Abstract

Binocular vision calibration is of great importance in 3D machine vision measurement. With respect to binocular vision calibration, the nonlinear optimization technique is a crucial step to improve the accuracy. The existing optimization methods mostly aim at minimizing the sum of reprojection errors for two cameras based on respective 2D image pixels coordinate. However, the subsequent measurement process is conducted in 3D coordinate system which is not consistent with the optimization coordinate system. Moreover, the error criterion with respect to optimization and measurement is different. The equal pixel distance error in 2D image plane leads to diverse 3D metric distance error at different position before the camera. To address these issues, we propose a precise calibration method for binocular vision system which is devoted to minimizing the metric distance error between the reconstructed point through optimal triangulation and the ground truth in 3D measurement coordinate system. In addition, the inherent epipolar constraint and constant distance constraint are combined to enhance the optimization process. To evaluate the performance of the proposed method, both simulative and real experiments have been carried out and the results show that the proposed method is reliable and efficient to improve measurement accuracy compared with conventional method.

© 2014 Optical Society of America

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2014 (1)

J. Wang, X. Wang, F. Liu, Y. Gong, H. Wang, Z. Qin, “Modeling of binocular stereo vision for remote coordinate measurement and fast calibration,” Opt. Lasers Eng. 54, 269–274 (2014).
[CrossRef]

2013 (4)

F. Zhou, Y. Wang, B. Peng, Y. Cui, “A novel way of understanding for calibrating stereo vision sensor constructed by a single camera and mirrors,” Meas. J. Int. Meas. Confed. 46(3), 1147–1160 (2013).
[CrossRef]

M. Xie, Z. Wei, G. Zhang, X. Wei, “A flexible technique for calibrating relative position and orientation of two cameras with no-overlapping FOV,” Meas. J. Int. Meas. Confed. 46(1), 34–44 (2013).
[CrossRef]

F. Zhou, Y. Cui, Y. Wang, L. Liu, H. Gao, “Accurate and robust estimation of camera parameters using RANSAC,” Opt. Lasers Eng. 51(3), 197–212 (2013).
[CrossRef]

F. Zhou, Y. Cui, H. Gao, Y. Wang, “Line-based camera calibration with lens distortion correction from a single image,” Opt. Lasers Eng. 51(12), 1332–1343 (2013).
[CrossRef]

2012 (4)

F. Zhou, Y. Cui, B. Peng, Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[CrossRef]

T. Xue, L. Qu, Z. Cao, T. Zhang, “Three-dimensional feature parameters measurement of bubbles in gas-liquid two-phase flow based on virtual stereo vision,” Flow Meas. Instrum. 27, 29–36 (2012).
[CrossRef]

Y. Zhao, X. Li, W. Li, “Binocular vision system calibration based on a one-dimensional target,” Appl. Opt. 51(16), 3338–3345 (2012).
[CrossRef] [PubMed]

H. Habe, Y. Nakamura, “Appearance-based parameter optimization for accurate stereo camera calibration,” Mach. Vis. Appl. 23(2), 313–325 (2012).
[CrossRef]

2011 (1)

J. Sun, Q. Liu, Z. Liu, G. Zhang, “A calibration method for stereo vision sensor with large FOV based on 1D targets,” Opt. Lasers Eng. 49(11), 1245–1250 (2011).
[CrossRef]

2010 (3)

S. Zhu, Y. Gao, “Noncontact 3-d coordinate measurement of cross-cutting feature points on the surface of a large-scale workpiece based on the machine vision method,” IEEE Trans. Instrum. Meas. 59(7), 1874–1887 (2010).
[CrossRef]

Z. Ren, J. Liao, L. Cai, “Three-dimensional measurement of small mechanical parts under a complicated background based on stereo vision,” Appl. Opt. 49(10), 1789–1801 (2010).
[CrossRef] [PubMed]

Y. Wan, Z. Miao, Z. Tang, “Robust and accurate fundamental matrix estimation with propagated matches,” Opt. Eng. 49(10), 107002 (2010).
[CrossRef]

2009 (2)

Y. Furukawa, J. Ponce, “Accurate camera calibration from multi-view stereo and bundle adjustment,” Int. J. Comput. Vis. 84(3), 257–268 (2009).
[CrossRef]

T. Dang, C. Hoffmann, C. Stiller, “Continuous stereo self-calibration by camera parameter tracking,” IEEE Trans. Image Process. 18(7), 1536–1550 (2009).
[CrossRef] [PubMed]

2008 (1)

P. F. Luo, J. Wu, “Easy calibration technique for stereo vision using a circle grid,” Opt. Eng. 47(3), 033607 (2008).
[CrossRef]

2007 (2)

T. Xue, B. Wu, J. G. Zhu, S. H. Ye, “Complete calibration of a structure-uniform stereovision sensor with free-position planar pattern,” Sens. Actuators A Phys. 135(1), 185–191 (2007).
[CrossRef]

F. Dornaika, “Self-calibration of a stereo rig using monocular epipolar geometries,” Pattern Recognit. 40(10), 2716–2729 (2007).
[CrossRef]

2003 (1)

M. Machacek, M. Sauter, T. Rosgen, “Two-step calibration of a stereo camera system for measurements in large volumes,” Meas. Sci. Technol. 14(9), 1631–1639 (2003).
[CrossRef]

2000 (2)

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[CrossRef]

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[CrossRef]

1998 (1)

Z. Zhang, “Determining the epipolar geometry and its uncertainty: A review,” Int. J. Comput. Vis. 27(2), 161–195 (1998).
[CrossRef]

1997 (1)

R. I. Hartley, P. Sturm, “Triangulation,” Comput. Vis. Image Underst. 68(2), 146–157 (1997).
[CrossRef]

1987 (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[CrossRef]

1981 (1)

H. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293(5828), 133–135 (1981).
[CrossRef]

1971 (1)

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. Remote Sensing 37, 855–866 (1971).

1966 (1)

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. Remote Sensing 32, 444–462 (1966).

Brown, D. C.

D. C. Brown, “Close-range camera calibration,” Photogramm. Eng. Remote Sensing 37, 855–866 (1971).

D. C. Brown, “Decentering distortion of lenses,” Photogramm. Eng. Remote Sensing 32, 444–462 (1966).

Cai, L.

Cao, Z.

T. Xue, L. Qu, Z. Cao, T. Zhang, “Three-dimensional feature parameters measurement of bubbles in gas-liquid two-phase flow based on virtual stereo vision,” Flow Meas. Instrum. 27, 29–36 (2012).
[CrossRef]

Cui, Y.

F. Zhou, Y. Wang, B. Peng, Y. Cui, “A novel way of understanding for calibrating stereo vision sensor constructed by a single camera and mirrors,” Meas. J. Int. Meas. Confed. 46(3), 1147–1160 (2013).
[CrossRef]

F. Zhou, Y. Cui, Y. Wang, L. Liu, H. Gao, “Accurate and robust estimation of camera parameters using RANSAC,” Opt. Lasers Eng. 51(3), 197–212 (2013).
[CrossRef]

F. Zhou, Y. Cui, H. Gao, Y. Wang, “Line-based camera calibration with lens distortion correction from a single image,” Opt. Lasers Eng. 51(12), 1332–1343 (2013).
[CrossRef]

F. Zhou, Y. Cui, B. Peng, Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[CrossRef]

Dang, T.

T. Dang, C. Hoffmann, C. Stiller, “Continuous stereo self-calibration by camera parameter tracking,” IEEE Trans. Image Process. 18(7), 1536–1550 (2009).
[CrossRef] [PubMed]

Dornaika, F.

F. Dornaika, “Self-calibration of a stereo rig using monocular epipolar geometries,” Pattern Recognit. 40(10), 2716–2729 (2007).
[CrossRef]

Furukawa, Y.

Y. Furukawa, J. Ponce, “Accurate camera calibration from multi-view stereo and bundle adjustment,” Int. J. Comput. Vis. 84(3), 257–268 (2009).
[CrossRef]

Gao, H.

F. Zhou, Y. Cui, Y. Wang, L. Liu, H. Gao, “Accurate and robust estimation of camera parameters using RANSAC,” Opt. Lasers Eng. 51(3), 197–212 (2013).
[CrossRef]

F. Zhou, Y. Cui, H. Gao, Y. Wang, “Line-based camera calibration with lens distortion correction from a single image,” Opt. Lasers Eng. 51(12), 1332–1343 (2013).
[CrossRef]

Gao, Y.

S. Zhu, Y. Gao, “Noncontact 3-d coordinate measurement of cross-cutting feature points on the surface of a large-scale workpiece based on the machine vision method,” IEEE Trans. Instrum. Meas. 59(7), 1874–1887 (2010).
[CrossRef]

Gong, Y.

J. Wang, X. Wang, F. Liu, Y. Gong, H. Wang, Z. Qin, “Modeling of binocular stereo vision for remote coordinate measurement and fast calibration,” Opt. Lasers Eng. 54, 269–274 (2014).
[CrossRef]

Habe, H.

H. Habe, Y. Nakamura, “Appearance-based parameter optimization for accurate stereo camera calibration,” Mach. Vis. Appl. 23(2), 313–325 (2012).
[CrossRef]

Hartley, R. I.

R. I. Hartley, P. Sturm, “Triangulation,” Comput. Vis. Image Underst. 68(2), 146–157 (1997).
[CrossRef]

Heikkila, J.

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pattern Anal. Mach. Intell. 22(10), 1066–1077 (2000).
[CrossRef]

Hoffmann, C.

T. Dang, C. Hoffmann, C. Stiller, “Continuous stereo self-calibration by camera parameter tracking,” IEEE Trans. Image Process. 18(7), 1536–1550 (2009).
[CrossRef] [PubMed]

Li, W.

Li, X.

Liao, J.

Liu, F.

J. Wang, X. Wang, F. Liu, Y. Gong, H. Wang, Z. Qin, “Modeling of binocular stereo vision for remote coordinate measurement and fast calibration,” Opt. Lasers Eng. 54, 269–274 (2014).
[CrossRef]

Liu, L.

F. Zhou, Y. Cui, Y. Wang, L. Liu, H. Gao, “Accurate and robust estimation of camera parameters using RANSAC,” Opt. Lasers Eng. 51(3), 197–212 (2013).
[CrossRef]

Liu, Q.

J. Sun, Q. Liu, Z. Liu, G. Zhang, “A calibration method for stereo vision sensor with large FOV based on 1D targets,” Opt. Lasers Eng. 49(11), 1245–1250 (2011).
[CrossRef]

Liu, Z.

J. Sun, Q. Liu, Z. Liu, G. Zhang, “A calibration method for stereo vision sensor with large FOV based on 1D targets,” Opt. Lasers Eng. 49(11), 1245–1250 (2011).
[CrossRef]

Longuet-Higgins, H.

H. Longuet-Higgins, “A computer algorithm for reconstructing a scene from two projections,” Nature 293(5828), 133–135 (1981).
[CrossRef]

Luo, P. F.

P. F. Luo, J. Wu, “Easy calibration technique for stereo vision using a circle grid,” Opt. Eng. 47(3), 033607 (2008).
[CrossRef]

Machacek, M.

M. Machacek, M. Sauter, T. Rosgen, “Two-step calibration of a stereo camera system for measurements in large volumes,” Meas. Sci. Technol. 14(9), 1631–1639 (2003).
[CrossRef]

Miao, Z.

Y. Wan, Z. Miao, Z. Tang, “Robust and accurate fundamental matrix estimation with propagated matches,” Opt. Eng. 49(10), 107002 (2010).
[CrossRef]

Nakamura, Y.

H. Habe, Y. Nakamura, “Appearance-based parameter optimization for accurate stereo camera calibration,” Mach. Vis. Appl. 23(2), 313–325 (2012).
[CrossRef]

Peng, B.

F. Zhou, Y. Wang, B. Peng, Y. Cui, “A novel way of understanding for calibrating stereo vision sensor constructed by a single camera and mirrors,” Meas. J. Int. Meas. Confed. 46(3), 1147–1160 (2013).
[CrossRef]

F. Zhou, Y. Cui, B. Peng, Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[CrossRef]

Ponce, J.

Y. Furukawa, J. Ponce, “Accurate camera calibration from multi-view stereo and bundle adjustment,” Int. J. Comput. Vis. 84(3), 257–268 (2009).
[CrossRef]

Qin, Z.

J. Wang, X. Wang, F. Liu, Y. Gong, H. Wang, Z. Qin, “Modeling of binocular stereo vision for remote coordinate measurement and fast calibration,” Opt. Lasers Eng. 54, 269–274 (2014).
[CrossRef]

Qu, L.

T. Xue, L. Qu, Z. Cao, T. Zhang, “Three-dimensional feature parameters measurement of bubbles in gas-liquid two-phase flow based on virtual stereo vision,” Flow Meas. Instrum. 27, 29–36 (2012).
[CrossRef]

Ren, Z.

Rosgen, T.

M. Machacek, M. Sauter, T. Rosgen, “Two-step calibration of a stereo camera system for measurements in large volumes,” Meas. Sci. Technol. 14(9), 1631–1639 (2003).
[CrossRef]

Sauter, M.

M. Machacek, M. Sauter, T. Rosgen, “Two-step calibration of a stereo camera system for measurements in large volumes,” Meas. Sci. Technol. 14(9), 1631–1639 (2003).
[CrossRef]

Stiller, C.

T. Dang, C. Hoffmann, C. Stiller, “Continuous stereo self-calibration by camera parameter tracking,” IEEE Trans. Image Process. 18(7), 1536–1550 (2009).
[CrossRef] [PubMed]

Sturm, P.

R. I. Hartley, P. Sturm, “Triangulation,” Comput. Vis. Image Underst. 68(2), 146–157 (1997).
[CrossRef]

Sun, J.

J. Sun, Q. Liu, Z. Liu, G. Zhang, “A calibration method for stereo vision sensor with large FOV based on 1D targets,” Opt. Lasers Eng. 49(11), 1245–1250 (2011).
[CrossRef]

Tang, Z.

Y. Wan, Z. Miao, Z. Tang, “Robust and accurate fundamental matrix estimation with propagated matches,” Opt. Eng. 49(10), 107002 (2010).
[CrossRef]

Tsai, R. Y.

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[CrossRef]

Wan, Y.

Y. Wan, Z. Miao, Z. Tang, “Robust and accurate fundamental matrix estimation with propagated matches,” Opt. Eng. 49(10), 107002 (2010).
[CrossRef]

Wang, H.

J. Wang, X. Wang, F. Liu, Y. Gong, H. Wang, Z. Qin, “Modeling of binocular stereo vision for remote coordinate measurement and fast calibration,” Opt. Lasers Eng. 54, 269–274 (2014).
[CrossRef]

Wang, J.

J. Wang, X. Wang, F. Liu, Y. Gong, H. Wang, Z. Qin, “Modeling of binocular stereo vision for remote coordinate measurement and fast calibration,” Opt. Lasers Eng. 54, 269–274 (2014).
[CrossRef]

Wang, X.

J. Wang, X. Wang, F. Liu, Y. Gong, H. Wang, Z. Qin, “Modeling of binocular stereo vision for remote coordinate measurement and fast calibration,” Opt. Lasers Eng. 54, 269–274 (2014).
[CrossRef]

Wang, Y.

F. Zhou, Y. Wang, B. Peng, Y. Cui, “A novel way of understanding for calibrating stereo vision sensor constructed by a single camera and mirrors,” Meas. J. Int. Meas. Confed. 46(3), 1147–1160 (2013).
[CrossRef]

F. Zhou, Y. Cui, Y. Wang, L. Liu, H. Gao, “Accurate and robust estimation of camera parameters using RANSAC,” Opt. Lasers Eng. 51(3), 197–212 (2013).
[CrossRef]

F. Zhou, Y. Cui, H. Gao, Y. Wang, “Line-based camera calibration with lens distortion correction from a single image,” Opt. Lasers Eng. 51(12), 1332–1343 (2013).
[CrossRef]

F. Zhou, Y. Cui, B. Peng, Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[CrossRef]

Wei, X.

M. Xie, Z. Wei, G. Zhang, X. Wei, “A flexible technique for calibrating relative position and orientation of two cameras with no-overlapping FOV,” Meas. J. Int. Meas. Confed. 46(1), 34–44 (2013).
[CrossRef]

Wei, Z.

M. Xie, Z. Wei, G. Zhang, X. Wei, “A flexible technique for calibrating relative position and orientation of two cameras with no-overlapping FOV,” Meas. J. Int. Meas. Confed. 46(1), 34–44 (2013).
[CrossRef]

Wu, B.

T. Xue, B. Wu, J. G. Zhu, S. H. Ye, “Complete calibration of a structure-uniform stereovision sensor with free-position planar pattern,” Sens. Actuators A Phys. 135(1), 185–191 (2007).
[CrossRef]

Wu, J.

P. F. Luo, J. Wu, “Easy calibration technique for stereo vision using a circle grid,” Opt. Eng. 47(3), 033607 (2008).
[CrossRef]

Xie, M.

M. Xie, Z. Wei, G. Zhang, X. Wei, “A flexible technique for calibrating relative position and orientation of two cameras with no-overlapping FOV,” Meas. J. Int. Meas. Confed. 46(1), 34–44 (2013).
[CrossRef]

Xue, T.

T. Xue, L. Qu, Z. Cao, T. Zhang, “Three-dimensional feature parameters measurement of bubbles in gas-liquid two-phase flow based on virtual stereo vision,” Flow Meas. Instrum. 27, 29–36 (2012).
[CrossRef]

T. Xue, B. Wu, J. G. Zhu, S. H. Ye, “Complete calibration of a structure-uniform stereovision sensor with free-position planar pattern,” Sens. Actuators A Phys. 135(1), 185–191 (2007).
[CrossRef]

Ye, S. H.

T. Xue, B. Wu, J. G. Zhu, S. H. Ye, “Complete calibration of a structure-uniform stereovision sensor with free-position planar pattern,” Sens. Actuators A Phys. 135(1), 185–191 (2007).
[CrossRef]

Zhang, G.

M. Xie, Z. Wei, G. Zhang, X. Wei, “A flexible technique for calibrating relative position and orientation of two cameras with no-overlapping FOV,” Meas. J. Int. Meas. Confed. 46(1), 34–44 (2013).
[CrossRef]

J. Sun, Q. Liu, Z. Liu, G. Zhang, “A calibration method for stereo vision sensor with large FOV based on 1D targets,” Opt. Lasers Eng. 49(11), 1245–1250 (2011).
[CrossRef]

Zhang, T.

T. Xue, L. Qu, Z. Cao, T. Zhang, “Three-dimensional feature parameters measurement of bubbles in gas-liquid two-phase flow based on virtual stereo vision,” Flow Meas. Instrum. 27, 29–36 (2012).
[CrossRef]

Zhang, Z.

Z. Zhang, “Determining the epipolar geometry and its uncertainty: A review,” Int. J. Comput. Vis. 27(2), 161–195 (1998).
[CrossRef]

Zhang, Z. Y.

Z. Y. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. Intell. 22(11), 1330–1334 (2000).
[CrossRef]

Zhao, Y.

Zhou, F.

F. Zhou, Y. Cui, Y. Wang, L. Liu, H. Gao, “Accurate and robust estimation of camera parameters using RANSAC,” Opt. Lasers Eng. 51(3), 197–212 (2013).
[CrossRef]

F. Zhou, Y. Cui, H. Gao, Y. Wang, “Line-based camera calibration with lens distortion correction from a single image,” Opt. Lasers Eng. 51(12), 1332–1343 (2013).
[CrossRef]

F. Zhou, Y. Wang, B. Peng, Y. Cui, “A novel way of understanding for calibrating stereo vision sensor constructed by a single camera and mirrors,” Meas. J. Int. Meas. Confed. 46(3), 1147–1160 (2013).
[CrossRef]

F. Zhou, Y. Cui, B. Peng, Y. Wang, “A novel optimization method of camera parameters used for vision measurement,” Opt. Laser Technol. 44(6), 1840–1849 (2012).
[CrossRef]

Zhu, J. G.

T. Xue, B. Wu, J. G. Zhu, S. H. Ye, “Complete calibration of a structure-uniform stereovision sensor with free-position planar pattern,” Sens. Actuators A Phys. 135(1), 185–191 (2007).
[CrossRef]

Zhu, S.

S. Zhu, Y. Gao, “Noncontact 3-d coordinate measurement of cross-cutting feature points on the surface of a large-scale workpiece based on the machine vision method,” IEEE Trans. Instrum. Meas. 59(7), 1874–1887 (2010).
[CrossRef]

Appl. Opt. (2)

Comput. Vis. Image Underst. (1)

R. I. Hartley, P. Sturm, “Triangulation,” Comput. Vis. Image Underst. 68(2), 146–157 (1997).
[CrossRef]

Flow Meas. Instrum. (1)

T. Xue, L. Qu, Z. Cao, T. Zhang, “Three-dimensional feature parameters measurement of bubbles in gas-liquid two-phase flow based on virtual stereo vision,” Flow Meas. Instrum. 27, 29–36 (2012).
[CrossRef]

IEEE J. Robot. Autom. (1)

R. Y. Tsai, “A versatile camera calibration technique for high-accuracy 3D machine vision metrology using off the-shelf TV cameras and lenses,” IEEE J. Robot. Autom. 3(4), 323–344 (1987).
[CrossRef]

IEEE Trans. Image Process. (1)

T. Dang, C. Hoffmann, C. Stiller, “Continuous stereo self-calibration by camera parameter tracking,” IEEE Trans. Image Process. 18(7), 1536–1550 (2009).
[CrossRef] [PubMed]

IEEE Trans. Instrum. Meas. (1)

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Figures (12)

Fig. 1
Fig. 1

The epipolar geometry.

Fig. 2
Fig. 2

3D optimization steps for binocular vision calibration.

Fig. 3
Fig. 3

Effects of pixel coordinates noise on structure parameters using different calibration techniques (initial solution, 2D optimizaion, 3D optimizaion): (a) rx ;(b) ry;(c) rz;(d) tx;(e) ty;(f) tz.

Fig. 4
Fig. 4

Effects of pixel coordinates noise on calibration accuracy using different calibration techniques (initial solution, 2D optimizaion, 3D optimizaion): (a) Ept ;(b) EF.

Fig. 5
Fig. 5

A sample of image pairs used for calibration [30]: (a) left image; (b) right image.

Fig. 6
Fig. 6

3D calibration error distribution and distance error statistics for training data using different technique: initial solution, 2D optimization and 3D optimization. (a)-(c): the 3D error distribution for all the points; (d)-(f): the 3D distance error statistics for all the points. The computation of 3D calibration error can be referred to Eq. (21).

Fig. 7
Fig. 7

Epipolar distance error statistics for training data using different technique: (a) initial solution; (b) 2D optimization; (c) 3D optimization. The computation of epipolar distance error can be referred to Eq. (22).

Fig. 8
Fig. 8

Comparative results of average measurement accuracy for testing data: (a) Ept; (b) EF.

Fig. 9
Fig. 9

3D calibration error distribution and distance error statistics for testing data using different techniques: 2D optimization and 3D optimization. (a)-(b): the 3D error distribution for all the points; (c)-(d): the 3D distance error statistics for all the points. The computation of 3D calibration error can be referred to Eq. (21).

Fig. 10
Fig. 10

Epipolar distance error statistics for testing data using different techniques: (a) 2D optimization; (b) 3D optimization. The computation of epipolar distance error can be referred to Eq. (22).

Fig. 11
Fig. 11

The feature points’ epipolar lines computed from the proposed method (depicted in the corrected image pair).

Fig. 12
Fig. 12

The reconstruction of all the feature points of testing data using the proposed method.

Tables (4)

Tables Icon

Table 1 Comparative result of intrinsic parameters and distortion coefficients for left camera

Tables Icon

Table 2 Comparative result of intrinsic parameters and distortion coefficients for right camera

Tables Icon

Table 3 Comparative result of structure parameters and calibration accuracy

Tables Icon

Table 4 Comparative results of measurement accuracy

Equations (22)

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λ l m ˜ l = A l ( R l | t l ) M ˜ , A l =[ f xl 0 u 0l 0 f yl v 0l 0 0 1 ]
λ r m ˜ r = A r ( R r | t r ) M ˜ , A r =[ f xr 0 u 0r 0 f yr v 0r 0 0 1 ]
m l d =[ 1+ k 1l r l 2 + k 2l r l 4 ] m l
m r d =[ 1+ k 1r r r 2 + k 2r r r 4 ] m r
m ˜ r T F m ˜ l =0with F= A r T [t] × R A l 1
R= R r R l 1 ,t= t r R r R l 1 t l
{ λ l m ˜ l = A l ( I|0 ) M ˜ λ r m ˜ r = A r ( R|t ) M ˜
d ( m l , m l ) 2 +d ( m r , m r ) 2
i=1 N j=1 L m l,i,j d m ^ l,i,j d ( A l , D l , R l,i , t l,i ) 2 + i=1 N j=1 L m r,i,j d m ^ r,i,j d ( A r , D r , R l,i , t l,i ,R,t) 2
R r =R R l , t r = R r R l 1 t l +t
m l d D l ( k 1l , k 2l ) m l , m r d D r ( k 1r , k 2r ) m r
m ˜ nl = A l 1 m ˜ l , m ˜ nr = A r 1 m ˜ r
m ˜ nr T E m ˜ nl =0with E= [t] × R
{ λ l m ˜ nl =( I|0 ) M ˜ λ r m ˜ nr =( R|t ) M ˜
d ( m nl , m nl ) 2 +d ( m nr , m nr ) 2 subject to m ˜ nr T E m ˜ nl =0
M ˜ =[ R l t l 0 T 1 ] M ˜ W
J 3D = i=1 N j=1 L M( R l,i , t l,i ) M ^ ( A l , A r , D l , D r ,R,t) 2
J e = i=1 N j=1 L (d ( m ˜ l,i,j , l l,i,j ) 2 +d ( m ˜ r,i,j , l r,i,j ) 2 )
J dis = p q L dis d( M ^ p , M ^ q ) 2
J= J 3D + J e + J dis
E pt = 1 n i=1 n M i M ^ i 2
E F = 1 2n i=1 n d ( m ˜ l,i , F T m ˜ r,i ) 2 +d ( m ˜ r,i ,F m ˜ l,i ) 2

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