Abstract

The omnidirectional structured-light vision measurement is significant for inner surface inspections. In existing systems, the camera and the projector are installed inside a glass tube, inevitably causing the refraction distortion. In this paper, we propose a measurement model of the omnidirectional structured-light vision and the corresponding calibration method. The model can correct the refraction distortion and realize the omnidirectional measurement. An aluminum tube with an internal diameter of 288.50 mm is measured by the system. The repeatability and precision reach 0.05 mm and 0.23 mm, respectively. The experimental results prove that the accuracy is improved by 7.9 times compared with the model ignoring distortion.

© 2019 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

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References

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  1. F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
    [Crossref]
  2. Y. Wang and R. Zhang, “In-pipe surface circular structured light 3D vision inspection system,” Infrared Laser Eng. 43(3), 891–896 (2014).
  3. Y. Zhu, Y. Gu, Y. Jin, and C. Zhai, “Flexible calibration method for an inner surface detector based on circle structured light,” Appl. Opt. 55(5), 1034–1039 (2016).
    [Crossref]
  4. T. Wu, S. Lu, and Y. Tang, “An in-pipe internal defects inspection system based on the active stereo omnidirectional vision sensor,” in 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) (IEEE, 2015), pp. 2637-2641.
  5. G. Zhang, J. He, and X. Li, “3D vision inspection for internal surface based on circle structured light,” Sens. Actuators A 122(1), 68–75 (2005).
    [Crossref]
  6. P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
    [Crossref]
  7. T. Yoshizawa and T. Wakayama, “Development of an inner profile measurement instrument using a ring beam device,” Proc. SPIE 7855, 78550B (2010).
    [Crossref]
  8. Z. Gong, Z. Liu, and G. Zhang, “Flexible method of refraction correction in vision measurement systems with multiple glass ports,” Opt. Express 25(2), 831–847 (2017).
    [Crossref]
  9. L. Huang, X. Zhao, S. Cai, and Y. Liu, “Plate refractive camera model and its applications,” J. Electron. Imaging 26(2), 023020 (2017).
    [Crossref]
  10. M. Feng, S. Huang, J. Wang, B. Yang, and T. Zheng, “Accurate calibration of a multi-camera system based on flat refractive geometry,” Appl. Opt. 56(35), 9724–9734 (2017).
    [Crossref]
  11. S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
    [Crossref]
  12. S.-Q. Li, X.-P. Xie, and Y.-J. Zhuang, “Research on the calibration technology of an underwater camera based on equivalent focal length,” Measurement 122, 275–283 (2018).
    [Crossref]
  13. C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
    [Crossref]
  14. Y. Fu and Y. Liu, “3D bubble reconstruction using multiple cameras and space carving method,” Meas. Sci. Technol. 29(7), 075206 (2018).
    [Crossref]
  15. S. Morinaka, F. Sakaue, J. Sato, K. Ishimaru, and N. Kawasaki, “3D reconstruction under light ray distortion from parametric focal cameras,” Pattern Recognit. Lett. to be published.
    [Crossref]
  16. J. Sun, G. Zhang, Q. Liu, and Z. Yang, “Universal Method for Calibrating Structured-light Vision Sensor on the Spot,” J. Mech. Eng. 45(03), 174–177 (2009).
    [Crossref]
  17. Z. Zhang, “A Flexible New Technique for Camera Calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
    [Crossref]

2018 (4)

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

S.-Q. Li, X.-P. Xie, and Y.-J. Zhuang, “Research on the calibration technology of an underwater camera based on equivalent focal length,” Measurement 122, 275–283 (2018).
[Crossref]

C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
[Crossref]

Y. Fu and Y. Liu, “3D bubble reconstruction using multiple cameras and space carving method,” Meas. Sci. Technol. 29(7), 075206 (2018).
[Crossref]

2017 (3)

2016 (1)

2015 (1)

P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
[Crossref]

2014 (1)

Y. Wang and R. Zhang, “In-pipe surface circular structured light 3D vision inspection system,” Infrared Laser Eng. 43(3), 891–896 (2014).

2013 (1)

F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
[Crossref]

2010 (1)

T. Yoshizawa and T. Wakayama, “Development of an inner profile measurement instrument using a ring beam device,” Proc. SPIE 7855, 78550B (2010).
[Crossref]

2009 (1)

J. Sun, G. Zhang, Q. Liu, and Z. Yang, “Universal Method for Calibrating Structured-light Vision Sensor on the Spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

2005 (1)

G. Zhang, J. He, and X. Li, “3D vision inspection for internal surface based on circle structured light,” Sens. Actuators A 122(1), 68–75 (2005).
[Crossref]

2000 (1)

Z. Zhang, “A Flexible New Technique for Camera Calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Albertazzi, A.

P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
[Crossref]

Buschinelli, P.

P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
[Crossref]

Cai, S.

L. Huang, X. Zhao, S. Cai, and Y. Liu, “Plate refractive camera model and its applications,” J. Electron. Imaging 26(2), 023020 (2017).
[Crossref]

Cui, Y.

F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
[Crossref]

Feng, M.

Feng, M. C.

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

Fu, Y.

Y. Fu and Y. Liu, “3D bubble reconstruction using multiple cameras and space carving method,” Meas. Sci. Technol. 29(7), 075206 (2018).
[Crossref]

Gong, Z.

Gu, Y.

He, J.

G. Zhang, J. He, and X. Li, “3D vision inspection for internal surface based on circle structured light,” Sens. Actuators A 122(1), 68–75 (2005).
[Crossref]

Huang, L.

L. Huang, X. Zhao, S. Cai, and Y. Liu, “Plate refractive camera model and its applications,” J. Electron. Imaging 26(2), 023020 (2017).
[Crossref]

Huang, S.

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

M. Feng, S. Huang, J. Wang, B. Yang, and T. Zheng, “Accurate calibration of a multi-camera system based on flat refractive geometry,” Appl. Opt. 56(35), 9724–9734 (2017).
[Crossref]

Ishimaru, K.

S. Morinaka, F. Sakaue, J. Sato, K. Ishimaru, and N. Kawasaki, “3D reconstruction under light ray distortion from parametric focal cameras,” Pattern Recognit. Lett. to be published.
[Crossref]

Jin, Y.

Kawasaki, N.

S. Morinaka, F. Sakaue, J. Sato, K. Ishimaru, and N. Kawasaki, “3D reconstruction under light ray distortion from parametric focal cameras,” Pattern Recognit. Lett. to be published.
[Crossref]

Li, F.

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

Li, J.

C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
[Crossref]

Li, S.-Q.

S.-Q. Li, X.-P. Xie, and Y.-J. Zhuang, “Research on the calibration technology of an underwater camera based on equivalent focal length,” Measurement 122, 275–283 (2018).
[Crossref]

Li, X.

G. Zhang, J. He, and X. Li, “3D vision inspection for internal surface based on circle structured light,” Sens. Actuators A 122(1), 68–75 (2005).
[Crossref]

Liu, Q.

J. Sun, G. Zhang, Q. Liu, and Z. Yang, “Universal Method for Calibrating Structured-light Vision Sensor on the Spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

Liu, Y.

Y. Fu and Y. Liu, “3D bubble reconstruction using multiple cameras and space carving method,” Meas. Sci. Technol. 29(7), 075206 (2018).
[Crossref]

L. Huang, X. Zhao, S. Cai, and Y. Liu, “Plate refractive camera model and its applications,” J. Electron. Imaging 26(2), 023020 (2017).
[Crossref]

Liu, Z.

Lu, S.

T. Wu, S. Lu, and Y. Tang, “An in-pipe internal defects inspection system based on the active stereo omnidirectional vision sensor,” in 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) (IEEE, 2015), pp. 2637-2641.

Morinaka, S.

S. Morinaka, F. Sakaue, J. Sato, K. Ishimaru, and N. Kawasaki, “3D reconstruction under light ray distortion from parametric focal cameras,” Pattern Recognit. Lett. to be published.
[Crossref]

Peng, B.

F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
[Crossref]

Pinto, T.

P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
[Crossref]

Sakaue, F.

S. Morinaka, F. Sakaue, J. Sato, K. Ishimaru, and N. Kawasaki, “3D reconstruction under light ray distortion from parametric focal cameras,” Pattern Recognit. Lett. to be published.
[Crossref]

Santos, J.

P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
[Crossref]

Sato, J.

S. Morinaka, F. Sakaue, J. Sato, K. Ishimaru, and N. Kawasaki, “3D reconstruction under light ray distortion from parametric focal cameras,” Pattern Recognit. Lett. to be published.
[Crossref]

Silva, F.

P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
[Crossref]

Sun, J.

J. Sun, G. Zhang, Q. Liu, and Z. Yang, “Universal Method for Calibrating Structured-light Vision Sensor on the Spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

Tan, H.

F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
[Crossref]

Tang, Y.

T. Wu, S. Lu, and Y. Tang, “An in-pipe internal defects inspection system based on the active stereo omnidirectional vision sensor,” in 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) (IEEE, 2015), pp. 2637-2641.

Tu, D.

C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
[Crossref]

Wakayama, T.

T. Yoshizawa and T. Wakayama, “Development of an inner profile measurement instrument using a ring beam device,” Proc. SPIE 7855, 78550B (2010).
[Crossref]

Wang, J.

Wang, J. Q.

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

Wang, Y.

Y. Wang and R. Zhang, “In-pipe surface circular structured light 3D vision inspection system,” Infrared Laser Eng. 43(3), 891–896 (2014).

F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
[Crossref]

Wu, T.

T. Wu, S. Lu, and Y. Tang, “An in-pipe internal defects inspection system based on the active stereo omnidirectional vision sensor,” in 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) (IEEE, 2015), pp. 2637-2641.

Xiao, L. F.

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

Xie, X.-P.

S.-Q. Li, X.-P. Xie, and Y.-J. Zhuang, “Research on the calibration technology of an underwater camera based on equivalent focal length,” Measurement 122, 275–283 (2018).
[Crossref]

Yang, B.

Yang, Z.

J. Sun, G. Zhang, Q. Liu, and Z. Yang, “Universal Method for Calibrating Structured-light Vision Sensor on the Spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

Yoshizawa, T.

T. Yoshizawa and T. Wakayama, “Development of an inner profile measurement instrument using a ring beam device,” Proc. SPIE 7855, 78550B (2010).
[Crossref]

Zhai, C.

Zhang, C.

C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
[Crossref]

Zhang, G.

Z. Gong, Z. Liu, and G. Zhang, “Flexible method of refraction correction in vision measurement systems with multiple glass ports,” Opt. Express 25(2), 831–847 (2017).
[Crossref]

J. Sun, G. Zhang, Q. Liu, and Z. Yang, “Universal Method for Calibrating Structured-light Vision Sensor on the Spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

G. Zhang, J. He, and X. Li, “3D vision inspection for internal surface based on circle structured light,” Sens. Actuators A 122(1), 68–75 (2005).
[Crossref]

Zhang, R.

Y. Wang and R. Zhang, “In-pipe surface circular structured light 3D vision inspection system,” Infrared Laser Eng. 43(3), 891–896 (2014).

Zhang, X.

C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
[Crossref]

Zhang, Z.

Z. Zhang, “A Flexible New Technique for Camera Calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Zhao, X.

L. Huang, X. Zhao, S. Cai, and Y. Liu, “Plate refractive camera model and its applications,” J. Electron. Imaging 26(2), 023020 (2017).
[Crossref]

Zheng, T.

Zheng, T. X.

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

Zhou, F.

F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
[Crossref]

Zhu, Y.

C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
[Crossref]

Y. Zhu, Y. Gu, Y. Jin, and C. Zhai, “Flexible calibration method for an inner surface detector based on circle structured light,” Appl. Opt. 55(5), 1034–1039 (2016).
[Crossref]

Zhuang, Y.-J.

S.-Q. Li, X.-P. Xie, and Y.-J. Zhuang, “Research on the calibration technology of an underwater camera based on equivalent focal length,” Measurement 122, 275–283 (2018).
[Crossref]

Appl. Opt. (2)

IEEE Trans. Pattern Anal. Mach. Intell. (1)

Z. Zhang, “A Flexible New Technique for Camera Calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[Crossref]

Infrared Laser Eng. (1)

Y. Wang and R. Zhang, “In-pipe surface circular structured light 3D vision inspection system,” Infrared Laser Eng. 43(3), 891–896 (2014).

IOP Conf. Ser. Mater. Sci. Eng. (1)

S. Huang, M. C. Feng, T. X. Zheng, F. Li, J. Q. Wang, and L. F. Xiao, “A Novel Multi-Camera Calibration Method based on Flat Refractive Geometry,” IOP Conf. Ser. Mater. Sci. Eng. 320(1), 012016 (2018).
[Crossref]

J. Electron. Imaging (1)

L. Huang, X. Zhao, S. Cai, and Y. Liu, “Plate refractive camera model and its applications,” J. Electron. Imaging 26(2), 023020 (2017).
[Crossref]

J. Mech. Eng. (1)

J. Sun, G. Zhang, Q. Liu, and Z. Yang, “Universal Method for Calibrating Structured-light Vision Sensor on the Spot,” J. Mech. Eng. 45(03), 174–177 (2009).
[Crossref]

J. Phys. Conf. Ser. (1)

P. Buschinelli, T. Pinto, F. Silva, J. Santos, and A. Albertazzi, “Laser Triangulation Profilometer for Inner Surface Inspection of 100 millimeters (4") Nominal Diameter,” J. Phys. Conf. Ser. 648, 012010 (2015).
[Crossref]

Meas. Sci. Technol. (2)

C. Zhang, X. Zhang, Y. Zhu, J. Li, and D. Tu, “Model and calibration of underwater stereo vision based on the light field,” Meas. Sci. Technol. 29(10), 105402 (2018).
[Crossref]

Y. Fu and Y. Liu, “3D bubble reconstruction using multiple cameras and space carving method,” Meas. Sci. Technol. 29(7), 075206 (2018).
[Crossref]

Measurement (1)

S.-Q. Li, X.-P. Xie, and Y.-J. Zhuang, “Research on the calibration technology of an underwater camera based on equivalent focal length,” Measurement 122, 275–283 (2018).
[Crossref]

Opt. Express (1)

Opt. Laser Technol. (1)

F. Zhou, B. Peng, Y. Cui, Y. Wang, and H. Tan, “A novel laser vision sensor for omnidirectional 3D measurement,” Opt. Laser Technol. 45(1), 1–12 (2013).
[Crossref]

Proc. SPIE (1)

T. Yoshizawa and T. Wakayama, “Development of an inner profile measurement instrument using a ring beam device,” Proc. SPIE 7855, 78550B (2010).
[Crossref]

Sens. Actuators A (1)

G. Zhang, J. He, and X. Li, “3D vision inspection for internal surface based on circle structured light,” Sens. Actuators A 122(1), 68–75 (2005).
[Crossref]

Other (2)

T. Wu, S. Lu, and Y. Tang, “An in-pipe internal defects inspection system based on the active stereo omnidirectional vision sensor,” in 2015 12th International Conference on Fuzzy Systems and Knowledge Discovery (FSKD) (IEEE, 2015), pp. 2637-2641.

S. Morinaka, F. Sakaue, J. Sato, K. Ishimaru, and N. Kawasaki, “3D reconstruction under light ray distortion from parametric focal cameras,” Pattern Recognit. Lett. to be published.
[Crossref]

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Figures (6)

Fig. 1.
Fig. 1. The measurement model of the omnidirectional structured-light vision.
Fig. 2.
Fig. 2. The two non-coplanar refraction planes (a) at ${Q_1}$. (b) at ${Q_0}$.
Fig. 3.
Fig. 3. Experimental devices. (a)Vision measurement system. (b)Calibration devices.
Fig. 4.
Fig. 4. Images captured by (a)the vision sensor camera; (b)the auxiliary camera.
Fig. 5.
Fig. 5. Measurement devices.
Fig. 6.
Fig. 6. The distorted points and the corrected ones (a) on the image plane; (b)in space.

Tables (5)

Tables Icon

Table 1. Calibration results of the parameters of the camera model

Tables Icon

Table 2. Calibration results of M, raxis, d and D

Tables Icon

Table 3. Results of the repeatability measurement mm

Tables Icon

Table 4. Results of the measurement experiments mm

Tables Icon

Table 5. Results of the comparison experiments mm

Equations (14)

Equations on this page are rendered with MathJax. Learn more.

||(xM)×raxis||=d
fi(x,y,z)=x~T[CSSTd2]x~
C=[r22+r32r1r2r1r3r1r2r12+r32r2r3r1r3r2r3r12+r22], S=CM.
n1=[CS]Q~1
fo(x,y,z)=x~T[CSSTD2]x~
n0=[CS]Q~0
rr=[n1×ren1n1×re×n1][01tanθ2]T
rr=[n1×ren1n1×re×n1][01tan(arcsin(nairnglasssinθ3))]T
ri=[n0×rrn0n0×rr×n0][01tan(arcsin(nairnglasssinθ0))]T
{riTx=0Aαx=0
j=1n||xcjxrj(M,raxis,d,D)||
sp~=HP~
Q~c=(Rt01)Q~i
Aα=[2.2580e(5)7.0373e(6)0.0016633.8265]

Metrics