## 3 Experiments and Result Analysis
### Real Experiments  
In order to verify the practicality of the proposed method, flange, oil gasket, and U-card are selected as the tested samples. The traditional method, the method of [15], and the method of this paper are, respectively, applied to detect the 3D contour of the above three samples. Firstly, the projection device projects the gratings with pitches of 34 and 36 pixels to the measured object, and eight phase-shifting images are obtained using the four-step phase-shifting procedure and the dual-frequency heterodyne method. The traditional method and the method of literature [15] are used to unwrap the wrapped phase and process the number of cycles correspondingly. Then the projection device projects gratings with pitches of 16, 20, and 26 pixels, respectively, to the measured object, and 12 phase-shifting images are obtained by the four-step phase-shifting procedure and three-frequency heterodyne technique. The technique in this paper is utilized to unwrap the wrapped phase and adjust the number of cycles by determining the threshold value. A group of data is taken across the center of the measured object to compare the absolute phase of the three techniques, as shown in Figures 5-7.  
Figure 4: Phase unwrapping by the method of this paper under different noises. Phase expansion by the method of this paper when no Gaussian noise is added (**a**). The phase expansion is performed by the method of the this paper when adding the first set of Gaussian noise (**b**). The phase expansion is performed by the method of the this paper when adding the second set of Gaussian noise (**c**).  
Figure 5: Absolute phase of flange acquired by different methods. Traditional method (**a**). Method of [15] (**b**). Method of this paper (**c**).  
Figure 6: Absolute phase of the oil gasket acquired by different methods. Traditional method (**a**). Method of [15] (**b**). Method of this paper (**c**).  
Figure 7: Absolute phase of U-card acquired by different methods. Traditional method (**a**). Method of [15] (**b**). Method of this paper (**c**).  
but there will still be part of the phase inaccuracy, which will still cause the decrease in the accuracy when the point cloud is calculated subsequently. Compared with the traditional method and the method of [15], the principle of three-frequency outlier deconvolution has a longer absolute phase period and more accurate data, and the combination of the improved method proposed in this paper can effectively remove the phase jump error, ensure the accuracy of the absolute phase, and improve the guarantee for the subsequent calculation of the point cloud, and it has certain anti-noise capability in the face of different noises. Figure 8a-c show the 3D reconstructed point clouds of the above three samples, respectively.  
In this paper, the internal and external parameters of the camera are obtained by using [PERSON]'s camera calibration technique [22]. Then the grating fringe is projected to the measured object, and the phase principal value of the measured object is obtained. The technique in this paper is used to expand the phase principal value, and finally the reconstructed 3D point cloud image of the measured object is obtained combined with the calibration parameters. In order to prove the applicability and universality of the technique, three mechanical workpieces including flange, oil gasket, and U-shaped card were selected for 3D profile measurement. Figure 8a-c are the 3D point cloud images representing the three kinds of mechanical workpiece. It can be clearly observed that the surface of the figure is flat and continuous, and the hollow, concave, and convex parts on the measured surface can be reconstructed appropriately, which indicates that the proposed technique has a good suppression effect on the jumping error of phase, and the improved technique has certain effectiveness.