Influence of sampling on three-dimensional surface shape measurement
doi: 10.37188/CO.EN-2024-0003
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摘要:
本文研究了抽样对三维形貌测量的影响。首先，利用傅立叶变换推出频谱表达式。在此基础上，分析了CCD像元的产生过程并给出了其表达式。然后，经抽样得到离散的变形条纹表达式，并推导出了其傅立叶频谱表达式，从而得到频域内无限重复的“频谱岛”。 最后，利用低通滤波器滤除高级频谱成份后仅保留其中一个基频成份，由逆傅立叶变换恢复信号强度。提出减小抽样间隔，即减小每根条纹抽样点数的方法，来增大抽样频率与光栅基频的比值
m ，使之在满足m >4的条件下能更准确地恢复物体的三维形貌。通过仿真和实验对基本原理进行验证。在仿真分析中，抽样间隔分别取8 pixels、4 pixels、2 pixels、1 pixel，后3种情况所得到的最大绝对误差值分别为第一种情况下的88.80%、38.38%和31.50%，平均绝对误差值分别为第一种情况下的71.84%、43.27%和32.26%。可见，抽样间隔越小，恢复效果越好。在实验中，取与仿真分析相同的4次抽样间隔，得到了与仿真分析相同的结论。结果表明：减小抽样间隔可提高三维形貌的测量精度，取得了更好的恢复效果。Abstract:In order to accurately measure an object’s three-dimensional surface shape, the influence of sampling on it was studied. First, on the basis of deriving spectra expressions through the Fourier transform, the generation of CCD pixels was analyzed, and its expression was given. Then, based on the discrete expression of deformation fringes obtained after sampling, its Fourier spectrum expression was derived, resulting in an infinitely repeated "spectra island" in the frequency domain. Finally, on the basis of using a low-pass filter to remove high-order harmonic components and retaining only one fundamental frequency component, the inverse Fourier transform was used to reconstruct the signal strength. A method of reducing the sampling interval, i.e., reducing the number of sampling points per fringe, was proposed to increase the ratio
$ m $ between the sampling frequency and the fundamental frequency of the grating. This was done to reconstruct the object’s surface shape more accurately under the condition of$m > 4$ . The basic principle was verified through simulation and experiment. In the simulation, the sampling intervals were 8 pixels, 4 pixels, 2 pixels, and 1 pixel, the maximum absolute error values obtained in the last three situations were 88.80%, 38.38%, and 31.50% in the first situation, respectively, and the corresponding average absolute error values are 71.84%, 43.27%, and 32.26%. It is demonstrated that the smaller the sampling interval, the better the recovery effect. Taking the same four sampling intervals in the experiment as in the simulation can also lead to the same conclusions. The simulated and experimental results show that reducing the sampling interval can improve the accuracy of object surface shape measurement and achieve better reconstruction results. -
Figure 4. The errors in the surface shape between the reconstructed and simulated objects
Figure 6. Results of object surface shape reconstructed under different sampling conditions
Table 1. Error between reconstructed object and simulated object
Sampling interval 8 pixels 4 pixels 2 pixels 1 pixels MAEV 1.3246 1.1762 0.5084 0.4173 AAEV 0.4758 0.3418 0.2059 0.1535 
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