-
传统N ≥ 5步相移双频条纹,可表示为[15]:
$$\begin{split} & {I_n}(x,y) = A(x,y) + \\ &{B_{\rm{h}}}(x,y)\cos [{\phi _{\rm{h}}}(x,y) + 2{\text{π}} n/N] + \\ & {B_{\rm{l}}}(x,y)\cos [{\phi _{\rm{l}}}(x,y) + 4{\text{π}} n/N] \end{split} $$ (1) 式中:A表示图像背景强度;Bh和 Bl分别表示高频分量和低频分量的条纹调制度;ϕh和ϕl分别表示高频相位和低频相位,其计算公式如下:
$${\phi _{\rm{h}}}(x,y) = {\rm{arctan}}\left[ {\frac{{\sum\nolimits_{n = 1}^N {{I_n}\sin (2{\text{π}} n/N)} }}{{\sum\nolimits_{n = 1}^N {{I_n}\cos (2{\text{π}} n/N)} }}} \right]$$ (2) $${\phi _{\rm{l}}}(x,y) = {\rm{arctan}}\left[ {\frac{{\sum\nolimits_{n = 1}^N {{I_n}\sin (4{\text{π}} n/N)} }}{{\sum\nolimits_{n = 1}^N {{I_n}\cos (4{\text{π}} n/N)} }}} \right]$$ (3) 上式计算得到的ϕh
和ϕl的取值范围为(−π,+π],因此ϕh和ϕl也称为截断相位,需要进行相位展开。假设ϕl只有一个周期,则Фl = ϕl。根据低频绝对相位Фl和高频绝对相位Фh的数学比例关系,可计算出ϕh所对应的条纹级次: $${K_{\rm{h}}}(x,y) = {\rm{round}}\left[ {\frac{{({f_{\rm{h}}}/{f_{\rm{l}}}) \times {\varPhi _{\rm{l}}} - {\phi _{\rm{h}}}}}{{2\pi }}} \right]$$ (4) 式中:fh和fl分别表示高频相位和低频相位的频率。进一步地,计算出: $${\varPhi _{\rm{h}}}(x,y) = {\phi _{\rm{h}}}(x,y) + 2{\text{π}} {K_{\rm{h}}}(x,y)$$ (5) 图1展示了传统双频条纹的相位展开原理,图中Th
和Tl分别表示ϕh和ϕl的周期。分析公式(4)得知, Фl参与计算时,需要放大γ = fh/fl 倍,这意味着Фl所包含的误差也放大了γ倍,容易导致部分 pixel 的Kh计算错误,进而引入相位展开误差至Фh。因此,传统双频条纹对噪声较为敏感,使得其应用范围受到了一定的限制。如何能够有效地降低比值γ,提高双频条纹的抗噪能力,是文中所要研究的内容。 -
对于测量范围内的某一物平面zmin,根据摄像机和投影仪的几何约束关系,可以建立该物平面的绝对相位图Фmin,称为最小相位图[12]。根据Фmin,可以对截断相位ϕ进行相位展开,其基本思想如图2所示。例如,在区间[A, B]内,满足条件0 < Фmin −ϕ < 2π,得到Ф = ϕ+2π;在区间[B, C]内,满足条件2π < Фmin −ϕ < 4π,得到Ф = ϕ+4π;依此类推,便可以将ϕ展开为Ф。因此,根据Фmin−ϕ的取值范围,便可以确定条纹级次。
$$K = {\rm{ceil}}\left( {\frac{{{\varPhi _{{\rm{min}}}} - \phi }}{{2\pi }}} \right)$$ (6) 式中:ceil()表示向上取整函数。需要注意,在相位域中,上述方法最大可测深度范围仅为2π。因此在实际测量过程中,被测物体需要尽可能地靠近物平面zmin。
-
针对传统双频条纹抗噪能力较差的问题,文中根据几何约束方法,对传统双频条纹进行改进,其相位展开原理如图3所示。改进双频条纹通过采用多个周期的低频相位(即fl >1),能够有效地降低比值γ,提升双频条纹的抗噪能力。由于低频相位包含多个周期,低频相位ϕl将发生截断,因此需要首先展开ϕl,再用于展开高频相位ϕh。改进双频条纹进行三维测量的具体实现流程如下:首先,利用五步相移算法计算出ϕh和ϕl;然后,根据几何约束关系,恢复出ϕl所对应的最小相位图Фmin;根据公式(6)计算ϕl所对应的条纹级次Kl,并对ϕl进行相位展开得到Фl;最后根据公式(4)计算ϕh的条纹级次Kh,并对ϕh进行相位展开得到Фh。注意,若直接采用几何约束方法展开ϕh,将导致高频相位域中的最大可测深度范围仅为2π。相比而言,文中首先利用几何约束方法展开ϕl,然后利用Фl展开ϕh,在低频相位域中的最大可测深度范围为2π,转换至高频相位域中的最大可测深度范围将达到2πγ。
Modified dual-frequency geometric constraint fringe projection for 3D shape measurement
-
摘要:
双频条纹投影已经广泛应用于三维形貌测量,然而其相位展开的准确性受噪声影响较大。文中提出了一种改进的双频几何约束条纹,通过提高低频相位的频率,有效地提升了相位展开的鲁棒性。在三维测量过程中,首先,利用五步相移算法计算出双频条纹的高频相位和低频相位。然后,利用几何约束方法展开低频相位。最后,采用双频算法展开高频相位,进而重建出物体的三维形貌。仿真和实验结果均表明,相对于传统双频条纹,改进的双频条纹具有更高的鲁棒性和适用性。
Abstract:Dual-frequency fringe projection methods have been widely used in three-dimensional (3D) shape measurement, but the phase unwrapping is very sensitive to random noises. A modified dual-frequency geometric constraint fringe was presented. The robustness of phase unwrapping can be effectively enhanced by improving the frequency of low-frequency phase. During the 3D shape measurement, firstly, the five-step phase-shifting algorithm was used to extract two wrapped phases. Secondly, the low-frequency phase was unwrapped based on the geometric constraint method. Finally, the dual-frequency algorithm was used to unwrap the high-frequency phase, and then the 3D shape could be reconstructed. Both simulations and experiments demonstrate that the modified dual-frequency fringe is more robust and applicable than the traditional one.
-
Key words:
- dual-frequency fringe /
- phase-shifting /
- geometric constraint /
- phase unwrapping
-
-
[1] Su X, Zhang Q. Dynamic 3-D shape measurement method: A review [J]. Optics and Lasers in Engineering, 2010, 48(2): 191−204. doi: 10.1016/j.optlaseng.2009.03.012 [2] Zhang S. High-speed 3D shape measurement with structured light methods: A review [J]. Optics and Lasers in Engineering, 2018, 106: 119−131. [3] Zuo C, Feng S, Huang L, et al. Phase shifting algorithms for fringe projection profilometry: A review [J]. Optics and Lasers in Engineering, 2018, 109: 23−59. doi: 10.1016/j.optlaseng.2018.04.019 [4] Zhang S. Absolute phase retrieval methods for digital fringe projection profilometry: A review [J]. Optics and Lasers in Engineering, 2018, 107: 28−37. doi: 10.1016/j.optlaseng.2018.03.003 [5] Wu Z, Guo W, Zhang Q. High-speed three-dimensional shape measurement based on shifting Gray-code light [J]. Optics Express, 2019, 27(16): 22631−22644. doi: 10.1364/OE.27.022631 [6] He X, Zheng D, Qian K, et al. Quaternary gray-code phase unwrapping for binary fringe projection profilometry [J]. Optics and Lasers in Engineering, 2019, 121: 358−368. doi: 10.1016/j.optlaseng.2019.04.009 [7] Wang Y, Zhang S. Novel phase-coding method for absolute phase retrieval [J]. Optics Letters, 2012, 37(11): 2067−2069. doi: 10.1364/OL.37.002067 [8] Li Biao, Wu Haitao, Zhang Jiancheng, et al. 3D shape measurement method combining sinusoidal pulse width modulation fringe with phase coding fringe [J]. Infrared and Laser Engineering, 2016, 45(6): 0617006. (in Chinese) [9] Zhang M, Chen Q, Tao T, et al. Robust and efficient multi-frequency temporal phase unwrapping: optimal fringe frequency and pattern sequence selection [J]. Optics Express, 2017, 25(17): 20381−20400. doi: 10.1364/OE.25.020381 [10] Han Xu, Wang Lin, Fu Yanjun. Phase unwrapping method based on dual-frequency heterodyne combined with phase encoding [J]. Infrared and Laser Engineering, 2019, 48(9): 0913003. (in Chinese) [11] Hyun J, Zhang S. Enhanced two-frequency phase-shifting method [J]. Applied Optics, 2016, 55(16): 4395−4401. doi: 10.1364/AO.55.004395 [12] An Y, Hyun J, Zhang S. Pixel-wise absolute phase unwrapping using geometric constraints of structured light system [J]. Optics Express, 2016, 24(16): 18445−18459. doi: 10.1364/OE.24.018445 [13] Ma M, Yao P, Deng H, et al. A simple and practical jump error removal method for fringe projection profilometry based on self-alignment technique [J]. Review of Scientific Instruments, 2018, 89(12): 123109. doi: 10.1063/1.5051635 [14] Li J, Su H, Su X. Two-frequency grating used in phase-measuring profilometry [J]. Applied Optics, 1997, 36(1): 277−280. doi: 10.1364/AO.36.000277 [15] Liu K, Wang Y, Lau D, et al. Dual-frequency pattern scheme for high-speed 3-D shape measurement [J]. Optics Express, 2010, 18(5): 5229−5244. doi: 10.1364/OE.18.005229 [16] Zuo C, Huang L, Zhang M, et al. Temporal phase unwrapping algorithms for fringe projection profilometry: A comparative review [J]. Optics and Lasers in Engineering, 2016, 85: 84−103. doi: 10.1016/j.optlaseng.2016.04.022
计量
- 文章访问数: 610
- HTML全文浏览量: 168
- 被引次数: 0