A high precision 3D reconstruction method for bend tube axis based on binocular stereo vision

Junhua Sun; Yue Zhang; Xiaoqi Cheng

doi:10.1364/OE.27.002292

Optics Express
Vol. 27,
Issue 3,
pp. 2292-2304
(2019)
•https://doi.org/10.1364/OE.27.002292

A high precision 3D reconstruction method for bend tube axis based on binocular stereo vision

Junhua Sun, Yue Zhang, and Xiaoqi Cheng

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Junhua Sun, Yue Zhang, and Xiaoqi Cheng, "A high precision 3D reconstruction method for bend tube axis based on binocular stereo vision," Opt. Express 27, 2292-2304 (2019)

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Table of Contents Category
- Instrumentation, Measurement, and Optical Sensors
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The topics in this list come from the Optics and Photonics Topics applied to this article.

About this Article
History
- Original Manuscript: October 25, 2018
- Revised Manuscript: December 13, 2018
- Manuscript Accepted: December 13, 2018
- Published: January 25, 2019

Abstract

Previous stereo-vision based methods for measuring bent tube axis usually approximate the centerlines of image contours as the axis projection, inevitably resulting in reconstruction errors. This phenomenon is more significant as the tube diameter increases. In this paper, a perspective projection model for any cross section of bent tube was established. Based on this model, a way to locate the precise projected position on image planes of points laying on axis was proposed and 3D coordinates of axis points are reconstructed by binocular stereo vision. We measured three bent tubes with different diameters. Compared with classical approaches, this method effectively reduces the reconstruction errors with measurement accuracy practically independent of diameter.

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References

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Year
Author
Publication

X. Ji and G. Gong, “Study of Realization Technologies for 3D Reconstruction of Superchargers′ Impellers,” J. Mech. Sci. Technol. 25(3), 322–325 (2006).
Y. Wei and A. Thornton, “Tube production and assembly systems: the impact of compliance and variability on yield,” presented at Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Baltimore, Maryland, 10–13 Sept. (2000).
Extreme Networks white paper, “Virtual metropolitan area networks,” (Extreme Networks, 2001), http://www.extremenetworks.com/technology/whitepapers/vMAN.asp .
F. Gu, H. Zhao, X. Zhou, J. Li, P. Bu, and Z. Zhao, “Photometric invariant stereo matching method,” Opt. Express 23(25), 31779–31792 (2015).
[Crossref] [PubMed]
Z. Liu, X. Li, F. Li, and G. Zhang, “Flexible dynamic measurement method of three-dimensional surface profilometry based on multiple vision sensors,” Opt. Express 23(1), 384–400 (2015).
[Crossref] [PubMed]
A. S. Metronor, Homepage, “Metronor One,” (Metronor, 2011–2018), http://www.metronor.com .
Gesellschaft für optische Messtechnik mbH Homepage, “ATOS Core – Optical 3D Scanner,” (GOM, 2011–2018), https:// www.gom.com/metrology-systems/atos/atos-core.html .
W. Bösemann, “Industrial photogrammetry-accepted METROLOGY tool or exotic niche,” presented at the International Archives of the Photogrammetry, Remote Sensing & Spatial Information Sciences, Prague, Czech Republic,12–19 July. (2016).
AICON 3D Systems GmbH Homepage, “Aicon-scanner,” (AICON, 2011–2018), https://www.aicon3d.com/de-DE/products/aicon-scanner .
AICON 3D Systems GmbH Homepage, “Tubeinspect,”(AICON, 2011–2018), https://www.aicon3d.com/de-DE/products/tubeinspect .
P. Jin, J. Liu, S. Liu, and X. Wang, “Automatic multi-stereo-vision reconstruction method of complicated tubes for industrial assembly,” Assem. Autom. 36(4), 362–375 (2016).
[Crossref]
S. Liu, J. Liu, P. Jin, and X. Wang, “Tube measurement based on stereo-vision: a review,” Int. J. Adv. Manuf. Technol. 92(5-8), 2017–2032 (2017).
[Crossref]
T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]
T. Zhang, J. Liu, C. Tang, and S. Liu, “3D reconstruction method of bent tube with NURBS curve,” in 2nd International Conference on Mechatronics and Applied Mechanics (ICMAM2012), C.K. Wang and J.Guo, ed. (Applied Mechanics and Materials, 2013), pp. 112–118.
[Crossref]
S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]
C. Doignon and M. de Mathelin, “A degenerate conic-based method for a direct fitting and 3-d pose of cylinders with a single perspective view,” in Proceedings of IEEE International Conference on Robotics and Automation, (IEEE, 2007), pp. 4220–4225.
[Crossref]
C. Liu and W. Hu, “Relative pose estimation for cylinder-shaped spacecrafts using single image,” in IEEE Transactions on Aerospace and Electronic Systems, (IEEE, 2014), pp. 3036–3056.
K. Wong, P. Mendonça, and R. Cipolla, “Reconstruction of surfaces of revolution from single uncalibrated views,” Image Vis. Comput. 22(10), 829–836 (2004).
[Crossref]
W. Bösemann, “The optical tube measurement system OLM photogrammetric methods used for industrial automation and process control,” presented at International Archives of Photogrammetry and Remote Sensing, Part B5, Vienna (1996).
I.-K. Lee, “Curve reconstruction from unorganized points,” Comput. Aided Geom. Des. 17(2), 161–177 (2000).
[Crossref]
I.-K. Lee and K.-J. Kim, “Shrinking: Another method for surface reconstruction,” in Geometric Modeling and Processing, 2004. Proceedings, (IEEE, 2004), pp. 259–266.
Caglioti and A. Giusti, “Reconstruction of canal surfaces from single images under exact perspective,” in European Conference on Computer Vision, A. Prinz, ed. (Springer, 2006), pp. 289–300.

2017 (3)

S. Liu, J. Liu, P. Jin, and X. Wang, “Tube measurement based on stereo-vision: a review,” Int. J. Adv. Manuf. Technol. 92(5-8), 2017–2032 (2017).
[Crossref]

T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]

S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]

2016 (1)

P. Jin, J. Liu, S. Liu, and X. Wang, “Automatic multi-stereo-vision reconstruction method of complicated tubes for industrial assembly,” Assem. Autom. 36(4), 362–375 (2016).
[Crossref]

2015 (2)

F. Gu, H. Zhao, X. Zhou, J. Li, P. Bu, and Z. Zhao, “Photometric invariant stereo matching method,” Opt. Express 23(25), 31779–31792 (2015).
[Crossref] [PubMed]

Z. Liu, X. Li, F. Li, and G. Zhang, “Flexible dynamic measurement method of three-dimensional surface profilometry based on multiple vision sensors,” Opt. Express 23(1), 384–400 (2015).
[Crossref] [PubMed]

2006 (1)

X. Ji and G. Gong, “Study of Realization Technologies for 3D Reconstruction of Superchargers′ Impellers,” J. Mech. Sci. Technol. 25(3), 322–325 (2006).

2004 (1)

K. Wong, P. Mendonça, and R. Cipolla, “Reconstruction of surfaces of revolution from single uncalibrated views,” Image Vis. Comput. 22(10), 829–836 (2004).
[Crossref]

2000 (1)

I.-K. Lee, “Curve reconstruction from unorganized points,” Comput. Aided Geom. Des. 17(2), 161–177 (2000).
[Crossref]

Bu, P.

F. Gu, H. Zhao, X. Zhou, J. Li, P. Bu, and Z. Zhao, “Photometric invariant stereo matching method,” Opt. Express 23(25), 31779–31792 (2015).
[Crossref] [PubMed]

Cipolla, R.

K. Wong, P. Mendonça, and R. Cipolla, “Reconstruction of surfaces of revolution from single uncalibrated views,” Image Vis. Comput. 22(10), 829–836 (2004).
[Crossref]

de Mathelin, M.

C. Doignon and M. de Mathelin, “A degenerate conic-based method for a direct fitting and 3-d pose of cylinders with a single perspective view,” in Proceedings of IEEE International Conference on Robotics and Automation, (IEEE, 2007), pp. 4220–4225.
[Crossref]

Doignon, C.

Gong, G.

X. Ji and G. Gong, “Study of Realization Technologies for 3D Reconstruction of Superchargers′ Impellers,” J. Mech. Sci. Technol. 25(3), 322–325 (2006).

Gu, F.

F. Gu, H. Zhao, X. Zhou, J. Li, P. Bu, and Z. Zhao, “Photometric invariant stereo matching method,” Opt. Express 23(25), 31779–31792 (2015).
[Crossref] [PubMed]

Ji, X.

X. Ji and G. Gong, “Study of Realization Technologies for 3D Reconstruction of Superchargers′ Impellers,” J. Mech. Sci. Technol. 25(3), 322–325 (2006).

Jin, P.

S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]

S. Liu, J. Liu, P. Jin, and X. Wang, “Tube measurement based on stereo-vision: a review,” Int. J. Adv. Manuf. Technol. 92(5-8), 2017–2032 (2017).
[Crossref]

T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]

P. Jin, J. Liu, S. Liu, and X. Wang, “Automatic multi-stereo-vision reconstruction method of complicated tubes for industrial assembly,” Assem. Autom. 36(4), 362–375 (2016).
[Crossref]

Lee, I.-K.

I.-K. Lee, “Curve reconstruction from unorganized points,” Comput. Aided Geom. Des. 17(2), 161–177 (2000).
[Crossref]

Li, F.

Li, J.

F. Gu, H. Zhao, X. Zhou, J. Li, P. Bu, and Z. Zhao, “Photometric invariant stereo matching method,” Opt. Express 23(25), 31779–31792 (2015).
[Crossref] [PubMed]

Li, X.

Liu, J.

S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]

T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]

S. Liu, J. Liu, P. Jin, and X. Wang, “Tube measurement based on stereo-vision: a review,” Int. J. Adv. Manuf. Technol. 92(5-8), 2017–2032 (2017).
[Crossref]

P. Jin, J. Liu, S. Liu, and X. Wang, “Automatic multi-stereo-vision reconstruction method of complicated tubes for industrial assembly,” Assem. Autom. 36(4), 362–375 (2016).
[Crossref]

Liu, S.

S. Liu, J. Liu, P. Jin, and X. Wang, “Tube measurement based on stereo-vision: a review,” Int. J. Adv. Manuf. Technol. 92(5-8), 2017–2032 (2017).
[Crossref]

T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]

S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]

P. Jin, J. Liu, S. Liu, and X. Wang, “Automatic multi-stereo-vision reconstruction method of complicated tubes for industrial assembly,” Assem. Autom. 36(4), 362–375 (2016).
[Crossref]

Liu, Z.

Mendonça, P.

K. Wong, P. Mendonça, and R. Cipolla, “Reconstruction of surfaces of revolution from single uncalibrated views,” Image Vis. Comput. 22(10), 829–836 (2004).
[Crossref]

Sun, P.

S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]

Tang, C.

T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]

Wang, X.

S. Liu, J. Liu, P. Jin, and X. Wang, “Tube measurement based on stereo-vision: a review,” Int. J. Adv. Manuf. Technol. 92(5-8), 2017–2032 (2017).
[Crossref]

S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]

P. Jin, J. Liu, S. Liu, and X. Wang, “Automatic multi-stereo-vision reconstruction method of complicated tubes for industrial assembly,” Assem. Autom. 36(4), 362–375 (2016).
[Crossref]

Wong, K.

K. Wong, P. Mendonça, and R. Cipolla, “Reconstruction of surfaces of revolution from single uncalibrated views,” Image Vis. Comput. 22(10), 829–836 (2004).
[Crossref]

Zhang, G.

Zhang, T.

T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]

Assem. Autom. (1)

P. Jin, J. Liu, S. Liu, and X. Wang, “Automatic multi-stereo-vision reconstruction method of complicated tubes for industrial assembly,” Assem. Autom. 36(4), 362–375 (2016).
[Crossref]

Chin. J. Mech. Eng. (1)

S. Liu, P. Jin, J. Liu, X. Wang, and P. Sun, “Accurate measurement method for tube’s endpoints based on machine vision,” Chin. J. Mech. Eng. 30(1), 152–163 (2017).
[Crossref]

Comput. Aided Geom. Des. (1)

I.-K. Lee, “Curve reconstruction from unorganized points,” Comput. Aided Geom. Des. 17(2), 161–177 (2000).
[Crossref]

Image Vis. Comput. (1)

K. Wong, P. Mendonça, and R. Cipolla, “Reconstruction of surfaces of revolution from single uncalibrated views,” Image Vis. Comput. 22(10), 829–836 (2004).
[Crossref]

Int. J. Adv. Manuf. Technol. (1)

S. Liu, J. Liu, P. Jin, and X. Wang, “Tube measurement based on stereo-vision: a review,” Int. J. Adv. Manuf. Technol. 92(5-8), 2017–2032 (2017).
[Crossref]

J. Mech. Sci. Technol. (1)

X. Ji and G. Gong, “Study of Realization Technologies for 3D Reconstruction of Superchargers′ Impellers,” J. Mech. Sci. Technol. 25(3), 322–325 (2006).

Measurement (1)

T. Zhang, J. Liu, S. Liu, C. Tang, and P. Jin, “A 3D reconstruction method for pipeline inspection based on multi-vision,” Measurement 98, 35–48 (2017).
[Crossref]

Opt. Express (2)

F. Gu, H. Zhao, X. Zhou, J. Li, P. Bu, and Z. Zhao, “Photometric invariant stereo matching method,” Opt. Express 23(25), 31779–31792 (2015).
[Crossref] [PubMed]

Other (13)

A. S. Metronor, Homepage, “Metronor One,” (Metronor, 2011–2018), http://www.metronor.com .

Gesellschaft für optische Messtechnik mbH Homepage, “ATOS Core – Optical 3D Scanner,” (GOM, 2011–2018), https:// www.gom.com/metrology-systems/atos/atos-core.html .

W. Bösemann, “Industrial photogrammetry-accepted METROLOGY tool or exotic niche,” presented at the International Archives of the Photogrammetry, Remote Sensing & Spatial Information Sciences, Prague, Czech Republic,12–19 July. (2016).

AICON 3D Systems GmbH Homepage, “Aicon-scanner,” (AICON, 2011–2018), https://www.aicon3d.com/de-DE/products/aicon-scanner .

AICON 3D Systems GmbH Homepage, “Tubeinspect,”(AICON, 2011–2018), https://www.aicon3d.com/de-DE/products/tubeinspect .

T. Zhang, J. Liu, C. Tang, and S. Liu, “3D reconstruction method of bent tube with NURBS curve,” in 2nd International Conference on Mechatronics and Applied Mechanics (ICMAM2012), C.K. Wang and J.Guo, ed. (Applied Mechanics and Materials, 2013), pp. 112–118.
[Crossref]

Y. Wei and A. Thornton, “Tube production and assembly systems: the impact of compliance and variability on yield,” presented at Design Engineering Technical Conferences and Computer and Information in Engineering Conference, Baltimore, Maryland, 10–13 Sept. (2000).

Extreme Networks white paper, “Virtual metropolitan area networks,” (Extreme Networks, 2001), http://www.extremenetworks.com/technology/whitepapers/vMAN.asp .

C. Liu and W. Hu, “Relative pose estimation for cylinder-shaped spacecrafts using single image,” in IEEE Transactions on Aerospace and Electronic Systems, (IEEE, 2014), pp. 3036–3056.

W. Bösemann, “The optical tube measurement system OLM photogrammetric methods used for industrial automation and process control,” presented at International Archives of Photogrammetry and Remote Sensing, Part B5, Vienna (1996).

I.-K. Lee and K.-J. Kim, “Shrinking: Another method for surface reconstruction,” in Geometric Modeling and Processing, 2004. Proceedings, (IEEE, 2004), pp. 259–266.

Caglioti and A. Giusti, “Reconstruction of canal surfaces from single images under exact perspective,” in European Conference on Computer Vision, A. Prinz, ed. (Springer, 2006), pp. 289–300.

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Fig. 1 Projection error of bent tube axis.

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Fig. 2 Perspective projection model of arbitrary cross section

n

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Fig. 3 Projected process of coupled contour points.

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Fig. 4 Procedure of bent tube axis reconstruction.

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Fig. 5 Binocular vision measurement system.

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Fig. 6 Result for tube with diameter about 6 mm. (a) Tube to be measured; (b) Reconstruction result.

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Fig. 7 Result for tube with diameter about 15 mm. (a) Tube to be measured; (b) Reconstruction result.

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Fig. 8 Result for tube with diameter about 24 mm. (a) Tube to be measured; (b) Reconstruction result.

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Fig. 9 Measurement results of bent tube images with noise.

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Tables (10)

Table 1 Calibration results of binocular stereo visual sensors

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Table 2 Results of the tube with about 6 mm diameter

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Table 3 Results of the tube with about 15 mm diameter

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Table 4 Results of the tube with about 24 mm diameter

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Table 5 Results of the tube with about 6 mm diameter by traditional method

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Table 6 Comparison results of two methods

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Table 7 Results of the tube with about 15mm diameter by traditional method

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Table 8 Comparison results of two methods

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Table 9 Results of the tube with about 24 mm diameter by traditional method

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Table 10 Comparison results of two methods

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Equations (11)

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

{\tilde{P}}_{1} = k_{1} {[\begin{matrix} A^{- 1} & 0 \end{matrix}]}^{T} {\tilde{P}}_{1}^{'} + {[\begin{matrix} 0 & 0 & 0 & 1 \end{matrix}]}^{T}

B = A^{- 1} = [\begin{matrix} b_{11} & b_{12} & b_{13} \\ b_{21} & b_{22} & b_{23} \\ b_{31} & b_{32} & b_{33} \end{matrix}]

[\begin{array}{l} X_{1} \\ Y_{1} \\ Z_{1} \end{array}] = [\begin{array}{l} k_{1} (b_{11} x_{1} + b_{12} y_{1} + b_{13}) \\ k_{1} (b_{21} x_{1} + b_{22} y_{1} + b_{23}) \\ k_{1} (b_{31} x_{1} + b_{32} y_{1} + b_{33}) \end{array}] = [\begin{array}{l} k_{1} γ_{11} \\ k_{1} γ_{12} \\ k_{1} γ_{13} \end{array}]

[\begin{array}{l} X_{1} \\ Y_{1} \\ Z_{1} \end{array}] = [\begin{array}{l} γ_{11} / (n_{1} γ_{11} + n_{2} γ_{12} + n_{3} γ_{13}) d \\ γ_{12} / (n_{1} γ_{11} + n_{2} γ_{12} + n_{3} γ_{13}) d \\ γ_{13} / (n_{1} γ_{11} + n_{2} γ_{12} + n_{3} γ_{13}) d \end{array}] = [\begin{array}{l} η_{11} d \\ η_{12} d \\ η_{13} d \end{array}]

{\tilde{P}}_{c} = k_{c} {[\begin{matrix} B & 0 \end{matrix}]}^{T} {\tilde{P}}_{c}^{'} + {[\begin{matrix} 0 & 0 & 0 & 1 \end{matrix}]}^{T}

[\begin{matrix} k_{c} n_{1} & k_{c} n_{2} & k_{c} n_{3} \\ T_{31} & T_{32} & T_{33} \\ k_{c} (η_{11} - η_{21}) & k_{c} (η_{12} - η_{22}) & k_{c} (η_{13} - η_{23}) \end{matrix}] [\begin{matrix} θ_{1} \\ θ_{2} \\ θ_{3} \end{matrix}] = d [\begin{matrix} 1 \\ 0 \\ Ω \end{matrix}]

{\begin{cases} B {\tilde{P}}_{c}^{'} = {[\begin{matrix} θ_{1} & θ_{2} & θ_{3} \end{matrix}]}^{T} \\ 1 / 2 (η_{11}^{2} - η_{21}^{2} + η_{12}^{2} - η_{22}^{2} + η_{13}^{2} - η_{23}^{2}) = Ω \end{cases}

{[\begin{matrix} ψ_{1} & ψ_{2} & ψ_{3} \end{matrix}]}^{T} = n - \frac{1}{Ω} {[\begin{matrix} η_{11} - η_{21} & η_{12} - η_{22} & η_{13} - η_{23} \end{matrix}]}^{T}

([\begin{array}{l} \begin{matrix} ψ_{1} & ψ_{2} & ψ_{3} \end{matrix} \\ \begin{matrix} T_{31} & T_{32} & T_{33} \end{matrix} \end{array}] B) {\tilde{P}}_{c}^{'} = 0

\frac{{\tilde{P}}_{1}^{'}^{T} ω {\tilde{V}}_{h}}{\sqrt{{\tilde{P}}_{1}^{'}^{T} ω {\tilde{P}}_{1}^{'}} \sqrt{{\tilde{V}}_{h}^{T} ω {\tilde{V}}_{h}}} = \frac{{\tilde{P}}_{2}^{'}^{T} ω {\tilde{V}}_{h}}{\sqrt{{\tilde{P}}_{2}^{'}^{T} ω {\tilde{P}}_{2}^{'}} \sqrt{{\tilde{V}}_{h}^{T} ω {\tilde{V}}_{h}}}

J_{c} (P_{1}^{'}, P_{2}^{'}) = {[arc \cos (\frac{{\tilde{P}}_{1}^{'}^{T} ω {\tilde{V}}_{h}}{\sqrt{{\tilde{P}}_{1}^{'}^{T} ω {\tilde{P}}_{1}^{'}} \sqrt{{\tilde{V}}_{h}^{T} ω {\tilde{V}}_{h}}}) - arc \cos (\frac{{\tilde{P}}_{2}^{'}^{T} ω {\tilde{V}}_{h}}{\sqrt{{\tilde{P}}_{2}^{'}^{T} ω {\tilde{P}}_{2}^{'}} \sqrt{{\tilde{V}}_{h}^{T} ω {\tilde{V}}_{h}}})]}^{2}

Parameters	Left camera	Right camera
Intrinsic matrix	$A_{l} = [\begin{matrix} 1964.23822 & 0 & 709.88099 \\ 0 & 1964.29461 & 548.68461 \\ 0 & 0 & 1 \end{matrix}]$	$A_{r} = [\begin{matrix} 1963.98789 & 0 & 699.9736 \\ 0 & 1964.0666 & 537.2184 \\ 0 & 0 & 1 \end{matrix}]$
Distortion coefficients	$k_{l 1} = -0 .1072$ $k_{l 2} = 0 .08346$ $k_{l 3} = 0.00001$ $k_{l 4} = - 0.00003$	$k_{r 1} = -0 .10365$ $k_{r 2} = 0 .13459$ $k_{r 3} = 0.00007$ $k_{r 4} = 0.00006$
Rotation vector	$O M = {[\begin{matrix} 0.02121 & 0.84063 & 0.02497 \end{matrix}]}^{T}$
Translation vector	$T_{0} = {[\begin{matrix} - 617.53569 & 10.44938 & 253.56603 \end{matrix}]}^{T}$

count	Angle1	Angle2	R1	R2
1	112.25	99.83	25.22	24.86
2	112.30	99.82	25.21	24.85
3	112.30	99.83	25.19	24.87
4	112.26	99.77	25.22	24.85
5	112.28	99.79	25.19	24.82
6	112.27	99.82	25.17	24.87
7	112.28	99.80	25.22	24.82
8	112.26	99.83	25.17	24.83
9	112.31	99.82	25.17	24.86
10	112.28	99.82	25.18	24.82
RMSE	0.020°	0.020°	0.022 mm	0.021 mm
RMSD	0.030°	0.031°	0.029 mm	0.029 mm

count	Angle1	Angle2	R1	R2
1	100.67	122.42	63.24	65.68
2	100.63	122.47	63.23	65.64
3	100.68	122.45	63.25	65.68
4	100.66	122.40	63.21	65.71
5	100.65	122.46	63.26	65.71
6	100.64	122.43	63.28	65.67
7	100.66	122.44	63.25	65.66
8	100.65	122.40	63.22	65.65
9	100.70	122.43	63.26	65.66
10	100.69	122.42	63.27	65.66
RMSE	0.022°	0.023°	0.019 mm	0.023 mm
RMSD	0.031°	0.029°	0.028 mm	0.029 mm

count	Angle1	Angle2	R1	R2
1	87.99	108.91	72.35	72.68
2	87.99	108.96	72.31	72.67
3	87.98	108.94	72.36	72.63
4	87.96	108.93	72.32	72.63
5	87.97	108.92	72.32	72.65
6	87.95	108.94	72.33	72.69
7	87.94	108.92	72.35	72.65
8	87.97	108.91	72.35	72.64
9	87.97	108.93	72.36	72.63
10	87.99	108.89	72.33	72.65
RMSE	0.017°	0.020°	0.018 mm	0.021 mm
RMSD	0.027°	0.031°	0.026 mm	0.029 mm

count	Angle1	Angle2	R1	R2
1	112.28	99.83	25.23	24.87
2	112.31	99.83	25.21	24.86
3	112.28	99.84	25.22	24.88
4	112.27	99.80	25.23	24.83
5	112.30	99.80	25.19	24.84
6	112.28	99.82	25.17	24.82
7	112.28	99.83	25.21	24.85
8	112.27	99.83	25.17	24.84
9	112.32	99.82	25.18	24.87
10	112.26	99.78	25.20	24.82
RMSE	0.019°	0.019°	0.023 mm	0.021 mm
RMSD	0.035°	0.034°	0.035 mm	0.032 mm

method	item	Angle1	Angle2	R1	R2
1	RMSD	0.031°	0.029°	0.030 mm	0.029 mm
2	RMSD	0.037°	0.038°	0.042 mm	0.038 mm
Increasing ratio		19.4%	31.0%	40%	31.0%

method	item	Angle1	Angle2	R1	R2
1	RMSD	0.027°	0.031°	0.026 mm	0.029 mm
2	RMSD	0.040°	0.044°	0.041 mm	0.041 mm
Increasing ratio		48.1%	41.9%	57.7%	41.4%

method	item	Angle1	Angle2	R1	R2
1	RMSD	0.031°	0.029°	0.030 mm	0.029 mm
2	RMSD	0.037°	0.038°	0.042 mm	0.038 mm
Increasing ratio		19.4%	31.0%	40%	31.0%

method	item	Angle1	Angle2	R1	R2
1	RMSD	0.027°	0.031°	0.026 mm	0.029 mm
2	RMSD	0.040°	0.044°	0.041 mm	0.041 mm
Increasing ratio		48.1%	41.9%	57.7%	41.4%

Abstract

References

2017 (3)

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2015 (2)

2006 (1)

2004 (1)

2000 (1)

Bu, P.

Cipolla, R.

de Mathelin, M.

Doignon, C.

Gong, G.

Gu, F.

Ji, X.

Jin, P.

Lee, I.-K.

Li, F.

Li, J.

Li, X.

Liu, J.

Liu, S.

Liu, Z.

Mendonça, P.

Sun, P.

Tang, C.

Wang, X.

Wong, K.

Zhang, G.

Zhang, T.

Zhao, H.

Zhao, Z.

Zhou, X.

Assem. Autom. (1)

Chin. J. Mech. Eng. (1)

Comput. Aided Geom. Des. (1)

Image Vis. Comput. (1)

Int. J. Adv. Manuf. Technol. (1)

J. Mech. Sci. Technol. (1)

Measurement (1)

Opt. Express (2)

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