Abstract

Based on the Fourier method, this paper deduces analytic formulae for interpolation bias in digital image correlation, explains the well-known sinusoidal-shaped curves of interpolation bias, and introduces the concept of interpolation bias kernel, which characterizes the frequency response of the interpolation bias and thus provides a measure of the subset matching quality of the interpolation algorithm. The interpolation bias kernel attributes the interpolation bias to aliasing effect of interpolation and indicates that high-frequency components are the major source of interpolation bias. Based on our theoretical results, a simple and effective interpolation bias prediction approach, which exploits the speckle spectrum and the interpolation transfer function, is proposed. Significant acceleration is attained, the effect of subset size is analyzed, and both numerical simulations and experimental results are found to agree with theoretical predictions. During the experiment, a novel experimental translation technique was developed that implements subpixel translation of a captured image through integer pixel translation on a computer screen. Owing to this remarkable technique, the influences of mechanical error and out-of-plane motion are eliminated, and complete interpolation bias curves as accurate as 0.01 pixel are attained by subpixel translation experiments.

© 2015 Optical Society of America

Full Article  |  PDF Article
OSA Recommended Articles
Noise-induced bias for convolution-based interpolation in digital image correlation

Yong Su, Qingchuan Zhang, Zeren Gao, and Xiaohai Xu
Opt. Express 24(2) 1175-1195 (2016)

Digital image correlation with reduced bias error based on digital signal upsampling theory

Wei Heng, Buwei Huo, Xinxing Shao, and Xiaoyuan He
Appl. Opt. 58(15) 3962-3973 (2019)

Study on subset size selection in digital image correlation for speckle patterns

Bing Pan, Huimin Xie, Zhaoyang Wang, Kemao Qian, and Zhiyong Wang
Opt. Express 16(10) 7037-7048 (2008)

References

  • View by:
  • |
  • |
  • |

  1. M. A. Sutton, J. J. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications (Springer Science & Business Media, 2009).
  2. B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
    [Crossref]
  3. F. Hild and S. Roux, “Digital image correlation: from displacement measurement to identification of elastic properties – a review,” Strain 42(2), 69–80 (2006).
    [Crossref]
  4. J. Goyens, J. Soons, P. Aerts, and J. Dirckx, “Finite-element modelling reveals force modulation of jaw adductors in stag beetles,” J. R. Soc. Interface 11(101), 20140908 (2014).
    [Crossref] [PubMed]
  5. R. Ghorbani, F. Matta, and M. A. Sutton, “Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation,” Exp. Mech. 55(1), 227–243 (2015).
    [Crossref]
  6. H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
    [Crossref]
  7. G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
    [Crossref]
  8. Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
    [Crossref]
  9. Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
    [Crossref]
  10. M. Pitter, C. See, J. Goh, and M. Somekh, “Focus errors and their correction in microscopic deformation analysis using correlation,” Opt. Express 10(23), 1361–1367 (2002).
    [Crossref] [PubMed]
  11. H. Yan and B. Pan, “Three-dimensional displacement measurement based on the combination of digital holography and digital image correlation,” Opt. Lett. 39(17), 5166–5169 (2014).
    [Crossref] [PubMed]
  12. M. Vo, Z. Y. Wang, B. Pan, and T. Y. Pan, “Hyper-accurate flexible calibration technique for fringe-projection-based three-dimensional imaging,” Opt. Express 20(15), 16926–16941 (2012).
    [Crossref]
  13. H. A. Bruck, S. R. Mcneill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989).
    [Crossref]
  14. M. A. Sutton, J. L. Turner, H. A. Bruck, and T. A. Chae, “Full-field representation of discretely sampled surface deformation for displacement and strain analysis,” Exp. Mech. 31(2), 168–177 (1991).
    [Crossref]
  15. M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
    [Crossref]
  16. B. Pan, H. Xie, Z. Wang, K. Qian, and Z. Wang, “Study on subset size selection in digital image correlation for speckle patterns,” Opt. Express 16(10), 7037–7048 (2008).
    [Crossref] [PubMed]
  17. H. W. Schreier and M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape functions,” Exp. Mech. 42(3), 303–310 (2002).
    [Crossref]
  18. W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain 41(4), 167–175 (2005).
    [Crossref]
  19. B. Pan, H. Xie, B. Xu, and F. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17(6), 1615–1621 (2006).
    [Crossref]
  20. H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39(11), 2915–2921 (2000).
    [Crossref]
  21. M. A. Sutton, S. R. Mcneill, J. S. Jang, and M. Babai, “Effects of subpixel image restoration on digital correlation error estimates,” Opt. Eng. 27(10), 870–877 (1988).
    [Crossref]
  22. S. Choi and S. P. Shah, “Measurement of deformations on concrete subjected to compression using image correlation,” Exp. Mech. 37(3), 307–313 (1997).
    [Crossref]
  23. M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
    [Crossref]
  24. G. K. Rohde, A. Aldroubi, and D. M. Healy., “Interpolation artifacts in sub-pixel image registration,” IEEE Trans. Image Process. 18(2), 333–345 (2009).
    [Crossref] [PubMed]
  25. L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
    [Crossref] [PubMed]
  26. P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015).
    [Crossref]
  27. B. Pan, “Bias error reduction of digital image correlation using Gaussian pre-filtering,” Opt. Lasers Eng. 51(10), 1161–1167 (2013).
    [Crossref]
  28. H. Haddadi and S. Belhabib, “Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique,” Opt. Lasers Eng. 46(2), 185–196 (2008).
    [Crossref]
  29. D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
    [Crossref]
  30. Y. Q. Wang, M. A. Sutton, H. A. Bruck, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009).
    [Crossref]
  31. P. L. Reu, “Experimental and numerical methods for exact subpixel shifting,” Exp. Mech. 51(4), 443–452 (2011).
    [Crossref]
  32. M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63(3), 207–224 (2005).
    [Crossref]
  33. J. Inglada, V. Muron, D. Pichard, and T. Feuvrier, “Analysis of artifacts in subpixel remote sensing image registration,” IEEE Trans. Geosci. Rem. Sens. 45(1), 254–264 (2007).
    [Crossref]
  34. L. Svilainis, K. Lukoseviciute, V. Dumbrava, and A. Chaziachmetovas, “Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain,” Measurement 46(10), 3950–3958 (2013).
    [Crossref]
  35. P. Thévenaz, T. Blu, and M. Unser, “Interpolation revisited,” IEEE Trans. Med. Imaging 19(7), 739–758 (2000).
    [Crossref] [PubMed]
  36. R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 29(6), 1153–1160 (1981).
    [Crossref]
  37. T. Blu, P. Thévenaz, and M. Unser, “MOMS: maximal-order interpolation of minimal support,” IEEE Trans. Image Process. 10(7), 1069–1080 (2001).
    [Crossref] [PubMed]
  38. S. K. Park and R. A. Schowengerdt, “Image sampling, reconstruction, and the effect of sample-scene phasing,” Appl. Opt. 21(17), 3142–3151 (1982).
    [Crossref] [PubMed]
  39. P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40(8), 1613–1620 (2001).
    [Crossref]
  40. 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]
  41. L. Yu and B. Pan, “The errors in digital image correlation due to overmatched shape functions,” Meas. Sci. Technol. 26(4), 045202 (2015).
    [Crossref]
  42. Y. Su, “Interpolation bias prediction in digital image correlation,” figshare (2015) [retrieved 10 July 2015] http://dx.doi.org/.
    [Crossref]

2015 (4)

R. Ghorbani, F. Matta, and M. A. Sutton, “Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation,” Exp. Mech. 55(1), 227–243 (2015).
[Crossref]

P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (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]

L. Yu and B. Pan, “The errors in digital image correlation due to overmatched shape functions,” Meas. Sci. Technol. 26(4), 045202 (2015).
[Crossref]

2014 (3)

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

H. Yan and B. Pan, “Three-dimensional displacement measurement based on the combination of digital holography and digital image correlation,” Opt. Lett. 39(17), 5166–5169 (2014).
[Crossref] [PubMed]

J. Goyens, J. Soons, P. Aerts, and J. Dirckx, “Finite-element modelling reveals force modulation of jaw adductors in stag beetles,” J. R. Soc. Interface 11(101), 20140908 (2014).
[Crossref] [PubMed]

2013 (2)

B. Pan, “Bias error reduction of digital image correlation using Gaussian pre-filtering,” Opt. Lasers Eng. 51(10), 1161–1167 (2013).
[Crossref]

L. Svilainis, K. Lukoseviciute, V. Dumbrava, and A. Chaziachmetovas, “Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain,” Measurement 46(10), 3950–3958 (2013).
[Crossref]

2012 (1)

2011 (2)

2009 (4)

Y. Q. Wang, M. A. Sutton, H. A. Bruck, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009).
[Crossref]

G. K. Rohde, A. Aldroubi, and D. M. Healy., “Interpolation artifacts in sub-pixel image registration,” IEEE Trans. Image Process. 18(2), 333–345 (2009).
[Crossref] [PubMed]

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

2008 (2)

B. Pan, H. Xie, Z. Wang, K. Qian, and Z. Wang, “Study on subset size selection in digital image correlation for speckle patterns,” Opt. Express 16(10), 7037–7048 (2008).
[Crossref] [PubMed]

H. Haddadi and S. Belhabib, “Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique,” Opt. Lasers Eng. 46(2), 185–196 (2008).
[Crossref]

2007 (2)

J. Inglada, V. Muron, D. Pichard, and T. Feuvrier, “Analysis of artifacts in subpixel remote sensing image registration,” IEEE Trans. Geosci. Rem. Sens. 45(1), 254–264 (2007).
[Crossref]

G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
[Crossref]

2006 (3)

F. Hild and S. Roux, “Digital image correlation: from displacement measurement to identification of elastic properties – a review,” Strain 42(2), 69–80 (2006).
[Crossref]

B. Pan, H. Xie, B. Xu, and F. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17(6), 1615–1621 (2006).
[Crossref]

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

2005 (4)

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63(3), 207–224 (2005).
[Crossref]

W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain 41(4), 167–175 (2005).
[Crossref]

Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
[Crossref]

Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
[Crossref]

2002 (2)

M. Pitter, C. See, J. Goh, and M. Somekh, “Focus errors and their correction in microscopic deformation analysis using correlation,” Opt. Express 10(23), 1361–1367 (2002).
[Crossref] [PubMed]

H. W. Schreier and M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape functions,” Exp. Mech. 42(3), 303–310 (2002).
[Crossref]

2001 (2)

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40(8), 1613–1620 (2001).
[Crossref]

T. Blu, P. Thévenaz, and M. Unser, “MOMS: maximal-order interpolation of minimal support,” IEEE Trans. Image Process. 10(7), 1069–1080 (2001).
[Crossref] [PubMed]

2000 (2)

P. Thévenaz, T. Blu, and M. Unser, “Interpolation revisited,” IEEE Trans. Med. Imaging 19(7), 739–758 (2000).
[Crossref] [PubMed]

H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39(11), 2915–2921 (2000).
[Crossref]

1999 (1)

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
[Crossref]

1997 (1)

S. Choi and S. P. Shah, “Measurement of deformations on concrete subjected to compression using image correlation,” Exp. Mech. 37(3), 307–313 (1997).
[Crossref]

1991 (1)

M. A. Sutton, J. L. Turner, H. A. Bruck, and T. A. Chae, “Full-field representation of discretely sampled surface deformation for displacement and strain analysis,” Exp. Mech. 31(2), 168–177 (1991).
[Crossref]

1989 (1)

H. A. Bruck, S. R. Mcneill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989).
[Crossref]

1988 (1)

M. A. Sutton, S. R. Mcneill, J. S. Jang, and M. Babai, “Effects of subpixel image restoration on digital correlation error estimates,” Opt. Eng. 27(10), 870–877 (1988).
[Crossref]

1982 (1)

1981 (1)

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 29(6), 1153–1160 (1981).
[Crossref]

Aerts, P.

J. Goyens, J. Soons, P. Aerts, and J. Dirckx, “Finite-element modelling reveals force modulation of jaw adductors in stag beetles,” J. R. Soc. Interface 11(101), 20140908 (2014).
[Crossref] [PubMed]

Aldroubi, A.

G. K. Rohde, A. Aldroubi, and D. M. Healy., “Interpolation artifacts in sub-pixel image registration,” IEEE Trans. Image Process. 18(2), 333–345 (2009).
[Crossref] [PubMed]

Asundi, A.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

Babai, M.

M. A. Sutton, S. R. Mcneill, J. S. Jang, and M. Babai, “Effects of subpixel image restoration on digital correlation error estimates,” Opt. Eng. 27(10), 870–877 (1988).
[Crossref]

Belhabib, S.

H. Haddadi and S. Belhabib, “Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique,” Opt. Lasers Eng. 46(2), 185–196 (2008).
[Crossref]

Blu, T.

T. Blu, P. Thévenaz, and M. Unser, “MOMS: maximal-order interpolation of minimal support,” IEEE Trans. Image Process. 10(7), 1069–1080 (2001).
[Crossref] [PubMed]

P. Thévenaz, T. Blu, and M. Unser, “Interpolation revisited,” IEEE Trans. Med. Imaging 19(7), 739–758 (2000).
[Crossref] [PubMed]

Bornert, M.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Bossuyt, S.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Braasch, J. R.

H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39(11), 2915–2921 (2000).
[Crossref]

Brémand, F.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Bruck, H. A.

Y. Q. Wang, M. A. Sutton, H. A. Bruck, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009).
[Crossref]

M. A. Sutton, J. L. Turner, H. A. Bruck, and T. A. Chae, “Full-field representation of discretely sampled surface deformation for displacement and strain analysis,” Exp. Mech. 31(2), 168–177 (1991).
[Crossref]

H. A. Bruck, S. R. Mcneill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989).
[Crossref]

Chae, T. A.

M. A. Sutton, J. L. Turner, H. A. Bruck, and T. A. Chae, “Full-field representation of discretely sampled surface deformation for displacement and strain analysis,” Exp. Mech. 31(2), 168–177 (1991).
[Crossref]

Chaziachmetovas, A.

L. Svilainis, K. Lukoseviciute, V. Dumbrava, and A. Chaziachmetovas, “Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain,” Measurement 46(10), 3950–3958 (2013).
[Crossref]

Chen, Z.

Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
[Crossref]

Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
[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]

Choi, S.

S. Choi and S. P. Shah, “Measurement of deformations on concrete subjected to compression using image correlation,” Exp. Mech. 37(3), 307–313 (1997).
[Crossref]

Cigada, A.

P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015).
[Crossref]

Dai, F.

B. Pan, H. Xie, B. Xu, and F. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17(6), 1615–1621 (2006).
[Crossref]

Dirckx, J.

J. Goyens, J. Soons, P. Aerts, and J. Dirckx, “Finite-element modelling reveals force modulation of jaw adductors in stag beetles,” J. R. Soc. Interface 11(101), 20140908 (2014).
[Crossref] [PubMed]

Doumalin, P.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Dumbrava, V.

L. Svilainis, K. Lukoseviciute, V. Dumbrava, and A. Chaziachmetovas, “Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain,” Measurement 46(10), 3950–3958 (2013).
[Crossref]

Dupré, J. C.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Fazzini, M.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Feuvrier, T.

J. Inglada, V. Muron, D. Pichard, and T. Feuvrier, “Analysis of artifacts in subpixel remote sensing image registration,” IEEE Trans. Geosci. Rem. Sens. 45(1), 254–264 (2007).
[Crossref]

Fu, D.

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[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]

Ghorbani, R.

R. Ghorbani, F. Matta, and M. A. Sutton, “Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation,” Exp. Mech. 55(1), 227–243 (2015).
[Crossref]

Goh, J.

Goodson, K. E.

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40(8), 1613–1620 (2001).
[Crossref]

Goyens, J.

J. Goyens, J. Soons, P. Aerts, and J. Dirckx, “Finite-element modelling reveals force modulation of jaw adductors in stag beetles,” J. R. Soc. Interface 11(101), 20140908 (2014).
[Crossref] [PubMed]

Grédiac, M.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Habraken, A. M.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Haddadi, H.

H. Haddadi and S. Belhabib, “Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique,” Opt. Lasers Eng. 46(2), 185–196 (2008).
[Crossref]

Healy, D. M.

G. K. Rohde, A. Aldroubi, and D. M. Healy., “Interpolation artifacts in sub-pixel image registration,” IEEE Trans. Image Process. 18(2), 333–345 (2009).
[Crossref] [PubMed]

Hild, F.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

F. Hild and S. Roux, “Digital image correlation: from displacement measurement to identification of elastic properties – a review,” Strain 42(2), 69–80 (2006).
[Crossref]

Hoang, T.

Huang, G.

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

Inglada, J.

J. Inglada, V. Muron, D. Pichard, and T. Feuvrier, “Analysis of artifacts in subpixel remote sensing image registration,” IEEE Trans. Geosci. Rem. Sens. 45(1), 254–264 (2007).
[Crossref]

Jang, J. S.

M. A. Sutton, S. R. Mcneill, J. S. Jang, and M. Babai, “Effects of subpixel image restoration on digital correlation error estimates,” Opt. Eng. 27(10), 870–877 (1988).
[Crossref]

Jiang, H.

Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
[Crossref]

Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
[Crossref]

Jiang, Z.

Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
[Crossref]

Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
[Crossref]

Ju, X. Y.

G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
[Crossref]

Kang, Y.

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

Keys, R. G.

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 29(6), 1153–1160 (1981).
[Crossref]

Lecompte, D.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Li, J.

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

Liu, H. W.

G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
[Crossref]

Lukoseviciute, K.

L. Svilainis, K. Lukoseviciute, V. Dumbrava, and A. Chaziachmetovas, “Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain,” Measurement 46(10), 3950–3958 (2013).
[Crossref]

Luu, L.

Ma, J.

Matta, F.

P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015).
[Crossref]

R. Ghorbani, F. Matta, and M. A. Sutton, “Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation,” Exp. Mech. 55(1), 227–243 (2015).
[Crossref]

Mazzoleni, P.

P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015).
[Crossref]

Mcneill, S. R.

H. A. Bruck, S. R. Mcneill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989).
[Crossref]

M. A. Sutton, S. R. Mcneill, J. S. Jang, and M. Babai, “Effects of subpixel image restoration on digital correlation error estimates,” Opt. Eng. 27(10), 870–877 (1988).
[Crossref]

Mistou, S.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Molimard, J.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Muron, V.

J. Inglada, V. Muron, D. Pichard, and T. Feuvrier, “Analysis of artifacts in subpixel remote sensing image registration,” IEEE Trans. Geosci. Rem. Sens. 45(1), 254–264 (2007).
[Crossref]

Okutomi, M.

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63(3), 207–224 (2005).
[Crossref]

Orteu, J. J.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Pan, B.

L. Yu and B. Pan, “The errors in digital image correlation due to overmatched shape functions,” Meas. Sci. Technol. 26(4), 045202 (2015).
[Crossref]

H. Yan and B. Pan, “Three-dimensional displacement measurement based on the combination of digital holography and digital image correlation,” Opt. Lett. 39(17), 5166–5169 (2014).
[Crossref] [PubMed]

B. Pan, “Bias error reduction of digital image correlation using Gaussian pre-filtering,” Opt. Lasers Eng. 51(10), 1161–1167 (2013).
[Crossref]

M. Vo, Z. Y. Wang, B. Pan, and T. Y. Pan, “Hyper-accurate flexible calibration technique for fringe-projection-based three-dimensional imaging,” Opt. Express 20(15), 16926–16941 (2012).
[Crossref]

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

B. Pan, H. Xie, Z. Wang, K. Qian, and Z. Wang, “Study on subset size selection in digital image correlation for speckle patterns,” Opt. Express 16(10), 7037–7048 (2008).
[Crossref] [PubMed]

B. Pan, H. Xie, B. Xu, and F. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17(6), 1615–1621 (2006).
[Crossref]

Pan, T. Y.

Park, S. K.

Peters, W. H.

H. A. Bruck, S. R. Mcneill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989).
[Crossref]

Pichard, D.

J. Inglada, V. Muron, D. Pichard, and T. Feuvrier, “Analysis of artifacts in subpixel remote sensing image registration,” IEEE Trans. Geosci. Rem. Sens. 45(1), 254–264 (2007).
[Crossref]

Pitter, M.

Qi, G.

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

Qian, K.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

B. Pan, H. Xie, Z. Wang, K. Qian, and Z. Wang, “Study on subset size selection in digital image correlation for speckle patterns,” Opt. Express 16(10), 7037–7048 (2008).
[Crossref] [PubMed]

Reu, P. L.

P. L. Reu, “Experimental and numerical methods for exact subpixel shifting,” Exp. Mech. 51(4), 443–452 (2011).
[Crossref]

Robert, L.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Rohde, G. K.

G. K. Rohde, A. Aldroubi, and D. M. Healy., “Interpolation artifacts in sub-pixel image registration,” IEEE Trans. Image Process. 18(2), 333–345 (2009).
[Crossref] [PubMed]

Roux, S.

F. Hild and S. Roux, “Digital image correlation: from displacement measurement to identification of elastic properties – a review,” Strain 42(2), 69–80 (2006).
[Crossref]

Schowengerdt, R. A.

Schreier, H. W.

Y. Q. Wang, M. A. Sutton, H. A. Bruck, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009).
[Crossref]

H. W. Schreier and M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape functions,” Exp. Mech. 42(3), 303–310 (2002).
[Crossref]

H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39(11), 2915–2921 (2000).
[Crossref]

See, C.

Shah, S. P.

S. Choi and S. P. Shah, “Measurement of deformations on concrete subjected to compression using image correlation,” Exp. Mech. 37(3), 307–313 (1997).
[Crossref]

Shimizu, M.

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63(3), 207–224 (2005).
[Crossref]

Smits, A.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Sol, H.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Somekh, M.

Song, H.

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

Soons, J.

J. Goyens, J. Soons, P. Aerts, and J. Dirckx, “Finite-element modelling reveals force modulation of jaw adductors in stag beetles,” J. R. Soc. Interface 11(101), 20140908 (2014).
[Crossref] [PubMed]

Su, 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]

Surrel, Y.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Sutton, M. A.

R. Ghorbani, F. Matta, and M. A. Sutton, “Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation,” Exp. Mech. 55(1), 227–243 (2015).
[Crossref]

P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015).
[Crossref]

Y. Q. Wang, M. A. Sutton, H. A. Bruck, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009).
[Crossref]

H. W. Schreier and M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape functions,” Exp. Mech. 42(3), 303–310 (2002).
[Crossref]

H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39(11), 2915–2921 (2000).
[Crossref]

M. A. Sutton, J. L. Turner, H. A. Bruck, and T. A. Chae, “Full-field representation of discretely sampled surface deformation for displacement and strain analysis,” Exp. Mech. 31(2), 168–177 (1991).
[Crossref]

H. A. Bruck, S. R. Mcneill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989).
[Crossref]

M. A. Sutton, S. R. Mcneill, J. S. Jang, and M. Babai, “Effects of subpixel image restoration on digital correlation error estimates,” Opt. Eng. 27(10), 870–877 (1988).
[Crossref]

Svilainis, L.

L. Svilainis, K. Lukoseviciute, V. Dumbrava, and A. Chaziachmetovas, “Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain,” Measurement 46(10), 3950–3958 (2013).
[Crossref]

Thévenaz, P.

T. Blu, P. Thévenaz, and M. Unser, “MOMS: maximal-order interpolation of minimal support,” IEEE Trans. Image Process. 10(7), 1069–1080 (2001).
[Crossref] [PubMed]

P. Thévenaz, T. Blu, and M. Unser, “Interpolation revisited,” IEEE Trans. Med. Imaging 19(7), 739–758 (2000).
[Crossref] [PubMed]

Tong, W.

W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain 41(4), 167–175 (2005).
[Crossref]

Turner, J. L.

M. A. Sutton, J. L. Turner, H. A. Bruck, and T. A. Chae, “Full-field representation of discretely sampled surface deformation for displacement and strain analysis,” Exp. Mech. 31(2), 168–177 (1991).
[Crossref]

Unser, M.

T. Blu, P. Thévenaz, and M. Unser, “MOMS: maximal-order interpolation of minimal support,” IEEE Trans. Image Process. 10(7), 1069–1080 (2001).
[Crossref] [PubMed]

P. Thévenaz, T. Blu, and M. Unser, “Interpolation revisited,” IEEE Trans. Med. Imaging 19(7), 739–758 (2000).
[Crossref] [PubMed]

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
[Crossref]

Vacher, P.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Van Hemelrijck, D.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Vantomme, J.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Vo, M.

Wang, Y. Q.

Y. Q. Wang, M. A. Sutton, H. A. Bruck, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009).
[Crossref]

Wang, Z.

Wang, Z. Y.

Wattrisse, B.

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

Wu, X.

Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
[Crossref]

Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
[Crossref]

Wu, X. P.

G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
[Crossref]

Xiang, G. F.

G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
[Crossref]

Xie, H.

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

B. Pan, H. Xie, Z. Wang, K. Qian, and Z. Wang, “Study on subset size selection in digital image correlation for speckle patterns,” Opt. Express 16(10), 7037–7048 (2008).
[Crossref] [PubMed]

B. Pan, H. Xie, B. Xu, and F. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17(6), 1615–1621 (2006).
[Crossref]

Xu, B.

B. Pan, H. Xie, B. Xu, and F. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17(6), 1615–1621 (2006).
[Crossref]

Xu, X.

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, H.

Yu, L.

L. Yu and B. Pan, “The errors in digital image correlation due to overmatched shape functions,” Meas. Sci. Technol. 26(4), 045202 (2015).
[Crossref]

Zappa, E.

P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015).
[Crossref]

Zhang, H.

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

Zhang, Q.

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]

Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
[Crossref]

Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
[Crossref]

Zhang, Q. C.

G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
[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]

Zhou, P.

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40(8), 1613–1620 (2001).
[Crossref]

Appl. Opt. (1)

Exp. Mech. (7)

R. Ghorbani, F. Matta, and M. A. Sutton, “Full-field deformation measurement and crack mapping on confined masonry walls using digital image correlation,” Exp. Mech. 55(1), 227–243 (2015).
[Crossref]

S. Choi and S. P. Shah, “Measurement of deformations on concrete subjected to compression using image correlation,” Exp. Mech. 37(3), 307–313 (1997).
[Crossref]

H. W. Schreier and M. A. Sutton, “Systematic errors in digital image correlation due to undermatched subset shape functions,” Exp. Mech. 42(3), 303–310 (2002).
[Crossref]

P. L. Reu, “Experimental and numerical methods for exact subpixel shifting,” Exp. Mech. 51(4), 443–452 (2011).
[Crossref]

H. A. Bruck, S. R. Mcneill, M. A. Sutton, and W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29(3), 261–267 (1989).
[Crossref]

M. A. Sutton, J. L. Turner, H. A. Bruck, and T. A. Chae, “Full-field representation of discretely sampled surface deformation for displacement and strain analysis,” Exp. Mech. 31(2), 168–177 (1991).
[Crossref]

M. Bornert, F. Brémand, P. Doumalin, J. C. Dupré, M. Fazzini, M. Grédiac, F. Hild, S. Mistou, J. Molimard, J. J. Orteu, L. Robert, Y. Surrel, P. Vacher, and B. Wattrisse, “Assessment of digital image correlation measurement errors: methodology and results,” Exp. Mech. 49(3), 353–370 (2009).
[Crossref]

IEEE Signal Process. Mag. (1)

M. Unser, “Splines: a perfect fit for signal and image processing,” IEEE Signal Process. Mag. 16(6), 22–38 (1999).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (1)

R. G. Keys, “Cubic convolution interpolation for digital image processing,” IEEE Trans. Acoust. Speech Signal Process. 29(6), 1153–1160 (1981).
[Crossref]

IEEE Trans. Geosci. Rem. Sens. (1)

J. Inglada, V. Muron, D. Pichard, and T. Feuvrier, “Analysis of artifacts in subpixel remote sensing image registration,” IEEE Trans. Geosci. Rem. Sens. 45(1), 254–264 (2007).
[Crossref]

IEEE Trans. Image Process. (2)

T. Blu, P. Thévenaz, and M. Unser, “MOMS: maximal-order interpolation of minimal support,” IEEE Trans. Image Process. 10(7), 1069–1080 (2001).
[Crossref] [PubMed]

G. K. Rohde, A. Aldroubi, and D. M. Healy., “Interpolation artifacts in sub-pixel image registration,” IEEE Trans. Image Process. 18(2), 333–345 (2009).
[Crossref] [PubMed]

IEEE Trans. Med. Imaging (1)

P. Thévenaz, T. Blu, and M. Unser, “Interpolation revisited,” IEEE Trans. Med. Imaging 19(7), 739–758 (2000).
[Crossref] [PubMed]

Int. J. Comput. Vis. (1)

M. Shimizu and M. Okutomi, “Sub-pixel estimation error cancellation on area-based matching,” Int. J. Comput. Vis. 63(3), 207–224 (2005).
[Crossref]

Int. J. Plast. (1)

Q. Zhang, Z. Jiang, H. Jiang, Z. Chen, and X. Wu, “On the propagation and pulsation of Portevin-Le Chatelier deformation bands: an experimental study with digital speckle pattern metrology,” Int. J. Plast. 21(11), 2150–2173 (2005).
[Crossref]

J. R. Soc. Interface (1)

J. Goyens, J. Soons, P. Aerts, and J. Dirckx, “Finite-element modelling reveals force modulation of jaw adductors in stag beetles,” J. R. Soc. Interface 11(101), 20140908 (2014).
[Crossref] [PubMed]

Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. (1)

Z. Jiang, Q. Zhang, H. Jiang, Z. Chen, and X. Wu, “Spatial characteristics of the Portevin-Le Chatelier deformation bands in Al-4 at%Cu polycrystals,” Mater. Sci. Eng. A-Struct, Mater. Prop. Microstruct. Process. 403(1–2), 154–164 (2005).
[Crossref]

Meas. Sci. Technol. (3)

B. Pan, K. Qian, H. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20(6), 062001 (2009).
[Crossref]

B. Pan, H. Xie, B. Xu, and F. Dai, “Performance of sub-pixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17(6), 1615–1621 (2006).
[Crossref]

L. Yu and B. Pan, “The errors in digital image correlation due to overmatched shape functions,” Meas. Sci. Technol. 26(4), 045202 (2015).
[Crossref]

Measurement (1)

L. Svilainis, K. Lukoseviciute, V. Dumbrava, and A. Chaziachmetovas, “Subsample interpolation bias error in time of flight estimation by direct correlation in digital domain,” Measurement 46(10), 3950–3958 (2013).
[Crossref]

Opt. Eng. (3)

H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39(11), 2915–2921 (2000).
[Crossref]

M. A. Sutton, S. R. Mcneill, J. S. Jang, and M. Babai, “Effects of subpixel image restoration on digital correlation error estimates,” Opt. Eng. 27(10), 870–877 (1988).
[Crossref]

P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation (DISC),” Opt. Eng. 40(8), 1613–1620 (2001).
[Crossref]

Opt. Express (3)

Opt. Lasers Eng. (5)

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]

P. Mazzoleni, F. Matta, E. Zappa, M. A. Sutton, and A. Cigada, “Gaussian pre-filtering for uncertainty minimization in digital image correlation using numerically-designed speckle patterns,” Opt. Lasers Eng. 66, 19–33 (2015).
[Crossref]

B. Pan, “Bias error reduction of digital image correlation using Gaussian pre-filtering,” Opt. Lasers Eng. 51(10), 1161–1167 (2013).
[Crossref]

H. Haddadi and S. Belhabib, “Use of rigid-body motion for the investigation and estimation of the measurement errors related to digital image correlation technique,” Opt. Lasers Eng. 46(2), 185–196 (2008).
[Crossref]

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. Van Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44(11), 1132–1145 (2006).
[Crossref]

Opt. Lett. (2)

Rock Mech. Rock Eng. (1)

H. Zhang, D. Fu, H. Song, Y. Kang, G. Huang, G. Qi, and J. Li, “Damage and fracture investigation of three-point bending notched sandstone beams by DIC and AE techniques,” Rock Mech. Rock Eng. 48(3), 1297–1303 (2014).
[Crossref]

Scr. Mater. (1)

G. F. Xiang, Q. C. Zhang, H. W. Liu, X. P. Wu, and X. Y. Ju, “Time-resolved deformation measurements of the Portevin–Le Chatelier bands,” Scr. Mater. 56(8), 721–724 (2007).
[Crossref]

Strain (3)

F. Hild and S. Roux, “Digital image correlation: from displacement measurement to identification of elastic properties – a review,” Strain 42(2), 69–80 (2006).
[Crossref]

Y. Q. Wang, M. A. Sutton, H. A. Bruck, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurements,” Strain 45(2), 160–178 (2009).
[Crossref]

W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain 41(4), 167–175 (2005).
[Crossref]

Other (2)

Y. Su, “Interpolation bias prediction in digital image correlation,” figshare (2015) [retrieved 10 July 2015] http://dx.doi.org/.
[Crossref]

M. A. Sutton, J. J. Orteu, and H. Schreier, Image Correlation for Shape, Motion and Deformation Measurements: Basic Concepts, Theory and Applications (Springer Science & Business Media, 2009).

Supplementary Material (4)

NameDescription
» Code 1       MATLAB implementation of the interpolation bias prediction algorithm
» Visualization 1: MOV (7503 KB)      experiment of fine speckle pattern
» Visualization 2: MOV (7503 KB)      experiment of medium speckle pattern
» Visualization 3: MOV (7503 KB)      experiment of coarse speckle pattern

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (10)

Fig. 1
Fig. 1 Translation simulation: (a) reference function; (b) sampled values of reference function; (c) translated function; (d) sampled values of translated function; (e) interpolation basis function; (f) reconstructed function.
Fig. 2
Fig. 2 (a) Interpolation basis function; (b) interpolation transfer function; (c) interpolation bias kernels; (d) aliasing effect of Keys interpolation.
Fig. 3
Fig. 3 1-D Synthetic speckle and interpolation bias: (a1)(b1)(c1) speckle patterns of radius 1.5, 2.0, 3.0 respectively; (a2)(b2)(c2) digital image correlation error, full estimate of interpolation bias [Eq. (12)] and sinusoidal approximation of interpolation bias [Eq. (15)] for radii 1.5, 2.0, 3.0.
Fig. 4
Fig. 4 Power spectrum (on a logarithmic scale) of speckle patterns with radii 1.5, 2.0, and 3.0 and the interpolation bias kernels of Keys, cubic BSpline and cubic OMOMS interpolation.
Fig. 5
Fig. 5 Traditional spray-painted speckle patterns (a) autocorrelation functions of different subsets; (b) experiment speckle image and correlation subsets; (c) interpolation bias by digital image correlation and theoretical prediction by sinusoidal approximation [Eq. (22)] of different subsets.
Fig. 6
Fig. 6 Speckle patterns of different subsets and the corresponding discrete Fourier transforms; the speckle patterns are magnified to identical size for clarification: (a1)(b1)(c1)(d1) speckle patterns of subsets 51, 101, 251, 501; (a2)(b2)(c2)(d2) discrete Fourier transforms of subsets 51, 101, 251, 501.
Fig. 7
Fig. 7 Numerically designed speckle patterns: (a1)(b1) speckle patterns and calculation subsets of coarse and fine patterns; (a2)(b2) discrete Fourier transforms of subsets of coarse and fine patterns; (a3) results of digital image correlation, band-limited approximation [Eq. (21)], and sinusoidal approximation [Eq. (22)] for coarse and fine patterns.
Fig. 8
Fig. 8 (a1)(b1) Close-ups of coarse and fine patterns; (a2)(b2) autocorrelation of coarse and fine patterns.
Fig. 9
Fig. 9 (a) Speckle pattern on the computer screen; (b) experimental setup.
Fig. 10
Fig. 10 (a1)(b1)(c1) Correlation subsets of fine, medium and coarse patterns (see Visualization 1, Visualization 2, and Visualization 3); (a2)(b2)(c2) translations calculated by digital image correlation; (a3)(b3)(c3) interpolation bias by digital image correlation and theoretical prediction by sinusoidal approximation [Eq. (22)]

Equations (22)

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

f T ( x )= k= c k r( xk ) ,
f T ( t )= k= f[ k ]φ( tk ) .
g( x )= n= g[ n ]φ( xn ) .
u=argmin n= [ f( n )g( n+u ) ] 2 .
u=argmin 1/2 1/2 | F ^ ( ν ) G ^ ( ν ) | 2 dν F ^ ( ν )= n= f( n ) e j2πνn , G ^ ( ν )= n= g( n+u ) e j2πνn
F ^ ( ν )= m= f ^ ( νm ) .
f( x ) F f ^ ( ν ) f( x u 0 ) F e j2πν u 0 f ^ ( ν ) f( x u 0 )comb( x ) F k= e j2π( νk ) u 0 f ^ ( νk ) g( x )=φ( x )[ f( x u 0 )comb( x ) ] F φ ^ ( ν ) k= e j2π( νk ) u 0 f ^ ( νk )
G ^ ( ν )= m= e j2π( νm )u φ ^ ( νm ) k= e j2π( νk ) u 0 f ^ ( νk ) .
u=argminΓ( u ) Γ( u )= 1/2 1/2 [ k= f ^ ( νk ) n= e j2π( νn ) u 0 f ^ * ( νn ) m= e j2π( νm )u φ ^ * ( νm ) k= f ^ * ( νk ) n= e j2π( νn ) u 0 f ^ ( νn ) m= e j2π( νm )u φ ^ ( νm ) + m 1 = m 2 = e j2π( m 2 m 1 )u φ ^ ( ν m 1 ) φ ^ * ( ν m 2 ) n 1 = n 2 = e j2π( n 1 n 2 ) u 0 f ^ ( ν n 1 ) f ^ * ( ν n 2 ) ] dν
Γ ( u )=0.
Γ ( u 0 )+ Γ ( u 0 ) u e 0.
u e Γ ( u 0 ) Γ ( u 0 ) Γ ( u 0 )=j2π 1/2 1/2 [ k= f ^ ( νk ) m= n= e j2π( mn ) u 0 ( νm ) f ^ * ( νn ) φ ^ * ( νm ) k= f ^ * ( νk ) m= n= e j2π( mn ) u 0 ( νm ) f ^ ( νn ) φ ^ ( νm ) + n 1 = n 2 = m 1 = m 2 = e j2π( n 1 n 2 + m 2 m 1 ) u 0 ( m 2 m 1 ) f ^ ( ν n 1 ) f ^ * ( ν n 2 ) φ ^ ( ν m 1 ) φ ^ * ( ν m 2 ) ]dν Γ ( u 0 )=4 π 2 1/2 1/2 [ k= f ^ ( νk ) m= n= e j2π( mn ) u 0 ( νm ) 2 f ^ * ( νn ) φ ^ * ( νm ) + k= f ^ * ( νk ) m= n= e j2π( mn ) u 0 ( νm ) 2 f ^ ( νn ) φ ^ ( νm ) n 1 = n 2 = m 1 = m 2 = e j2π( n 1 n 2 + m 2 m 1 ) u 0 ( m 2 m 1 ) 2 f ^ ( ν n 1 ) f ^ * ( ν n 2 ) φ ^ ( ν m 1 ) φ ^ * ( ν m 2 ) ]dν
Γ ( u 0 )4 π 2 1/2 1/2 ν 2 ( φ ^ ( ν )+ φ ^ * ( ν ) ) | f ^ ( ν ) | 2 dν .
u e Γ ( u 0 ) Γ ( u 0 ) Γ ( u 0 )=4π 1/2 1/2 [ m= ( νm ) φ ^ ( νm )sin( 2πm u 0 ) + 1 2 m 1 = m 2 = ( m 2 m 1 ) φ ^ ( ν m 1 ) φ ^ ( ν m 2 )sin( 2π( m 2 m 1 ) u 0 ) ] | f ^ ( ν ) | 2 dν Γ ( u 0 )8 π 2 1/2 1/2 ν 2 φ ^ ( ν ) | f ^ ( ν ) | 2 dν
u e Csin2π u 0 C= 1 2π 1/2 1/2 [ ( ν1 ) φ ^ ( ν1 )( ν+1 ) φ ^ ( ν+1 )+ φ ^ ( ν ) φ ^ ( ν+1 )+ φ ^ ( ν ) φ ^ ( ν1 ) ] | f ^ ( ν ) | 2 dν 1/2 1/2 ν 2 φ ^ ( ν ) | f ^ ( ν ) | 2 dν
E ib ( ν )=( ν1 ) φ ^ ( ν1 )( ν+1 ) φ ^ ( ν+1 )+ φ ^ ( ν ) φ ^ ( ν+1 )+ φ ^ ( ν ) φ ^ ( ν1 )
{ E ib Keys ( ν )> E ib BSpline ( ν ),| ν |<0.39 E ib BSpline ( ν )> E ib OMOMS ( ν ),| ν |<0.35
f( x )= k=0 N I k exp[ ( x x k ) 2 r 2 ] ,
f ^ ( ν )= π r e π 2 r 2 ν 2 k=0 N I k e j2πν x k .
Ω [ k m n e j2π( mn ) u 0 φ ^ * ( νm ) f ^ ( νk ) f ^ * ( νn )( νm ) k m n e j2π( mn ) u 0 φ ^ ( νm ) f ^ * ( νk ) f ^ ( νn )( νm ) + n 1 n 2 m 1 m 2 e j2π( m 2 m 1 + n 1 n 2 ) u 0 φ ^ ( ν m 1 ) φ ^ * ( ν m 2 ) f ^ ( ν n 1 ) f ^ * ( ν n 2 )( m 2 m 1 ) ] dν =j2π{ Ω [ k m n e j2π( mn ) u 0 φ ^ * ( νm ) f ^ ( νk ) f ^ * ( νn )( νm )( νm ) + k m n e j2π( mn ) u 0 φ ^ ( νm ) f ^ * ( νk ) f ^ ( νn )( νm )( νm ) n 1 n 2 m 1 m 2 e j2π( m 2 m 1 + n 1 n 2 ) u 0 φ ^ ( ν m 1 ) φ ^ * ( ν m 2 ) f ^ ( ν n 1 ) f ^ * ( ν n 2 )( m 2 m 1 )( m 2 m 1 ) ] dν } u e
u e 1 2π 1/2 1/2 1/2 1/2 [ k= l= ( ν x k ) φ ^ ( ν x k, ν y l )sin( 2πk u 0 ) + 1 2 k 1 = k 2 = l 1 = l 2 = ( k 2 k 1 ) φ ^ ( ν x k 1 , ν y l 1 ) φ ^ ( ν x k 2 , ν y l 2 )sin( 2π( k 2 k 1 ) u 0 ) ] | f ^ ( ν x , ν y ) | 2 d ν x d ν y 1/2 1/2 1/2 1/2 ν x 2 φ ^ ( ν x , ν y ) | f ^ ( ν x , ν y ) | 2 d ν x d ν y
u e Csin( 2π u 0 ) C= 1 2π 1/2 1/2 1/2 1/2 E ib ( ν x , ν y ) | f ^ ( ν x , ν y ) | 2 d ν x d ν y 1/2 1/2 1/2 1/2 ν x 2 φ ^ ( ν x , ν y ) | f ^ ( ν x , ν y ) | 2 d ν x d ν y E ib ( ν x , ν y )=( ν x 1 ) φ ^ ( ν x 1, ν y )( ν x +1 ) φ ^ ( ν x +1, ν y )+ φ ^ ( ν x , ν y ) φ ^ ( ν x 1, ν y )+ φ ^ ( ν x , ν y ) φ ^ ( ν x +1, ν y )

Metrics