Abstract

The accuracy of strain measurement using a common optical extensometer with two-dimensional (2D) digital image correlation (DIC) is not sufficient for experimental applications due to the effect of out-of-plane motion. Although three-dimensional (3D) DIC can measure all three components of displacement without introducing in-plane displacement errors, 3D-DIC requires the stringent synchronization between two digital cameras and requires complicated system calibration of binocular stereovision, which makes the measurement rather inconvenient. To solve the problems described above, this paper proposes a self-calibration single-lens 3D video extensometer for non-contact, non-destructive and high-accuracy strain measurement. In the established video extensometer, a single-lens 3D imaging system with a prism and two mirrors is constructed to acquire stereo images of the test sample surface, so the problems of synchronization and out-of-plane displacement can be solved easily. Moreover, a speckle-based self-calibration method which calibrates the single-lens stereo system using the reference speckle image of the specimen instead of the calibration targets is proposed, which will make the system more convenient to be used without complicated calibration. Furthermore, an efficient and robust inverse compositional Gauss-Newton algorithm combined with a robust stereo matching stage is employed to achieve high-accuracy and real-time subset-based stereo matching. Tensile tests of an Al-alloy specimen were performed to demonstrate the feasibility and effectiveness of the proposed self-calibration single-lens 3D video extensometer.

© 2016 Optical Society of America

Full Article  |  PDF Article
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References

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    [Crossref]
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    [Crossref]
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    [Crossref]
  27. Z. Zhang, “A flexible new technique for camera calibration,” IEEE Trans. Pattern Anal. Mach. 22(11), 1330–1334 (2000).
    [Crossref]
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    [Crossref] [PubMed]
  30. B. Pan, “Reliability-guided digital image correlation for image deformation measurement,” Appl. Opt. 48(8), 1535–1542 (2009).
    [Crossref] [PubMed]
  31. Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
    [Crossref]
  32. X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
    [Crossref]
  33. B. Wang and B. Pan, “Random errors in digital image correlation due to matched or overmatched shape functions,” Exp. Mech. 55(9), 1717–1727 (2015).
    [Crossref]
  34. S. Baker and I. Matthews, “Lucas-kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56(3), 221–255 (2004).
    [Crossref]
  35. B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53(7), 1277–1289 (2013).
    [Crossref]
  36. Y. Su, Q. Zhang, X. Xu, and Z. Gao, “Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors,” Opt. Lasers Eng. 86, 132–142 (2016).
    [Crossref]
  37. X. Shao, X. Dai, Z. Chen, and X. He, “Real-time 3D digital image correlation method and its application in human pulse monitoring,” Appl. Opt. 55(4), 696–704 (2016).
    [Crossref] [PubMed]
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    [Crossref]
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    [Crossref]

2016 (9)

H. Gustafson, G. Siegmund, and P. Cripton, “Comparison of strain rosettes and digital image correlation for measuring vertebral body strain,” J. Biomech. Eng. 138, 054501 (2016).

S. Biswal and A. Ramaswamy, “Measurement of existing prestressing force in concrete structures through an embedded vibrating beam strain gauge,” Measurement 83, 10–19 (2016).
[Crossref]

R. Wu, C. Kong, K. Li, and D. Zhang, “Real-time digital image correlation for dynamic strain measurement,” Exp. Mech. 56(5), 833–843 (2016).
[Crossref]

B. Pan and L. Tian, “Advanced video extensometer for non-contact, real-time, high-accuracy strain measurement,” Opt. Express 24(17), 19082–19093 (2016).
[Crossref] [PubMed]

F. Zhu, P. Bai, H. Shi, Z. Jiang, D. Lei, and X. He, “Enhancement of strain measurement accuracy using optical extensometer by application of dual-reflector imaging,” Meas. Sci. Technol. 27(6), 065007 (2016).
[Crossref]

L. F. Wu, J. G. Zhu, H. M. Xie, and Q. Zhang, “An accurate method for shape retrieval and displacement measurement using bi-prism-based single lens 3D digital image correlation,” Exp. Mech. 56(9), 1611–1624 (2016).
[Crossref]

X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
[Crossref]

Y. Su, Q. Zhang, X. Xu, and Z. Gao, “Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors,” Opt. Lasers Eng. 86, 132–142 (2016).
[Crossref]

X. Shao, X. Dai, Z. Chen, and X. He, “Real-time 3D digital image correlation method and its application in human pulse monitoring,” Appl. Opt. 55(4), 696–704 (2016).
[Crossref] [PubMed]

2015 (7)

Z. Chen, C. Quan, F. Zhu, and X. He, “A method to transfer speckle patterns for digital image correlation,” Meas. Sci. Technol. 26(9), 095201 (2015).
[Crossref]

B. Wang and B. Pan, “Random errors in digital image correlation due to matched or overmatched shape functions,” Exp. Mech. 55(9), 1717–1727 (2015).
[Crossref]

Z. Pan, S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction-assisted image correlation for three-dimensional surface profiling,” Exp. Mech. 55(1), 155–165 (2015).
[Crossref]

Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
[Crossref]

X. Shao, X. Dai, and X. He, “Noise robustness and parallel computation of the inverse compositional Gauss–Newton algorithm in digital image correlation,” Opt. Lasers Eng. 71, 9–19 (2015).
[Crossref]

P. Bai, F. Zhu, and X. He, “Optical extensometer and elimination of the effect of out-of-plane motions,” Opt. Lasers Eng. 65, 28–37 (2015).
[Crossref]

G. Gao, S. Huang, K. Xia, and Z. Li, “Application of digital image correlation (DIC) in dynamic notched semi-circular bend (NSCB) tests,” Exp. Mech. 55(1), 95–104 (2015).
[Crossref]

2014 (1)

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements using low-cost imaging lenses: implementation of a generalized compensation method,” Meas. Sci. Technol. 25(2), 025001 (2014).
[Crossref]

2013 (4)

S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimensional digital image correlation,” Exp. Mech. 53(5), 755–765 (2013).
[Crossref]

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech. 53(9), 1719–1733 (2013).
[Crossref]

K. Genovese, L. Casaletto, J. A. Rayas, V. Flores, and A. Martinez, “Stereo-Digital Image Correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53(7), 1277–1289 (2013).
[Crossref]

2012 (1)

2011 (1)

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

2010 (1)

2009 (3)

L. Chen, G. M. Treece, J. E. Lindop, A. H. Gee, and R. W. Prager, “A quality-guided displacement tracking algorithm for ultrasonic elasticity imaging,” Med. Image Anal. 13(2), 286–296 (2009).
[Crossref] [PubMed]

B. Pan, “Reliability-guided digital image correlation for image deformation measurement,” Appl. Opt. 48(8), 1535–1542 (2009).
[Crossref] [PubMed]

Y. H. Huang, L. Liu, T. W. Yeung, and Y. Y. Hung, “Real-time monitoring of clamping force of a bolted joint by use of automatic digital image correlation,” Opt. Laser Technol. 41(4), 408–414 (2009).
[Crossref]

2008 (2)

M. A. Sutton, J. H. Yan, V. Tiwari, H. W. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. 46(10), 746–757 (2008).
[Crossref]

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

2004 (4)

R. L. Edwards, G. Coles, and W. N. Sharpe., “Comparison of tensile and bulge tests for thin-film silicon nitride,” Exp. Mech. 44(1), 49–54 (2004).
[Crossref]

D. Zhang, D. Arola, P. G. Charalambides, and M. C. L. Patterson, “On the mechanical behavior of carbon-carbon optic grids determined using a bi-axial optical extensometer,” J. Mater. Sci. 39(14), 4495–4505 (2004).
[Crossref]

R. Völkl and B. Fischer, “Mechanical testing of ultra-high temperature alloys,” Exp. Mech. 44(2), 121–127 (2004).
[Crossref]

S. Baker and I. Matthews, “Lucas-kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56(3), 221–255 (2004).
[Crossref]

2001 (1)

S. Boyd, N. Shrive, G. Wohl, R. Müller, and R. Zernicke, “Measurement of cancellous bone strain during mechanical tests using a new extensometer device,” Med. Eng. Phys. 23(6), 411–416 (2001).
[Crossref] [PubMed]

2000 (3)

T. Namazu, Y. Isono, and T. Tanaka, “Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM,” J. Microelectromech. Syst. 9(4), 450–459 (2000).
[Crossref]

D. Coimbra, R. Greenwood, and K. Kendall, “Tensile testing of ceramic fibres by video extensometry,” J. Mater. Sci. 35(13), 3341–3345 (2000).
[Crossref]

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

1986 (1)

H. W. Reinhardt, H. A. W. Cornelissen, and D. A. Hordijk, “Tensile tests and failure analysis of concrete,” J. Struct. Eng. 112(11), 2462–2477 (1986).
[Crossref]

Arola, D.

D. Zhang, D. Arola, P. G. Charalambides, and M. C. L. Patterson, “On the mechanical behavior of carbon-carbon optic grids determined using a bi-axial optical extensometer,” J. Mater. Sci. 39(14), 4495–4505 (2004).
[Crossref]

Bai, P.

F. Zhu, P. Bai, H. Shi, Z. Jiang, D. Lei, and X. He, “Enhancement of strain measurement accuracy using optical extensometer by application of dual-reflector imaging,” Meas. Sci. Technol. 27(6), 065007 (2016).
[Crossref]

P. Bai, F. Zhu, and X. He, “Optical extensometer and elimination of the effect of out-of-plane motions,” Opt. Lasers Eng. 65, 28–37 (2015).
[Crossref]

Baker, S.

S. Baker and I. Matthews, “Lucas-kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56(3), 221–255 (2004).
[Crossref]

Biswal, S.

S. Biswal and A. Ramaswamy, “Measurement of existing prestressing force in concrete structures through an embedded vibrating beam strain gauge,” Measurement 83, 10–19 (2016).
[Crossref]

Boyd, S.

S. Boyd, N. Shrive, G. Wohl, R. Müller, and R. Zernicke, “Measurement of cancellous bone strain during mechanical tests using a new extensometer device,” Med. Eng. Phys. 23(6), 411–416 (2001).
[Crossref] [PubMed]

Casaletto, L.

K. Genovese, L. Casaletto, J. A. Rayas, V. Flores, and A. Martinez, “Stereo-Digital Image Correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Charalambides, P. G.

D. Zhang, D. Arola, P. G. Charalambides, and M. C. L. Patterson, “On the mechanical behavior of carbon-carbon optic grids determined using a bi-axial optical extensometer,” J. Mater. Sci. 39(14), 4495–4505 (2004).
[Crossref]

Chen, L.

L. Chen, G. M. Treece, J. E. Lindop, A. H. Gee, and R. W. Prager, “A quality-guided displacement tracking algorithm for ultrasonic elasticity imaging,” Med. Image Anal. 13(2), 286–296 (2009).
[Crossref] [PubMed]

Chen, P. C. Y.

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

Chen, Y. Q.

Chen, Z.

X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
[Crossref]

X. Shao, X. Dai, Z. Chen, and X. He, “Real-time 3D digital image correlation method and its application in human pulse monitoring,” Appl. Opt. 55(4), 696–704 (2016).
[Crossref] [PubMed]

Z. Chen, C. Quan, F. Zhu, and X. He, “A method to transfer speckle patterns for digital image correlation,” Meas. Sci. Technol. 26(9), 095201 (2015).
[Crossref]

Cheng, T.

Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
[Crossref]

Chew, C. M.

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

Coimbra, D.

D. Coimbra, R. Greenwood, and K. Kendall, “Tensile testing of ceramic fibres by video extensometry,” J. Mater. Sci. 35(13), 3341–3345 (2000).
[Crossref]

Coles, G.

R. L. Edwards, G. Coles, and W. N. Sharpe., “Comparison of tensile and bulge tests for thin-film silicon nitride,” Exp. Mech. 44(1), 49–54 (2004).
[Crossref]

Cornelissen, H. A. W.

H. W. Reinhardt, H. A. W. Cornelissen, and D. A. Hordijk, “Tensile tests and failure analysis of concrete,” J. Struct. Eng. 112(11), 2462–2477 (1986).
[Crossref]

Cripton, P.

H. Gustafson, G. Siegmund, and P. Cripton, “Comparison of strain rosettes and digital image correlation for measuring vertebral body strain,” J. Biomech. Eng. 138, 054501 (2016).

Dai, X.

X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
[Crossref]

X. Shao, X. Dai, Z. Chen, and X. He, “Real-time 3D digital image correlation method and its application in human pulse monitoring,” Appl. Opt. 55(4), 696–704 (2016).
[Crossref] [PubMed]

X. Shao, X. Dai, and X. He, “Noise robustness and parallel computation of the inverse compositional Gauss–Newton algorithm in digital image correlation,” Opt. Lasers Eng. 71, 9–19 (2015).
[Crossref]

Dai, Y.

X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
[Crossref]

Dong, S.

X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
[Crossref]

Edwards, R. L.

R. L. Edwards, G. Coles, and W. N. Sharpe., “Comparison of tensile and bulge tests for thin-film silicon nitride,” Exp. Mech. 44(1), 49–54 (2004).
[Crossref]

Fischer, B.

R. Völkl and B. Fischer, “Mechanical testing of ultra-high temperature alloys,” Exp. Mech. 44(2), 121–127 (2004).
[Crossref]

Flores, V.

K. Genovese, L. Casaletto, J. A. Rayas, V. Flores, and A. Martinez, “Stereo-Digital Image Correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Gao, G.

G. Gao, S. Huang, K. Xia, and Z. Li, “Application of digital image correlation (DIC) in dynamic notched semi-circular bend (NSCB) tests,” Exp. Mech. 55(1), 95–104 (2015).
[Crossref]

Gao, Y.

Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
[Crossref]

Gao, Z.

Y. Su, Q. Zhang, X. Xu, and Z. Gao, “Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors,” Opt. Lasers Eng. 86, 132–142 (2016).
[Crossref]

Gdoutou, A.

Z. Pan, S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction-assisted image correlation for three-dimensional surface profiling,” Exp. Mech. 55(1), 155–165 (2015).
[Crossref]

S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimensional digital image correlation,” Exp. Mech. 53(5), 755–765 (2013).
[Crossref]

Gee, A. H.

L. Chen, G. M. Treece, J. E. Lindop, A. H. Gee, and R. W. Prager, “A quality-guided displacement tracking algorithm for ultrasonic elasticity imaging,” Med. Image Anal. 13(2), 286–296 (2009).
[Crossref] [PubMed]

Genovese, K.

K. Genovese, L. Casaletto, J. A. Rayas, V. Flores, and A. Martinez, “Stereo-Digital Image Correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Greenwood, R.

D. Coimbra, R. Greenwood, and K. Kendall, “Tensile testing of ceramic fibres by video extensometry,” J. Mater. Sci. 35(13), 3341–3345 (2000).
[Crossref]

Gustafson, H.

H. Gustafson, G. Siegmund, and P. Cripton, “Comparison of strain rosettes and digital image correlation for measuring vertebral body strain,” J. Biomech. Eng. 138, 054501 (2016).

He, X.

X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
[Crossref]

X. Shao, X. Dai, Z. Chen, and X. He, “Real-time 3D digital image correlation method and its application in human pulse monitoring,” Appl. Opt. 55(4), 696–704 (2016).
[Crossref] [PubMed]

F. Zhu, P. Bai, H. Shi, Z. Jiang, D. Lei, and X. He, “Enhancement of strain measurement accuracy using optical extensometer by application of dual-reflector imaging,” Meas. Sci. Technol. 27(6), 065007 (2016).
[Crossref]

X. Shao, X. Dai, and X. He, “Noise robustness and parallel computation of the inverse compositional Gauss–Newton algorithm in digital image correlation,” Opt. Lasers Eng. 71, 9–19 (2015).
[Crossref]

P. Bai, F. Zhu, and X. He, “Optical extensometer and elimination of the effect of out-of-plane motions,” Opt. Lasers Eng. 65, 28–37 (2015).
[Crossref]

Z. Chen, C. Quan, F. Zhu, and X. He, “A method to transfer speckle patterns for digital image correlation,” Meas. Sci. Technol. 26(9), 095201 (2015).
[Crossref]

Ho, H. N.

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

Hordijk, D. A.

H. W. Reinhardt, H. A. W. Cornelissen, and D. A. Hordijk, “Tensile tests and failure analysis of concrete,” J. Struct. Eng. 112(11), 2462–2477 (1986).
[Crossref]

Huang, S.

G. Gao, S. Huang, K. Xia, and Z. Li, “Application of digital image correlation (DIC) in dynamic notched semi-circular bend (NSCB) tests,” Exp. Mech. 55(1), 95–104 (2015).
[Crossref]

Huang, Y. H.

Y. H. Huang, L. Liu, T. W. Yeung, and Y. Y. Hung, “Real-time monitoring of clamping force of a bolted joint by use of automatic digital image correlation,” Opt. Laser Technol. 41(4), 408–414 (2009).
[Crossref]

Hung, Y. Y.

Y. H. Huang, L. Liu, T. W. Yeung, and Y. Y. Hung, “Real-time monitoring of clamping force of a bolted joint by use of automatic digital image correlation,” Opt. Laser Technol. 41(4), 408–414 (2009).
[Crossref]

Isono, Y.

T. Namazu, Y. Isono, and T. Tanaka, “Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM,” J. Microelectromech. Syst. 9(4), 450–459 (2000).
[Crossref]

Jeyapalina, S.

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

Jiang, Z.

F. Zhu, P. Bai, H. Shi, Z. Jiang, D. Lei, and X. He, “Enhancement of strain measurement accuracy using optical extensometer by application of dual-reflector imaging,” Meas. Sci. Technol. 27(6), 065007 (2016).
[Crossref]

Justusson, B.

Ke, X. D.

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

Kendall, K.

D. Coimbra, R. Greenwood, and K. Kendall, “Tensile testing of ceramic fibres by video extensometry,” J. Mater. Sci. 35(13), 3341–3345 (2000).
[Crossref]

Kong, C.

R. Wu, C. Kong, K. Li, and D. Zhang, “Real-time digital image correlation for dynamic strain measurement,” Exp. Mech. 56(5), 833–843 (2016).
[Crossref]

Lei, D.

F. Zhu, P. Bai, H. Shi, Z. Jiang, D. Lei, and X. He, “Enhancement of strain measurement accuracy using optical extensometer by application of dual-reflector imaging,” Meas. Sci. Technol. 27(6), 065007 (2016).
[Crossref]

Li, K.

R. Wu, C. Kong, K. Li, and D. Zhang, “Real-time digital image correlation for dynamic strain measurement,” Exp. Mech. 56(5), 833–843 (2016).
[Crossref]

B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53(7), 1277–1289 (2013).
[Crossref]

Li, Z.

G. Gao, S. Huang, K. Xia, and Z. Li, “Application of digital image correlation (DIC) in dynamic notched semi-circular bend (NSCB) tests,” Exp. Mech. 55(1), 95–104 (2015).
[Crossref]

Lim, B. H.

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

Lim, K. H.

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

Lindop, J. E.

L. Chen, G. M. Treece, J. E. Lindop, A. H. Gee, and R. W. Prager, “A quality-guided displacement tracking algorithm for ultrasonic elasticity imaging,” Med. Image Anal. 13(2), 286–296 (2009).
[Crossref] [PubMed]

Liu, L.

Y. H. Huang, L. Liu, T. W. Yeung, and Y. Y. Hung, “Real-time monitoring of clamping force of a bolted joint by use of automatic digital image correlation,” Opt. Laser Technol. 41(4), 408–414 (2009).
[Crossref]

Martinez, A.

K. Genovese, L. Casaletto, J. A. Rayas, V. Flores, and A. Martinez, “Stereo-Digital Image Correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Matthews, I.

S. Baker and I. Matthews, “Lucas-kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56(3), 221–255 (2004).
[Crossref]

Miller, T. J.

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

Müller, R.

S. Boyd, N. Shrive, G. Wohl, R. Müller, and R. Zernicke, “Measurement of cancellous bone strain during mechanical tests using a new extensometer device,” Med. Eng. Phys. 23(6), 411–416 (2001).
[Crossref] [PubMed]

Namazu, T.

T. Namazu, Y. Isono, and T. Tanaka, “Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM,” J. Microelectromech. Syst. 9(4), 450–459 (2000).
[Crossref]

Orteu, J. J.

M. A. Sutton, J. H. Yan, V. Tiwari, H. W. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. 46(10), 746–757 (2008).
[Crossref]

Pan, B.

B. Pan and L. Tian, “Advanced video extensometer for non-contact, real-time, high-accuracy strain measurement,” Opt. Express 24(17), 19082–19093 (2016).
[Crossref] [PubMed]

B. Wang and B. Pan, “Random errors in digital image correlation due to matched or overmatched shape functions,” Exp. Mech. 55(9), 1717–1727 (2015).
[Crossref]

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements using low-cost imaging lenses: implementation of a generalized compensation method,” Meas. Sci. Technol. 25(2), 025001 (2014).
[Crossref]

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech. 53(9), 1719–1733 (2013).
[Crossref]

B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53(7), 1277–1289 (2013).
[Crossref]

Y. Zhou, B. Pan, and Y. Q. Chen, “Large deformation measurement using digital image correlation: a fully automated approach,” Appl. Opt. 51(31), 7674–7683 (2012).
[Crossref] [PubMed]

B. Pan, “Reliability-guided digital image correlation for image deformation measurement,” Appl. Opt. 48(8), 1535–1542 (2009).
[Crossref] [PubMed]

Pan, Z.

Z. Pan, S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction-assisted image correlation for three-dimensional surface profiling,” Exp. Mech. 55(1), 155–165 (2015).
[Crossref]

Pankow, M.

Patterson, M. C. L.

D. Zhang, D. Arola, P. G. Charalambides, and M. C. L. Patterson, “On the mechanical behavior of carbon-carbon optic grids determined using a bi-axial optical extensometer,” J. Mater. Sci. 39(14), 4495–4505 (2004).
[Crossref]

Prager, R. W.

L. Chen, G. M. Treece, J. E. Lindop, A. H. Gee, and R. W. Prager, “A quality-guided displacement tracking algorithm for ultrasonic elasticity imaging,” Med. Image Anal. 13(2), 286–296 (2009).
[Crossref] [PubMed]

Quan, C.

Z. Chen, C. Quan, F. Zhu, and X. He, “A method to transfer speckle patterns for digital image correlation,” Meas. Sci. Technol. 26(9), 095201 (2015).
[Crossref]

Ramaswamy, A.

S. Biswal and A. Ramaswamy, “Measurement of existing prestressing force in concrete structures through an embedded vibrating beam strain gauge,” Measurement 83, 10–19 (2016).
[Crossref]

Rappel, J. K.

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

Ravichandran, G.

Z. Pan, S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction-assisted image correlation for three-dimensional surface profiling,” Exp. Mech. 55(1), 155–165 (2015).
[Crossref]

S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimensional digital image correlation,” Exp. Mech. 53(5), 755–765 (2013).
[Crossref]

Rayas, J. A.

K. Genovese, L. Casaletto, J. A. Rayas, V. Flores, and A. Martinez, “Stereo-Digital Image Correlation (DIC) measurements with a single camera using a biprism,” Opt. Lasers Eng. 51(3), 278–285 (2013).
[Crossref]

Reinhardt, H. W.

H. W. Reinhardt, H. A. W. Cornelissen, and D. A. Hordijk, “Tensile tests and failure analysis of concrete,” J. Struct. Eng. 112(11), 2462–2477 (1986).
[Crossref]

Reu, P. L.

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

Schreier, H. W.

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

M. A. Sutton, J. H. Yan, V. Tiwari, H. W. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. 46(10), 746–757 (2008).
[Crossref]

Shao, X.

X. Shao, X. Dai, Z. Chen, Y. Dai, S. Dong, and X. He, “Calibration of stereo-digital image correlation for deformation measurement of large engineering components,” Meas. Sci. Technol. 27(12), 125010 (2016).
[Crossref]

X. Shao, X. Dai, Z. Chen, and X. He, “Real-time 3D digital image correlation method and its application in human pulse monitoring,” Appl. Opt. 55(4), 696–704 (2016).
[Crossref] [PubMed]

X. Shao, X. Dai, and X. He, “Noise robustness and parallel computation of the inverse compositional Gauss–Newton algorithm in digital image correlation,” Opt. Lasers Eng. 71, 9–19 (2015).
[Crossref]

Sharpe, W. N.

R. L. Edwards, G. Coles, and W. N. Sharpe., “Comparison of tensile and bulge tests for thin-film silicon nitride,” Exp. Mech. 44(1), 49–54 (2004).
[Crossref]

Shi, H.

F. Zhu, P. Bai, H. Shi, Z. Jiang, D. Lei, and X. He, “Enhancement of strain measurement accuracy using optical extensometer by application of dual-reflector imaging,” Meas. Sci. Technol. 27(6), 065007 (2016).
[Crossref]

Shrive, N.

S. Boyd, N. Shrive, G. Wohl, R. Müller, and R. Zernicke, “Measurement of cancellous bone strain during mechanical tests using a new extensometer device,” Med. Eng. Phys. 23(6), 411–416 (2001).
[Crossref] [PubMed]

Siegmund, G.

H. Gustafson, G. Siegmund, and P. Cripton, “Comparison of strain rosettes and digital image correlation for measuring vertebral body strain,” J. Biomech. Eng. 138, 054501 (2016).

Su, Y.

Y. Su, Q. Zhang, X. Xu, and Z. Gao, “Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors,” Opt. Lasers Eng. 86, 132–142 (2016).
[Crossref]

Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
[Crossref]

Sutton, M. A.

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

M. A. Sutton, J. H. Yan, V. Tiwari, H. W. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. 46(10), 746–757 (2008).
[Crossref]

Tanaka, T.

T. Namazu, Y. Isono, and T. Tanaka, “Evaluation of size effect on mechanical properties of single crystal silicon by nanoscale bending test using AFM,” J. Microelectromech. Syst. 9(4), 450–459 (2000).
[Crossref]

Tian, L.

Tiwari, V.

M. A. Sutton, J. H. Yan, V. Tiwari, H. W. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. 46(10), 746–757 (2008).
[Crossref]

Tong, W.

B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53(7), 1277–1289 (2013).
[Crossref]

Treece, G. M.

L. Chen, G. M. Treece, J. E. Lindop, A. H. Gee, and R. W. Prager, “A quality-guided displacement tracking algorithm for ultrasonic elasticity imaging,” Med. Image Anal. 13(2), 286–296 (2009).
[Crossref] [PubMed]

Völkl, R.

R. Völkl and B. Fischer, “Mechanical testing of ultra-high temperature alloys,” Exp. Mech. 44(2), 121–127 (2004).
[Crossref]

Waas, A. M.

Wang, B.

B. Wang and B. Pan, “Random errors in digital image correlation due to matched or overmatched shape functions,” Exp. Mech. 55(9), 1717–1727 (2015).
[Crossref]

Wang, Y. Q.

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

Wohl, G.

S. Boyd, N. Shrive, G. Wohl, R. Müller, and R. Zernicke, “Measurement of cancellous bone strain during mechanical tests using a new extensometer device,” Med. Eng. Phys. 23(6), 411–416 (2001).
[Crossref] [PubMed]

Wu, D.

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements using low-cost imaging lenses: implementation of a generalized compensation method,” Meas. Sci. Technol. 25(2), 025001 (2014).
[Crossref]

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech. 53(9), 1719–1733 (2013).
[Crossref]

Wu, L. F.

L. F. Wu, J. G. Zhu, H. M. Xie, and Q. Zhang, “An accurate method for shape retrieval and displacement measurement using bi-prism-based single lens 3D digital image correlation,” Exp. Mech. 56(9), 1611–1624 (2016).
[Crossref]

Wu, R.

R. Wu, C. Kong, K. Li, and D. Zhang, “Real-time digital image correlation for dynamic strain measurement,” Exp. Mech. 56(5), 833–843 (2016).
[Crossref]

Xia, K.

G. Gao, S. Huang, K. Xia, and Z. Li, “Application of digital image correlation (DIC) in dynamic notched semi-circular bend (NSCB) tests,” Exp. Mech. 55(1), 95–104 (2015).
[Crossref]

Xia, S.

Z. Pan, S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction-assisted image correlation for three-dimensional surface profiling,” Exp. Mech. 55(1), 155–165 (2015).
[Crossref]

S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimensional digital image correlation,” Exp. Mech. 53(5), 755–765 (2013).
[Crossref]

Xie, H. M.

L. F. Wu, J. G. Zhu, H. M. Xie, and Q. Zhang, “An accurate method for shape retrieval and displacement measurement using bi-prism-based single lens 3D digital image correlation,” Exp. Mech. 56(9), 1611–1624 (2016).
[Crossref]

Xu, X.

Y. Su, Q. Zhang, X. Xu, and Z. Gao, “Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors,” Opt. Lasers Eng. 86, 132–142 (2016).
[Crossref]

Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
[Crossref]

Yan, J. H.

M. A. Sutton, J. H. Yan, V. Tiwari, H. W. Schreier, and J. J. Orteu, “The effect of out-of-plane motion on 2D and 3D digital image correlation measurements,” Opt. Lasers Eng. 46(10), 746–757 (2008).
[Crossref]

Yeung, T. W.

Y. H. Huang, L. Liu, T. W. Yeung, and Y. Y. Hung, “Real-time monitoring of clamping force of a bolted joint by use of automatic digital image correlation,” Opt. Laser Technol. 41(4), 408–414 (2009).
[Crossref]

Yu, L.

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements using low-cost imaging lenses: implementation of a generalized compensation method,” Meas. Sci. Technol. 25(2), 025001 (2014).
[Crossref]

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech. 53(9), 1719–1733 (2013).
[Crossref]

Zernicke, R.

S. Boyd, N. Shrive, G. Wohl, R. Müller, and R. Zernicke, “Measurement of cancellous bone strain during mechanical tests using a new extensometer device,” Med. Eng. Phys. 23(6), 411–416 (2001).
[Crossref] [PubMed]

Zhang, D.

R. Wu, C. Kong, K. Li, and D. Zhang, “Real-time digital image correlation for dynamic strain measurement,” Exp. Mech. 56(5), 833–843 (2016).
[Crossref]

D. Zhang, D. Arola, P. G. Charalambides, and M. C. L. Patterson, “On the mechanical behavior of carbon-carbon optic grids determined using a bi-axial optical extensometer,” J. Mater. Sci. 39(14), 4495–4505 (2004).
[Crossref]

Zhang, Q.

L. F. Wu, J. G. Zhu, H. M. Xie, and Q. Zhang, “An accurate method for shape retrieval and displacement measurement using bi-prism-based single lens 3D digital image correlation,” Exp. Mech. 56(9), 1611–1624 (2016).
[Crossref]

Y. Su, Q. Zhang, X. Xu, and Z. Gao, “Quality assessment of speckle patterns for DIC by consideration of both systematic errors and random errors,” Opt. Lasers Eng. 86, 132–142 (2016).
[Crossref]

Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
[Crossref]

Zhang, Y.

Y. Gao, T. Cheng, Y. Su, X. Xu, Y. Zhang, and Q. Zhang, “High-efficiency and high-accuracy digital image correlation for three-dimensional measurement,” Opt. Lasers Eng. 65, 73–80 (2015).
[Crossref]

Zhang, Z.

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

Zhou, Y.

Zhu, F.

F. Zhu, P. Bai, H. Shi, Z. Jiang, D. Lei, and X. He, “Enhancement of strain measurement accuracy using optical extensometer by application of dual-reflector imaging,” Meas. Sci. Technol. 27(6), 065007 (2016).
[Crossref]

P. Bai, F. Zhu, and X. He, “Optical extensometer and elimination of the effect of out-of-plane motions,” Opt. Lasers Eng. 65, 28–37 (2015).
[Crossref]

Z. Chen, C. Quan, F. Zhu, and X. He, “A method to transfer speckle patterns for digital image correlation,” Meas. Sci. Technol. 26(9), 095201 (2015).
[Crossref]

Zhu, J. G.

L. F. Wu, J. G. Zhu, H. M. Xie, and Q. Zhang, “An accurate method for shape retrieval and displacement measurement using bi-prism-based single lens 3D digital image correlation,” Exp. Mech. 56(9), 1611–1624 (2016).
[Crossref]

Appl. Opt. (4)

Exp. Mech. (11)

R. L. Edwards, G. Coles, and W. N. Sharpe., “Comparison of tensile and bulge tests for thin-film silicon nitride,” Exp. Mech. 44(1), 49–54 (2004).
[Crossref]

G. Gao, S. Huang, K. Xia, and Z. Li, “Application of digital image correlation (DIC) in dynamic notched semi-circular bend (NSCB) tests,” Exp. Mech. 55(1), 95–104 (2015).
[Crossref]

R. Völkl and B. Fischer, “Mechanical testing of ultra-high temperature alloys,” Exp. Mech. 44(2), 121–127 (2004).
[Crossref]

R. Wu, C. Kong, K. Li, and D. Zhang, “Real-time digital image correlation for dynamic strain measurement,” Exp. Mech. 56(5), 833–843 (2016).
[Crossref]

S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction assisted image correlation: a novel method for measuring three-dimensional deformation using two-dimensional digital image correlation,” Exp. Mech. 53(5), 755–765 (2013).
[Crossref]

Z. Pan, S. Xia, A. Gdoutou, and G. Ravichandran, “Diffraction-assisted image correlation for three-dimensional surface profiling,” Exp. Mech. 55(1), 155–165 (2015).
[Crossref]

L. F. Wu, J. G. Zhu, H. M. Xie, and Q. Zhang, “An accurate method for shape retrieval and displacement measurement using bi-prism-based single lens 3D digital image correlation,” Exp. Mech. 56(9), 1611–1624 (2016).
[Crossref]

B. Pan, L. Yu, and D. Wu, “High-accuracy 2D digital image correlation measurements with bilateral telecentric lenses: error analysis and experimental verification,” Exp. Mech. 53(9), 1719–1733 (2013).
[Crossref]

B. Wang and B. Pan, “Random errors in digital image correlation due to matched or overmatched shape functions,” Exp. Mech. 55(9), 1717–1727 (2015).
[Crossref]

B. Pan, K. Li, and W. Tong, “Fast, robust and accurate digital image correlation calculation without redundant computations,” Exp. Mech. 53(7), 1277–1289 (2013).
[Crossref]

Y. Q. Wang, M. A. Sutton, X. D. Ke, H. W. Schreier, P. L. Reu, and T. J. Miller, “On error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 405–422 (2011).
[Crossref]

IEEE Trans. Pattern Anal. Mach. (1)

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

Int. J. Comput. Vis. (1)

S. Baker and I. Matthews, “Lucas-kanade 20 years on: A unifying framework,” Int. J. Comput. Vis. 56(3), 221–255 (2004).
[Crossref]

J. Biomech. (1)

K. H. Lim, C. M. Chew, P. C. Y. Chen, S. Jeyapalina, H. N. Ho, J. K. Rappel, and B. H. Lim, “New extensometer to measure in vivo uniaxial mechanical properties of human skin,” J. Biomech. 41(5), 931–936 (2008).
[Crossref] [PubMed]

J. Biomech. Eng. (1)

H. Gustafson, G. Siegmund, and P. Cripton, “Comparison of strain rosettes and digital image correlation for measuring vertebral body strain,” J. Biomech. Eng. 138, 054501 (2016).

J. Mater. Sci. (2)

D. Coimbra, R. Greenwood, and K. Kendall, “Tensile testing of ceramic fibres by video extensometry,” J. Mater. Sci. 35(13), 3341–3345 (2000).
[Crossref]

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Opt. Express (1)

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Figures (11)

Fig. 1
Fig. 1 (a) Schematic diagram of the single-lens 3D video extensometer and (b) optical arrangement of the reflection stereo imaging device.
Fig. 2
Fig. 2 The actual setup of the single-lens 3D video extensometer.
Fig. 3
Fig. 3 Working principle of the single-lens 3D video extensometer.
Fig. 4
Fig. 4 Example of speckle-based calibration of single-lens stereo system.
Fig. 5
Fig. 5 Robust stereo matching stage without accumulated errors.
Fig. 6
Fig. 6 Flowchart of real-time strain measurement using the proposed 3D video extensometer.
Fig. 7
Fig. 7 The schematic of the experimental setup: (a) the geometric size of the Al-alloy specimen, (b) the positive side of the painted specimen, (c) the negative side of the specimen adhered with two stage gauges, (d) the field experimental setup.
Fig. 8
Fig. 8 Reconstructed 3D points in ROC after non-linear least-squares optimization.
Fig. 9
Fig. 9 Static errors of the proposed single-lens 3D video extensometer.
Fig. 10
Fig. 10 Comparisons of measured strain between strain gauge and the proposed single-lens 3D video extensometer.
Fig. 11
Fig. 11 Comparisons of measured strain errors between 2D video extensometer and the proposed single-lens 3D video extensometer.

Tables (1)

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Table 1 The initial value and calibrated parameters using the proposed speckle-based calibration method

Equations (2)

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ε= ΔL L = P 1 ' P 2 ' ¯ P 1 P 2 ¯ P 1 P 2 ¯
C st = i=1 N m 1 i m ^ 1 i (A,k, M i ) 2 + i=1 N m 2 i m ^ 2 i (A,k,R,t, M i ) 2

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