# Motion-Induced Phase Error Compensation Using Three-Stream Neural Networks
## 3 Experiment
### Quantitative Evaluation  
To verify the performance of the proposed method, we first evaluated the trained model in the validation set, which is generated by a virtual FPP system. In Figure 5, the fist shaped object in the first row fronted towards the projector and translated from right to left. One of the captured fringe patterns is shown in Figure 5a. Since the poses in every frame were recorded and fixed for generating the ground truth, the phase error compared with 12-step phase-shift can be calculated conveniently. The phase errors of the traditional three-step phase-shifting algorithm and the proposed method are listed in the left and right of Figure 5b. The error distributions along the marked red dotted line are shown in Figure 5c. Figure 5d-f show the corresponding results of two owl sculptures with more complex details and faster movement speed. Figure 5g-i show the results of a hat-shaped object rotated along \(x\) axis. Figure 5j-l show this for two industrial components rotated along the \(y\) axis. Figure 5m-r are the results of a Buddhist sculpture and two owl sculptures both translated and rotated around their centroids. It can be seen from Figure 5 that the phase errors of the three-step PSP show a double frequency distribution and have an offset compared with their own ground truth, which is consistent with the Equation (9). The RMS of phase errors of the proposed method are reduced by 75%-93% compared with that of the traditional three-step PSP. The results show that the proposed method can successfully suppress the motion-induced error caused by different motion types. Then, we tested the trained model in an actual FPP system shown in Figure 2a to evaluate the performance of the presented model in a real scene. In Figure 6, a standard ceramic plate shown in Figure 6a placed on a motorized linear stage translated in-depth direction. The movement speed of the translation stage was 1 mm/step; one fringe pattern was projected for each step. The camera was triggered synchronously by the projector and captured a deformed fringe pattern for each step. To quantitatively analyze the influence of motion-induced error, the ground truth was obtained by projecting 12-step fringe patterns at each position of the stage and shown in Figure 6b. The three-step fringe patterns modulated by ovine plate were generated by selecting the first, fifth, and ninth images of the 12-step PSP. The 3D shape was reconstructed using the multiple-frequency phase unwrapping method [39] and triangular stereo calibration model [35]. As shown in Figure 6c,d, the result of the three-step PSP has obvious periodical distribution in surface, while the proposed method significantly reduces the fluctuations and the bias (shown in Figure 6e) caused by direct-current (DC) component in Equation (9). The root mean square (RMS) error in height (Z) dimension of the three-step PSP and the proposed method are 1.07 mm and 0.12 mm, respectively. The motion-induced error has been decreased about 90%. Figure 4: Loss curves of the training stage. Figure 5: Measurement results of the scenes of different motion types in the validation set. (**a**) One fringe pattern of a translational fist-shaped object. (**b**) Phase error of three-step PSP (left) and proposed method (right), and their RMS errors (top). (**c**) Section line of phase errors along the red dotted line marked in (**b**). (**d–r**) Corresponding results of translational two owl sculptures, one rotating hat-shaped object, two rotating components, one translational and rotating Buddhist sculpture, and owl sculptures. Two standard spheres were measured under the same condition with ceramic plate. The diameters of two standard spheres are 50.7991 mm and 50.7970 mm, respectively, and the center distance is 100.2537 mm, as shown in Figure 7a. One of the deformed fringe patterns is shown in Figure 7b. Figure 7c,d illustrate the reconstructed results of traditional three-step PSP and our method, respectively.