Abstract

A flexible reconstruction method, which is based on the line re-projection errors of the laser plane, is presented for the profile recovery. The bi-cuboid references are designed to cover the large view-field of the camera. The local intrinsic and extrinsic parameter matrices of the camera are initially contributed by the RQ decomposition. Then the balance model is demonstrated to obtain the global parameter matrices in view of the refined projection in the camera coordinate system. The flexible laser plane is solved by the Plücker matrices of the projection laser lines that are generated from the homographies of the cubical references and the global parameter matrices. Furthermore, the laser plane and global parameter matrices are improved by the cost function that is constructed by the re-projection errors of the parameterized laser lines on the references. The reconstruction experiments are performed to verify the validity and the accuracy of the optimization method and the initial method. The impact factors of the measurement distance, the reference distance and the test distance are investigated in the experiments. The average reconstruction errors are 1.14 mm, 1.13 mm, 1.15 mm and 1.17 mm in the four groups of experiments, which shows the good application prospect of the profile reconstructions.

© 2017 Optical Society of America

Full Article  |  PDF Article
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References

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2017 (5)

2016 (3)

G. Xu, X. Zhang, J. Su, X. Li, and A. Zheng, “Solution approach of a laser plane based on Plücker matrices of the projective lines on a flexible 2D target,” Appl. Opt. 55(10), 2653–2656 (2016).
[PubMed]

A. Glowacz and Z. Glowacz, “Diagnostics of stator faults of the single-phase induction motor using thermal images, moasos and selected classifiers,” Measurement 93, 86–93 (2016).

H. Zhang, K. Iijima, J. Huang, G. P. Walcott, and J. M. Rogers, “Optical mapping of membrane potential and epicardial deformation in beating hearts,” Biophys. J. 111(2), 438–451 (2016).
[PubMed]

2015 (1)

C. Wei, C. Sihai, L. Dong, and J. Guohua, “A compact two-dimensional laser scanner based on piezoelectric actuators,” Rev. Sci. Instrum. 86(1), 013102 (2015).
[PubMed]

2014 (1)

R. S. Lim, H. M. La, and W. Sheng, “A robotic crack inspection and mapping system for bridge deck maintenance,” IEEE Trans. Autom. Sci. Eng. 11(2), 367–378 (2014).

2013 (1)

2012 (3)

F. J. Brosed, J. Santolaria, J. J. Aguilar, and D. Guillomía, “Laser triangulation sensor and six axes anthropomorphic robot manipulator modelling for the measurement of complex geometry products,” Robot. Com.- Int. Manuf. 28(6), 660–671 (2012).

X. Cui, K. B. Lim, Q. Guo, and D. Wang, “Accurate geometrical optics model for single-lens stereovision system using a prism,” J. Opt. Soc. Am. A 29(9), 1828–1837 (2012).
[PubMed]

J. E. Ha, “Extrinsic calibration of a camera and laser range finder using a new calibration structure of a plane with a triangular hole,” Int. J. Control Autom. 10(6), 1240–1244 (2012).

2011 (3)

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).

J. Santolaria, J. J. Aguilar, D. Guillomía, and C. Cajal, “A crenellated target-based calibration method for laser triangulation sensors integration in articulated measurement arms,” Robot. Com.- Int. Manuf. 27(2), 282–291 (2011).

C. H. Lee, Y. C. Lim, S. Kwon, and J. H. Lee, “Stereo vision-based vehicle detection using a road feature and disparity histogram,” Opt. Eng. 50(2), 027004 (2011).

2009 (1)

J. Santolaria, J. J. Pastor, F. J. Brosed, and J. J. Aguilar, “A one-step intrinsic and extrinsic calibration method for laser line scanner operation in coordinate measuring machines,” Meas. Sci. Technol. 20(4), 045107 (2009).

2006 (1)

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45(8), 083601 (2006).

2005 (1)

M. Bleyer and M. Gelautz, “A layered stereo matching algorithm using image segmentation and global visibility constraints,” ISPRS J. Photogramm. 59(3), 128–150 (2005).

2004 (1)

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pat. Anal. Mach. Intell. 26(7), 892–899 (2004).
[PubMed]

2002 (1)

H. Hirschmüller, P. R. Innocent, and J. Garibaldi, “Real-time correlation-based stereo vision with reduced border errors,” Int. J. Comput. Vis. 47(1–3), 229–246 (2002).

2000 (3)

C. L. Zitnick and T. Kanade, “A cooperative algorithm for stereo matching and occlusion detection,” IEEE Trans. Pat. Anal. 22(7), 675–684 (2000).

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pat. Anal. 22(11), 1330–1334 (2000).

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pat. Anal. 22(10), 1066–1077 (2000).

1998 (1)

C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pat. Anal. 20(2), 113–125 (1998).

1987 (1)

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

Abdel-Aziz, Y. I.

Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation into object space coordinates in close-range photogrammetry,” in Proceedings of the Symposium on Close-Range Photogrammetry (Falls Church, VA, USA, 1971), pp. 1–18.

Agis, K. Ö.

Aguilar, J. J.

F. J. Brosed, J. Santolaria, J. J. Aguilar, and D. Guillomía, “Laser triangulation sensor and six axes anthropomorphic robot manipulator modelling for the measurement of complex geometry products,” Robot. Com.- Int. Manuf. 28(6), 660–671 (2012).

J. Santolaria, J. J. Aguilar, D. Guillomía, and C. Cajal, “A crenellated target-based calibration method for laser triangulation sensors integration in articulated measurement arms,” Robot. Com.- Int. Manuf. 27(2), 282–291 (2011).

J. Santolaria, J. J. Pastor, F. J. Brosed, and J. J. Aguilar, “A one-step intrinsic and extrinsic calibration method for laser line scanner operation in coordinate measuring machines,” Meas. Sci. Technol. 20(4), 045107 (2009).

An, Y.

Bleyer, M.

M. Bleyer and M. Gelautz, “A layered stereo matching algorithm using image segmentation and global visibility constraints,” ISPRS J. Photogramm. 59(3), 128–150 (2005).

Brosed, F. J.

F. J. Brosed, J. Santolaria, J. J. Aguilar, and D. Guillomía, “Laser triangulation sensor and six axes anthropomorphic robot manipulator modelling for the measurement of complex geometry products,” Robot. Com.- Int. Manuf. 28(6), 660–671 (2012).

J. Santolaria, J. J. Pastor, F. J. Brosed, and J. J. Aguilar, “A one-step intrinsic and extrinsic calibration method for laser line scanner operation in coordinate measuring machines,” Meas. Sci. Technol. 20(4), 045107 (2009).

Cajal, C.

J. Santolaria, J. J. Aguilar, D. Guillomía, and C. Cajal, “A crenellated target-based calibration method for laser triangulation sensors integration in articulated measurement arms,” Robot. Com.- Int. Manuf. 27(2), 282–291 (2011).

Cui, X.

Dong, L.

C. Wei, C. Sihai, L. Dong, and J. Guohua, “A compact two-dimensional laser scanner based on piezoelectric actuators,” Rev. Sci. Instrum. 86(1), 013102 (2015).
[PubMed]

Garibaldi, J.

H. Hirschmüller, P. R. Innocent, and J. Garibaldi, “Real-time correlation-based stereo vision with reduced border errors,” Int. J. Comput. Vis. 47(1–3), 229–246 (2002).

Gelautz, M.

M. Bleyer and M. Gelautz, “A layered stereo matching algorithm using image segmentation and global visibility constraints,” ISPRS J. Photogramm. 59(3), 128–150 (2005).

Geng, J.

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).

Glowacz, A.

A. Głowacz and Z. Głowacz, “Diagnosis of the three-phase induction motor using thermal imaging,” Infrared Phys. Technol. 81, 7–16 (2017).

A. Glowacz and Z. Glowacz, “Diagnostics of stator faults of the single-phase induction motor using thermal images, moasos and selected classifiers,” Measurement 93, 86–93 (2016).

Glowacz, Z.

A. Głowacz and Z. Głowacz, “Diagnosis of the three-phase induction motor using thermal imaging,” Infrared Phys. Technol. 81, 7–16 (2017).

A. Glowacz and Z. Glowacz, “Diagnostics of stator faults of the single-phase induction motor using thermal images, moasos and selected classifiers,” Measurement 93, 86–93 (2016).

Guillomía, D.

F. J. Brosed, J. Santolaria, J. J. Aguilar, and D. Guillomía, “Laser triangulation sensor and six axes anthropomorphic robot manipulator modelling for the measurement of complex geometry products,” Robot. Com.- Int. Manuf. 28(6), 660–671 (2012).

J. Santolaria, J. J. Aguilar, D. Guillomía, and C. Cajal, “A crenellated target-based calibration method for laser triangulation sensors integration in articulated measurement arms,” Robot. Com.- Int. Manuf. 27(2), 282–291 (2011).

Guo, Q.

Guohua, J.

C. Wei, C. Sihai, L. Dong, and J. Guohua, “A compact two-dimensional laser scanner based on piezoelectric actuators,” Rev. Sci. Instrum. 86(1), 013102 (2015).
[PubMed]

Ha, J. E.

J. E. Ha, “Extrinsic calibration of a camera and laser range finder using a new calibration structure of a plane with a triangular hole,” Int. J. Control Autom. 10(6), 1240–1244 (2012).

Harris, C.

C. Harris and M. Stephens, “A combined corner and edge detector,” in Proceedings of the 4th Alvey Vision Conference (1988), pp. 147–151.

Heikkila, J.

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pat. Anal. 22(10), 1066–1077 (2000).

Hirschmüller, H.

H. Hirschmüller, P. R. Innocent, and J. Garibaldi, “Real-time correlation-based stereo vision with reduced border errors,” Int. J. Comput. Vis. 47(1–3), 229–246 (2002).

Huang, J.

H. Zhang, K. Iijima, J. Huang, G. P. Walcott, and J. M. Rogers, “Optical mapping of membrane potential and epicardial deformation in beating hearts,” Biophys. J. 111(2), 438–451 (2016).
[PubMed]

Huang, P. S.

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45(8), 083601 (2006).

Iijima, K.

H. Zhang, K. Iijima, J. Huang, G. P. Walcott, and J. M. Rogers, “Optical mapping of membrane potential and epicardial deformation in beating hearts,” Biophys. J. 111(2), 438–451 (2016).
[PubMed]

Innocent, P. R.

H. Hirschmüller, P. R. Innocent, and J. Garibaldi, “Real-time correlation-based stereo vision with reduced border errors,” Int. J. Comput. Vis. 47(1–3), 229–246 (2002).

Kanade, T.

C. L. Zitnick and T. Kanade, “A cooperative algorithm for stereo matching and occlusion detection,” IEEE Trans. Pat. Anal. 22(7), 675–684 (2000).

Karara, H. M.

Y. I. Abdel-Aziz and H. M. Karara, “Direct linear transformation into object space coordinates in close-range photogrammetry,” in Proceedings of the Symposium on Close-Range Photogrammetry (Falls Church, VA, USA, 1971), pp. 1–18.

Kim, D.

Kwon, S.

C. H. Lee, Y. C. Lim, S. Kwon, and J. H. Lee, “Stereo vision-based vehicle detection using a road feature and disparity histogram,” Opt. Eng. 50(2), 027004 (2011).

La, H. M.

R. S. Lim, H. M. La, and W. Sheng, “A robotic crack inspection and mapping system for bridge deck maintenance,” IEEE Trans. Autom. Sci. Eng. 11(2), 367–378 (2014).

Lee, C. H.

C. H. Lee, Y. C. Lim, S. Kwon, and J. H. Lee, “Stereo vision-based vehicle detection using a road feature and disparity histogram,” Opt. Eng. 50(2), 027004 (2011).

Lee, J. H.

C. H. Lee, Y. C. Lim, S. Kwon, and J. H. Lee, “Stereo vision-based vehicle detection using a road feature and disparity histogram,” Opt. Eng. 50(2), 027004 (2011).

Lee, S.

Li, W.

Li, X.

G. Xu, J. Yuan, X. Li, and J. Su, “3D reconstruction of laser projective point with projection invariant generated from five points on 2D target,” Sci. Rep. 7(1), 7049 (2017).
[PubMed]

G. Xu, X. Zhang, J. Su, X. Li, and A. Zheng, “Solution approach of a laser plane based on Plücker matrices of the projective lines on a flexible 2D target,” Appl. Opt. 55(10), 2653–2656 (2016).
[PubMed]

Lim, K. B.

Lim, R. S.

R. S. Lim, H. M. La, and W. Sheng, “A robotic crack inspection and mapping system for bridge deck maintenance,” IEEE Trans. Autom. Sci. Eng. 11(2), 367–378 (2014).

Lim, Y. C.

C. H. Lee, Y. C. Lim, S. Kwon, and J. H. Lee, “Stereo vision-based vehicle detection using a road feature and disparity histogram,” Opt. Eng. 50(2), 027004 (2011).

Liu, H.

Özcan, M.

Özgürün, B.

Pastor, J. J.

J. Santolaria, J. J. Pastor, F. J. Brosed, and J. J. Aguilar, “A one-step intrinsic and extrinsic calibration method for laser line scanner operation in coordinate measuring machines,” Meas. Sci. Technol. 20(4), 045107 (2009).

Rogers, J. M.

H. Zhang, K. Iijima, J. Huang, G. P. Walcott, and J. M. Rogers, “Optical mapping of membrane potential and epicardial deformation in beating hearts,” Biophys. J. 111(2), 438–451 (2016).
[PubMed]

Santolaria, J.

F. J. Brosed, J. Santolaria, J. J. Aguilar, and D. Guillomía, “Laser triangulation sensor and six axes anthropomorphic robot manipulator modelling for the measurement of complex geometry products,” Robot. Com.- Int. Manuf. 28(6), 660–671 (2012).

J. Santolaria, J. J. Aguilar, D. Guillomía, and C. Cajal, “A crenellated target-based calibration method for laser triangulation sensors integration in articulated measurement arms,” Robot. Com.- Int. Manuf. 27(2), 282–291 (2011).

J. Santolaria, J. J. Pastor, F. J. Brosed, and J. J. Aguilar, “A one-step intrinsic and extrinsic calibration method for laser line scanner operation in coordinate measuring machines,” Meas. Sci. Technol. 20(4), 045107 (2009).

Shan, S.

Sheng, W.

R. S. Lim, H. M. La, and W. Sheng, “A robotic crack inspection and mapping system for bridge deck maintenance,” IEEE Trans. Autom. Sci. Eng. 11(2), 367–378 (2014).

Sihai, C.

C. Wei, C. Sihai, L. Dong, and J. Guohua, “A compact two-dimensional laser scanner based on piezoelectric actuators,” Rev. Sci. Instrum. 86(1), 013102 (2015).
[PubMed]

Steger, C.

C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pat. Anal. 20(2), 113–125 (1998).

Stephens, M.

C. Harris and M. Stephens, “A combined corner and edge detector,” in Proceedings of the 4th Alvey Vision Conference (1988), pp. 147–151.

Su, J.

G. Xu, J. Yuan, X. Li, and J. Su, “3D reconstruction of laser projective point with projection invariant generated from five points on 2D target,” Sci. Rep. 7(1), 7049 (2017).
[PubMed]

G. Xu, X. Zhang, J. Su, X. Li, and A. Zheng, “Solution approach of a laser plane based on Plücker matrices of the projective lines on a flexible 2D target,” Appl. Opt. 55(10), 2653–2656 (2016).
[PubMed]

Tayyar, D. Ö.

Tsai, R.

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

Walcott, G. P.

H. Zhang, K. Iijima, J. Huang, G. P. Walcott, and J. M. Rogers, “Optical mapping of membrane potential and epicardial deformation in beating hearts,” Biophys. J. 111(2), 438–451 (2016).
[PubMed]

Wang, D.

Wei, C.

C. Wei, C. Sihai, L. Dong, and J. Guohua, “A compact two-dimensional laser scanner based on piezoelectric actuators,” Rev. Sci. Instrum. 86(1), 013102 (2015).
[PubMed]

Xu, G.

G. Xu, J. Yuan, X. Li, and J. Su, “3D reconstruction of laser projective point with projection invariant generated from five points on 2D target,” Sci. Rep. 7(1), 7049 (2017).
[PubMed]

G. Xu, X. Zhang, J. Su, X. Li, and A. Zheng, “Solution approach of a laser plane based on Plücker matrices of the projective lines on a flexible 2D target,” Appl. Opt. 55(10), 2653–2656 (2016).
[PubMed]

Yuan, J.

G. Xu, J. Yuan, X. Li, and J. Su, “3D reconstruction of laser projective point with projection invariant generated from five points on 2D target,” Sci. Rep. 7(1), 7049 (2017).
[PubMed]

Zhang, H.

H. Zhang, K. Iijima, J. Huang, G. P. Walcott, and J. M. Rogers, “Optical mapping of membrane potential and epicardial deformation in beating hearts,” Biophys. J. 111(2), 438–451 (2016).
[PubMed]

Zhang, S.

Zhang, X.

Zhang, Z.

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pat. Anal. Mach. Intell. 26(7), 892–899 (2004).
[PubMed]

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pat. Anal. 22(11), 1330–1334 (2000).

Zheng, A.

Zitnick, C. L.

C. L. Zitnick and T. Kanade, “A cooperative algorithm for stereo matching and occlusion detection,” IEEE Trans. Pat. Anal. 22(7), 675–684 (2000).

Adv. Opt. Photonics (1)

J. Geng, “Structured-light 3D surface imaging: a tutorial,” Adv. Opt. Photonics 3(2), 128–160 (2011).

Appl. Opt. (4)

Biophys. J. (1)

H. Zhang, K. Iijima, J. Huang, G. P. Walcott, and J. M. Rogers, “Optical mapping of membrane potential and epicardial deformation in beating hearts,” Biophys. J. 111(2), 438–451 (2016).
[PubMed]

IEEE Trans. Autom. Sci. Eng. (2)

R. S. Lim, H. M. La, and W. Sheng, “A robotic crack inspection and mapping system for bridge deck maintenance,” IEEE Trans. Autom. Sci. Eng. 11(2), 367–378 (2014).

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

IEEE Trans. Pat. Anal. (4)

Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pat. Anal. 22(11), 1330–1334 (2000).

C. L. Zitnick and T. Kanade, “A cooperative algorithm for stereo matching and occlusion detection,” IEEE Trans. Pat. Anal. 22(7), 675–684 (2000).

J. Heikkila, “Geometric camera calibration using circular control points,” IEEE Trans. Pat. Anal. 22(10), 1066–1077 (2000).

C. Steger, “An unbiased detector of curvilinear structures,” IEEE Trans. Pat. Anal. 20(2), 113–125 (1998).

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

Z. Zhang, “Camera calibration with one-dimensional objects,” IEEE Trans. Pat. Anal. Mach. Intell. 26(7), 892–899 (2004).
[PubMed]

Infrared Phys. Technol. (1)

A. Głowacz and Z. Głowacz, “Diagnosis of the three-phase induction motor using thermal imaging,” Infrared Phys. Technol. 81, 7–16 (2017).

Int. J. Comput. Vis. (1)

H. Hirschmüller, P. R. Innocent, and J. Garibaldi, “Real-time correlation-based stereo vision with reduced border errors,” Int. J. Comput. Vis. 47(1–3), 229–246 (2002).

Int. J. Control Autom. (1)

J. E. Ha, “Extrinsic calibration of a camera and laser range finder using a new calibration structure of a plane with a triangular hole,” Int. J. Control Autom. 10(6), 1240–1244 (2012).

ISPRS J. Photogramm. (1)

M. Bleyer and M. Gelautz, “A layered stereo matching algorithm using image segmentation and global visibility constraints,” ISPRS J. Photogramm. 59(3), 128–150 (2005).

J. Opt. Soc. Am. A (2)

Meas. Sci. Technol. (1)

J. Santolaria, J. J. Pastor, F. J. Brosed, and J. J. Aguilar, “A one-step intrinsic and extrinsic calibration method for laser line scanner operation in coordinate measuring machines,” Meas. Sci. Technol. 20(4), 045107 (2009).

Measurement (1)

A. Glowacz and Z. Glowacz, “Diagnostics of stator faults of the single-phase induction motor using thermal images, moasos and selected classifiers,” Measurement 93, 86–93 (2016).

Opt. Eng. (2)

C. H. Lee, Y. C. Lim, S. Kwon, and J. H. Lee, “Stereo vision-based vehicle detection using a road feature and disparity histogram,” Opt. Eng. 50(2), 027004 (2011).

S. Zhang and P. S. Huang, “Novel method for structured light system calibration,” Opt. Eng. 45(8), 083601 (2006).

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C. Wei, C. Sihai, L. Dong, and J. Guohua, “A compact two-dimensional laser scanner based on piezoelectric actuators,” Rev. Sci. Instrum. 86(1), 013102 (2015).
[PubMed]

Robot. Com.- Int. Manuf. (2)

F. J. Brosed, J. Santolaria, J. J. Aguilar, and D. Guillomía, “Laser triangulation sensor and six axes anthropomorphic robot manipulator modelling for the measurement of complex geometry products,” Robot. Com.- Int. Manuf. 28(6), 660–671 (2012).

J. Santolaria, J. J. Aguilar, D. Guillomía, and C. Cajal, “A crenellated target-based calibration method for laser triangulation sensors integration in articulated measurement arms,” Robot. Com.- Int. Manuf. 27(2), 282–291 (2011).

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G. Xu, J. Yuan, X. Li, and J. Su, “3D reconstruction of laser projective point with projection invariant generated from five points on 2D target,” Sci. Rep. 7(1), 7049 (2017).
[PubMed]

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Figures (8)

Fig. 1
Fig. 1

Profile reconstruction method with two cuboid references and a flexible laser plane.

Fig. 2
Fig. 2

The solution diagram of the global intrinsic parameter matrix and the global extrinsic parameter matrix.

Fig. 3
Fig. 3

The solution diagram of the initial solution of the flexible laser plane.

Fig. 4
Fig. 4

The solution diagram of the optimization solution of the flexible laser plane.

Fig. 5
Fig. 5

Verification method with the benchmark ruler.

Fig. 6
Fig. 6

Profile reconstruction results adopting the parameterized re-projection errors of laser lines from the bi-cuboid references. (a) The experiment setup of the reconstruction method. (b) The verification experiment. (c) The captured image of the car model. (d) The reconstruction results of the car model. (e) The captured image of the square box. (f) The reconstruction results of the square box. (g) The captured image of the cylindrical tube. (h) The reconstruction results of the cylindrical tube. (i) The captured image of the flat surface. (j) The reconstruction results of the flat surface.

Fig. 7
Fig. 7

Reconstruction errors of the initial method and the optimization method. λj is the reconstruction error. IN indicates the initial method. OP indicates the optimization method. MD indicates measurement distance, mm. RD indicates reference distance, mm. (a) MD = 700, RD = 150. (b) MD = 700, RD = 200. (c) MD = 700, RD = 250. (d) MD = 700, RD = 300. (e) MD = 800, RD = 150. (f) MD = 800, RD = 200. (g) MD = 800, RD = 250. (h) MD = 800, RD = 300. (i) MD = 900, RD = 150. (j) MD = 900, RD = 200. (k) MD = 900, RD = 250. (l) MD = 900, RD = 300. (m) MD = 1000, RD = 150. (n) MD = 1000, RD = 200. (o) MD = 1000, RD = 250. (p) MD = 1000, RD = 300.

Fig. 8
Fig. 8

Relative errors of the initial method and the optimization method. δj is the reconstruction error. IN indicates the initial method. OP indicates the optimization method. MD indicates measurement distance, mm. RD indicates reference distance, mm. (a) MD = 700, RD = 150. (b) MD = 700, RD = 200. (c) MD = 700, RD = 250. (d) MD = 700, RD = 300. (e) MD = 800, RD = 150. (f) MD = 800, RD = 200. (g) MD = 800, RD = 250. (h) MD = 800, RD = 300. (i) MD = 900, RD = 150. (j) MD = 900, RD = 200. (k) MD = 900, RD = 250. (l) MD = 900, RD = 300. (m) MD = 1000, RD = 150. (n) MD = 1000, RD = 200. (o) MD = 1000, RD = 250. (p) MD = 1000, RD = 300.

Tables (1)

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Table 1 Statistic data of the reconstruction experiments.

Equations (18)

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δ m x i (I,m) =P m X i (W,m)
Pm= K m (L) [R m (L) , t m (L) ]
X i (C,m) =[ R m (L) , t m (L) ] X i (W,m)
δ x i (I) = K (G) X i (C)
[ X i (C) 0 Y i (C) Z i (C) 0 0 Y i (C) 0 0 Z i (C) ][ α β γ u v ]=[ x i (I) Z i (C) y i (I) Z i (C) ]
[ α,β,γ,u,v ] T = ( B T B) 1 B T b
[R m (G) , t m (G) ]= ( K (G) ) 1 Pm
l j (m,n) = (H (W,m,n) ) T ( l j (m,n) )
f j (m,1) = J m T [1,0,0,0] T
f j (m,2) = J m T [0,1,0,0] T
e j (m,1) = J m T [0, ( l j (m,1) ) x , ( l j (m,1) ) y , ( l j (m,1) ) z ] T
e j (m,2) = J m T [ ( l j (m,2) ) x ,0, ( l j (m,2) ) y , ( l j (m,2) ) z ] T
L j (m,n)* = e j (m,n) ( f j (m,n) ) T f j (m,n) ( e j (m,n) ) T = [ l j (m,n) ] 4×4
[ L j (1,1) , L j (1,2) , L j (2,1) , L j (2,2) ] T Π j =0
L j (C,m,n)* ( Π j , R m (G) , t m (G) )= Π j [ f j (m,n) ( R m (G) , t m (G) )] T [ f j (m,n) ( R m (G) , t m (G) )] Π j T = [ l j (C,m,n) ] 4×4
[( l ˜ j (m,n) ( Π j , K (G) , R m (G) , t m (G) ) ) ] × ={ K (G) [R m (G) , t m (G) ]}{ L j (C,m,n) ( Π j , R m (G) , t m (G) )} { K (G) [R m (G) , t m (G) ]} T
f( Π j , K (G) , R 1 (G) , t 1 (G) , R 2 (G) , t 2 (G) )= ( l ˜ j (1,1) ( Π j , K (G) , R 1 (G) , t 1 (G) ) ) ( l j (1,1) ) 2 + ( l ˜ j (1,2) ( Π j , K (G) , R 1 (G) , t 1 (G) ) ) ( l j (1,2) ) 2 + ( l ˜ j (2,1) ( Π j , K (G) , R 2 (G) , t 2 (G) ) ) ( l j (2,1) ) 2 + ( l ˜ j (2,2) ( Π j , K (G) , R 2 (G) , t 2 (G) ) ) ( l j (2,2) ) 2
Π ^ j T X i (O) =0