## 3 Numerical Simulations and Experiments  
### Numerical Simulations  
In off-axis PMD configuration, the measured surface is not perpendicular to the optical axis of the camera lens, as shown in Figure 1. The experimental deformed patterns of three typical surfaces under test, including the convex, the concave, and the flat, are separately recorded by the camera as shown in Figure 3a-c when a straight fringe pattern is displayed on the screen. In order to verify the method in Section 2 mentioned above, the three cases are simulated, where the normal of the measured surface is not parallel to the optical axis of the camera lens, a convex sphere, and a concave sphere, the radius of curvature and diameter of which are 300 mm and 120 mm, respectively; additionally a free-form surface with the area of 120 mm \(\times\) 120 mm and sampling points of 120 \(\times\) 120. Their fringe patterns acquired by the camera and coordinate points \(M(x_{m},y_{m},z_{m})\) obtained by using the iterative method in the tracing model are shown in Figure 3a-f, respectively. Figure 3: Experiment patterns and ray tracing patterns. (**a–c**) Fringe patterns experimentally acquired by the configuration in Figure 1 for a convex, a concave, and a free-form surface, respectively; (**d–f**) coordinate points obtained by ray tracing with the configuration in Figure 1 for a convex, a concave, and a free-form surface, respectively. The simulation demonstrates that these quadrilateral geometries in Figure 3a-c acquired by the CCD camera are similar to the results of ray tracing in Figure 3d-f. The keystone distortion and the shape of patterns agree with the experiments. For the three elements, their coordinates and the slope of each sampling point are obtained by ray tracing at the angle of \(8^{\circ}\), \(11^{\circ}\), \(14^{\circ}\), \(17^{\circ}\), and \(20^{\circ}\), respectively. The simulated surface is then reconstructed using the improved Southwell algorithm and the RBF interpolation algorithm, respectively. Random noise with a mean value of 0.017 mm [27] is added to the screen coordinates. The reconstructed residual errors are differences between the reconstructed surface and the ground truth, the RMS and PV (Peak Value) values of the residual errors for the two algorithms at different angles are shown in the Appendix of Tables A1 and A2, and the RMS values of the residual error of the two methods are multiplied by \(10^{6}\) and taken logarithmically and plotted as Figures 4 and 5. For both convex and concave spheres, the RMS value of the reconstruction residual error of the RBF interpolation algorithm is approximately \(10^{-6}\) mm, while the PV value is approximately \(10^{-5}\) mm. In contrast, the RMS value of the reconstruction residual error of the improved Southwell algorithm is approximately \(10^{-4}\) mm, while the PV value is approximately \(10^{-3}\) mm. The simulation results indicate that the RBF interpolation algorithm offers nearly one order of magnitude better reconstruction accuracy for this quadrilateral geometry, and the keystone distortion hardly impacts the accuracy of reconstruction. A surface function defined as Equation (11) which is more frequently used [10; 17], is combined with a plane to represent a common free-form surface to verify the advantages of RBF, and the height distribution of a simulated free-form surface seen in Figure 6.  
\[\mathrm{W}=0.5x^{2}-0.3\times\{3(1-x)^{2}e^{[-x^{2}-(y+1)^{2}]}-10\Big{(} \frac{x}{5}-x^{3}-y^{5}\Big{)}e^{(-x^{2}-y^{2})}-\frac{1}{3}e^{[-(x+1)^{2}-y^ {2}]}\} \tag{11}\]  
Figure 4: RMS of residual error a convex sphere using Southwell and RBF with different angles. Figure 5: RMS of residual error of a concave sphere using Southwell and RBF with different angles.