## 4 Experiments
### Experimental Results on Synthetic Images  
The experiments employ the images of a semi-spherical surface and sinusoidal surface, which are the benchmark images in some studies [39; 40]. The semi-spherical and sinusoidal surfaces are generated by the surface formula, which is shown in Figures 9a and 10a, respectively. The diameter of the semi-spherical is five mm. The amplitudes of the sinusoidal surface are 2.5 mm and 0.5 mm, respectively, and the frequency difference is five times. The two synthetic images are formed by the Lambert model, where the vector of the light source is (0, 0, 1). Figures 9b and 10b illustrate the synthetic images of the two original surfaces, while the reconstructed surfaces are shown in Figures 9c and 10c, respectively. Figures 9d and 10d show the errors between the original surfaces and reconstructed surfaces. It can be seen that the reconstructed surfaces are similar to the original surfaces, with the maximum error values being less than 0.154 mm and 0.0178 mm, respectively. The reconstruction errors mainly occur near the boundary, while the errors of the other points are negligible. This is due to the discontinuity of the boundary surface resulting in the abnormal change in the gray scale. At the same time, it is apparent from the figure that although there is ambiguous gray-scale information in the synthetic image, the peaks and valleys of the surface can be accurately recovered using the proposed correction method. To evaluate the performance of the proposed reconstruction algorithm, the comparison of the original surface and the reconstructed surface is studied using the mean absolute error (MAE) and root mean square error (RMSE). The results are shown in Table 1.  
Figure 9: Simulation results of the semi-spherical surface: (**a**) Semi-spherical surface; (**b**) Synthetic image; (**c**) Reconstructed surface; and (**d**) Reconstruction error. We extracted the profiles of the same position from the above-mentioned topography and the profile between the reconstructed topography and measured topography, which were compared. The results are shown in Figure 14.  
Figure 11: Experimental results for turning surface image: (**a**) Turning surface image; (**b**) Measured result; (**c**) Reconstruction result of Horn and Brooks algorithm; and (**d**) Reconstruction result of proposed algorithm. Figure 12: Experimental results for milling surface image: (**a**) Milling surface image; (**b**) Measured result; (**c**) Reconstruction result of Horn and Brooks algorithm; and (**d**) Reconstruction result of proposed algorithm. It can be seen that the profiles reconstructed by the Horn and Brooks algorithm are different to the measured profile, and the deviations between them are larger. Hence, the reconstruction method based on the Horn and Brooks algorithm is not suitable for machined surface topography. On the contrary, the spacing and height of peaks and valleys between the profiles that were reconstructed by the proposed method and measured profiles are similar. Due to the interference of mutual reflection, secondary reflection, noise, or stray radiation, which are inevitable in the imaging process, the results in some details of the profiles are different. However, from the distribution of peaks and valleys, the profiles reconstructed by the proposed method are very close to the measured profile. The arithmetic average roughness (\(Ra\)), maximum height of the profiles (\(Rz\)), and root mean square roughness (\(Rq\)) are used to compare the profile reconstructed by the proposed method and measured profile. The results are shown in Table 2.  
As shown in Table 2, the relative error between the reconstructed profile and the measured profile is small: less than 10%. It indicates that the reconstructed data are very close to the measured data, and the reconstruction accuracy can meet the requirements of surface topography measurements. Therefore, the experimental results demonstrate that the proposed algorithm can accurately and effectively reconstruct the m