extracting the principal phase value of the target, unwrapping the phase, and 3D reconstruction. The phase-shifting approach is a common approach to extract the phase, but the acquired phase is wrapped in \([-\pi,\pi]\), which is not unique at this time, so the wrapped phase must be unwrapped to obtain the full-field real phase field. Currently, approaches of unwrapping phases are classified in principle into the time domain and the space domain. Spatial phase unwrapping [4] can be completed with only one phase figure, but the phase unwrapping operation of each point is related to its surrounding adjacent points. Once the phase of one point is unwrapped incorrectly because of phase aliasing and phase noise, the error will be transmitted to the phase unwrapping process of all subsequent points, resulting in a reduction in measurement accuracy. On the other hand, temporal phase unwrapping refers to the phase unwrapping of the phase diagrams obtained for the same pixel point at different times to obtain the absolute phase. In temporal phase unwrapping, the phase of each point is only related to the point at the same position on the phase diagrams at different frequencies, not to its adjacent points, so even if the phase of one point is unwrapped incorrectly due to the phase mixing and phase noise, these errors are not transmitted. The multi-frequency heterodyne phase unwrapping approach, as a kind of temporal phase unwrapping approach, is widely used because of its wide applicability and high precision. In the heterodyne phase-solving approach, three grating fringes of different frequencies are used to obtain phase values at three frequencies by a three-step phase-shifting methodology. Then, the heterodyne operation is performed on the phase values of these three frequencies to obtain two heterodyne phase maps. These two heterodyne phase maps can be further processed to obtain a phase solution wrap. Finally, the absolute phase value is calculated by the phase solution parcel diagram. However, the phase jump will occur after directly utilizing this methodology to solve the wrapped phase, so the results need to be corrected. Aiming at the jumping errors generated in unwrapping by the multi-frequency heterodyne methodology, [PERSON] et al. [5] modified the principle of multi-frequency heterodyne to correct the unwrapping phase error, but this methodology is only limited to the dual-frequency heterodyne, and the phase errors generated by the multi-frequency heterodyne cannot be completely eliminated. [PERSON] et al. [6] extended the methodology in [5] and derived the methodology of the multi-frequency heterodyne phase solution, but the phase jumping error after using the solution cannot be completely avoided. [PERSON] et al. [7] improved the methodology proposed in the literature [5], but there exist many constraints, and the grating fringe of the third frequency has a large pitch, which could easily lead to the loss of details and could not improve the measurement accuracy. [PERSON] et al. [8] analyzed the neighborhood of the phase error points and corrected the errors. This methodology works better for objects with simple and gentle surfaces, but it ignores the independence of each pixel point during phase unwrapping, so it is not suitable for measuring objects with complex surfaces. [PERSON] et al. [9] proposed a phase unwrapping methodology for phase-shifting projected fringe profilometry that does not require additional projections to identify fringe orders. The pattern used for phase extraction can be directly used for phase unwrapping.