Abstract

In digital image correlation (DIC), the noise-induced bias is significant if the noise level is high or the contrast of the image is low. However, existing methods for the estimation of the noise-induced bias are merely applicable to traditional interpolation methods such as linear and cubic interpolation, but are not applicable to generalized interpolation methods such as BSpline and OMOMS. Both traditional interpolation and generalized interpolation belong to convolution-based interpolation. Considering the widely use of generalized interpolation, this paper presents a theoretical analysis of noise-induced bias for convolution-based interpolation. A sinusoidal approximate formula for noise-induced bias is derived; this formula motivates an estimating strategy which is with speed, ease, and accuracy; furthermore, based on this formula, the mechanism of sophisticated interpolation methods generally reducing noise-induced bias is revealed. The validity of the theoretical analysis is established by both numerical simulations and actual subpixel translation experiment. Compared to existing methods, formulae provided by this paper are simpler, briefer, and more general. In addition, a more intuitionistic explanation of the cause of noise-induced bias is provided by quantitatively characterized the position-dependence of noise variability in the spatial domain.

© 2016 Optical Society of America

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References

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    [Crossref]
  27. M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image representation and interpolation,” IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991).
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    [Crossref]
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    [Crossref]
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    [Crossref]

2015 (2)

X. Xu, Y. Su, Y. Cai, T. Cheng, and Q. Zhang, “Effects of various shape functions and subset size in local deformation measurements using DIC,” Exp. Mech. 55(8), 1575–1590 (2015).
[Crossref]

Y. Su, Q. Zhang, Z. Gao, X. Xu, and X. Wu, “Fourier-based interpolation bias prediction in digital image correlation,” Opt. Express 23(15), 19242–19260 (2015).
[Crossref] [PubMed]

2013 (1)

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

2011 (3)

2009 (3)

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]

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. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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 (1)

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)

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]

M. S. Kirugulige, H. V. Tippur, and T. S. Denney, “Measurement of transient deformations using digital image correlation method and high-speed photography: application to dynamic fracture,” Appl. Opt. 46(22), 5083–5096 (2007).
[Crossref] [PubMed]

2006 (3)

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]

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]

2005 (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]

2002 (2)

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]

E. Meijering, “A chronology of interpolation: from ancient astronomy to modern signal and image processing,” Proc. IEEE 90(3), 319–342 (2002).
[Crossref]

2001 (1)

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]

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]

1991 (2)

M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image representation and interpolation,” IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991).
[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]

1988 (1)

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

1985 (1)

T. C. Chu, W. F. Ranson, and M. A. Sutton, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[Crossref]

1982 (1)

W. H. Peters and W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21(3), 427–431 (1982).
[Crossref]

1981 (2)

I. Yamaguchi, “A laser-speckle strain gauge,” J. Phys. E Sci. Instrum. 14(11), 1270–1273 (1981).
[Crossref]

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

Aldroubi, A.

M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image representation and interpolation,” IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991).
[Crossref]

Amiot, F.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

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. 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.

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

Bornert, M.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Bremand, F.

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Cai, Y.

X. Xu, Y. Su, Y. Cai, T. Cheng, and Q. Zhang, “Effects of various shape functions and subset size in local deformation measurements using DIC,” Exp. Mech. 55(8), 1575–1590 (2015).
[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]

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]

Cheng, T.

X. Xu, Y. Su, Y. Cai, T. Cheng, and Q. Zhang, “Effects of various shape functions and subset size in local deformation measurements using DIC,” Exp. Mech. 55(8), 1575–1590 (2015).
[Crossref]

Chu, T. C.

T. C. Chu, W. F. Ranson, and M. A. Sutton, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[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]

Denney, T. S.

Doumalin, P.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Dupre, J. C.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Eden, M.

M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image representation and interpolation,” IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991).
[Crossref]

Fazzini, M.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Gao, J.

Gao, Z.

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]

Grediac, M.

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Hild, F.

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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.

Jang, J.

M. A. Sutton, S. R. McNeill, J. 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]

Jiang, 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]

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]

Ke, X. D.

X. D. Ke, H. W. Schreier, M. A. Sutton, and Y. Q. Wang, “Error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 423–441 (2011).
[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]

Kirugulige, M. S.

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]

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]

Liu, Z.

Luu, L.

Ma, J.

McNeill, S. R.

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

Meijering, E.

E. Meijering, “A chronology of interpolation: from ancient astronomy to modern signal and image processing,” Proc. IEEE 90(3), 319–342 (2002).
[Crossref]

Mistou, S.

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Orteu, J. J.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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.

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]

Peters, W. H.

W. H. Peters and W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21(3), 427–431 (1982).
[Crossref]

Poilane, C.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[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]

Ranson, W. F.

T. C. Chu, W. F. Ranson, and M. A. Sutton, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[Crossref]

W. H. Peters and W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21(3), 427–431 (1982).
[Crossref]

Robert, L.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Rotinat, R.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

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]

Schreier, H. W.

X. D. Ke, H. W. Schreier, M. A. Sutton, and Y. Q. Wang, “Error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 423–441 (2011).
[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]

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]

Su, Y.

X. Xu, Y. Su, Y. Cai, T. Cheng, and Q. Zhang, “Effects of various shape functions and subset size in local deformation measurements using DIC,” Exp. Mech. 55(8), 1575–1590 (2015).
[Crossref]

Y. Su, Q. Zhang, Z. Gao, X. Xu, and X. Wu, “Fourier-based interpolation bias prediction in digital image correlation,” Opt. Express 23(15), 19242–19260 (2015).
[Crossref] [PubMed]

Surrel, Y.

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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.

X. D. Ke, H. W. Schreier, M. A. Sutton, and Y. Q. Wang, “Error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 423–441 (2011).
[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]

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

T. C. Chu, W. F. Ranson, and M. A. Sutton, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[Crossref]

Thévenaz, P.

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

Tippur, H. V.

Toussaint, E.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[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.

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]

M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image representation and interpolation,” IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991).
[Crossref]

Vacher, P.

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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.

X. D. Ke, H. W. Schreier, M. A. Sutton, and Y. Q. Wang, “Error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 423–441 (2011).
[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]

Wang, Z.

Wattrisse, B.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

Wienin, J. S.

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

Wu, X.

Y. Su, Q. Zhang, Z. Gao, X. Xu, and X. Wu, “Fourier-based interpolation bias prediction in digital image correlation,” Opt. Express 23(15), 19242–19260 (2015).
[Crossref] [PubMed]

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, 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. Su, Q. Zhang, Z. Gao, X. Xu, and X. Wu, “Fourier-based interpolation bias prediction in digital image correlation,” Opt. Express 23(15), 19242–19260 (2015).
[Crossref] [PubMed]

X. Xu, Y. Su, Y. Cai, T. Cheng, and Q. Zhang, “Effects of various shape functions and subset size in local deformation measurements using DIC,” Exp. Mech. 55(8), 1575–1590 (2015).
[Crossref]

Yamaguchi, I.

I. Yamaguchi, “A laser-speckle strain gauge,” J. Phys. E Sci. Instrum. 14(11), 1270–1273 (1981).
[Crossref]

Zhang, Q.

Y. Su, Q. Zhang, Z. Gao, X. Xu, and X. Wu, “Fourier-based interpolation bias prediction in digital image correlation,” Opt. Express 23(15), 19242–19260 (2015).
[Crossref] [PubMed]

X. Xu, Y. Su, Y. Cai, T. Cheng, and Q. Zhang, “Effects of various shape functions and subset size in local deformation measurements using DIC,” Exp. Mech. 55(8), 1575–1590 (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]

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]

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. (6)

T. C. Chu, W. F. Ranson, and M. A. Sutton, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25(3), 232–244 (1985).
[Crossref]

M. Bornert, F. Bremand, P. Doumalin, J. C. Dupre, M. Fazzini, M. Grediac, 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]

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. 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]

X. Xu, Y. Su, Y. Cai, T. Cheng, and Q. Zhang, “Effects of various shape functions and subset size in local deformation measurements using DIC,” Exp. Mech. 55(8), 1575–1590 (2015).
[Crossref]

X. D. Ke, H. W. Schreier, M. A. Sutton, and Y. Q. Wang, “Error assessment in stereo-based deformation measurements,” Exp. Mech. 51(4), 423–441 (2011).
[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. Med. Imaging (1)

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

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

M. Unser, A. Aldroubi, and M. Eden, “Fast B-spline transforms for continuous image representation and interpolation,” IEEE Trans. Pattern Anal. Mach. Intell. 13(3), 277–285 (1991).
[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. Phys. E Sci. Instrum. (1)

I. Yamaguchi, “A laser-speckle strain gauge,” J. Phys. E Sci. Instrum. 14(11), 1270–1273 (1981).
[Crossref]

Meas. Sci. Technol. (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]

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]

Opt. Eng. (4)

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]

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]

W. H. Peters and W. F. Ranson, “Digital imaging techniques in experimental stress analysis,” Opt. Eng. 21(3), 427–431 (1982).
[Crossref]

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

Opt. Express (2)

Opt. Lasers Eng. (2)

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]

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]

Opt. Lett. (1)

Proc. IEEE (1)

E. Meijering, “A chronology of interpolation: from ancient astronomy to modern signal and image processing,” Proc. IEEE 90(3), 319–342 (2002).
[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]

F. Amiot, M. Bornert, P. Doumalin, J. C. Dupre, M. Fazzini, J. J. Orteu, C. Poilane, L. Robert, R. Rotinat, E. Toussaint, B. Wattrisse, and J. S. Wienin, “Assessment of digital image correlation measurement accuracy in the ultimate error regime: main results of a collaborative benchmark,” Strain 49(6), 483–496 (2013).
[Crossref]

Other (2)

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).

J. W. Goodman, Introduction to Fourier Optics (Roberts and Company Publishers, 2005).

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Figures (10)

Fig. 1
Fig. 1 Comparison of traditional interpolation and generalized interpolation. (a1) Interpolation bases of traditional interpolation, including linear interpolation and Keys interpolation, superimposed on the desired sinc function. (a2) Cardinal interpolation bases of generalized interpolation, including cubic BSpline, cubic OMOMS, quintic BSpline, and septic BSpline, superimposed on the desired sinc function. (b) Interpolation transfer functions of several interpolation methods, including linear interpolation, Keys interpolation, cubic BSpline, cubic OMOMS, quintic BSpline, septic BSpline, and the transfer function of sinc interpolation – the ideal low-pass filter.
Fig. 2
Fig. 2 1-dimensional speckle patterns utilized in the numerical tests. The speckle radii R of these Gaussian patterns [see Eq. (23)] are (a) 1.5, (b) 2.0, and (c) 3.0 respectively.
Fig. 3
Fig. 3 Systematic errors correspond to (Left) cubic BSpline and (Right) quintic BSpline interpolation at different noise level (noise standard deviations range from 0 to 0.05). From the top to the bottom: systematic errors correspond to speckle patterns shown in Fig. 2(a), 2(b), and 2(c) respectively.
Fig. 4
Fig. 4 Noise-induced bias correspond to (Left) cubic BSpline and (Right) quintic BSpline interpolation at different noise level (noise standard deviations range from 0.01 to 0.05). From the top to the bottom: noise-induced bias for speckle patterns shown in Figs. 2(a), 2(b), and 2(c) respectively.
Fig. 5
Fig. 5 Interpolation-noise coupling function and its corresponding first order Fourier series.
Fig. 6
Fig. 6 Subpixel translation experiment. (a) Experiment set-up. (b) A speckle pattern was displayed on a computer screen; then the pattern was translated using an image processing software. (c) Schematic of the principle: integer pixel translation on the computer screen will introduce a subpixel position in the camera sensor plane.
Fig. 7
Fig. 7 Computational flow chart. (a) Bias without noise: the deformed images are averaged first to obtain a virtual noiseless deform image; then the average deform image correlates with the reference image. (b) Bias with noise: the noisy deformed images correlate with the reference image; then the measured displacements are averaged.
Fig. 8
Fig. 8 Subpixel translation experimental. From left to right, results for speckle patterns with different speckle size. From top to bottom, (a1)-(c1) Subsets for correlation: the speckle size are roughly 4, 6, and 8 pixels respectively. (a2)-(c2) Standard deviations of the image noise: it can be seen that the noise level depend on the image intensity. (a3)-(c3) Systematic errors with and without noise (denoted as noisy and noiseless in the figure): the systematic errors without noise are the interpolation bias; the systematic errors with noise include both interpolation bias and noise-induced bias [Eq. (16)]. The DIC results were obtained following the computational flow chart shown in Fig. 7. The theory estimations were based on the interpolation bias prediction formula in [21] and the noise-induced bias estimation formula Eq. (26). (a4)-(c4) Noise-induced bias: the DIC results were obtained by subtracting interpolation bias from the systematic errors, the theoretical estimations were based on Eq. (26).
Fig. 9
Fig. 9 The position-dependence of noise. The standard deviation of added noise at integer positions is 0.01. The theoretical estimations are based on Eq. (33).
Fig. 10
Fig. 10 (a1)-(a3) Sum of squared intensity gradients correspond to speckle patterns shown in Fig. 2; (b) ECOS(ν).

Tables (1)

Tables Icon

Table 1 Fourier series coefficients of interpolation-noise coupling function.

Equations (47)

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

f( x )= k= f( k )sinc( xk ) .
f T ( x )= k= f( k ) φ int ( xk ) .
f T ( x )= k= c( k )φ( xk ) ,
f( k 0 )= k= c( k )φ( k 0 k ) .
φ int ( x )= k= p( k )φ( xk ) ,
P( z )= 1 k= φ( k ) z k .
g( x )= k= f( k u 0 ) φ int ( xk ) ,
u=argmin C SSD ( u ), C SSD ( u )= n= [ f( n )g( n+u ) ] 2 .
f noise ( n )=f( n )+ ε f ( n ).
g noise ( n+u )= k= [ f( k u 0 )+ ε g ( k ) ] φ int ( n+uk ) =g( n+u )+ k= ε g ( k ) φ int ( n+uk ) =g( n+u )+ ε g ( n+u ),
C SSD ( u )= n= [ f noise ( n )g noise ( n+u ) ] 2 =Γ( u )+Λ( u ); Γ( u )= n= [ f( n )g( n+u ) ] 2 , Λ( u )=2 n= [ f( n )g( n+u ) ][ ε f ( n ) ε g ( n+u ) ] + n= [ ε f ( n ) ε g ( n+u ) ] 2 ,
Γ ( u )+ Λ ( u )=0.
u e Γ ( u 0 )+ Λ ( u 0 ) Γ ( u 0 )+ Λ ( u 0 ) .
E( u e ) Γ ( u 0 )+E( Λ ( u 0 ) ) Γ ( u 0 )+E( Λ ( u 0 ) ) , VA( u e ) VA( Λ ( u 0 ) ) [ Γ ( u 0 )+E( Λ ( u 0 ) ) ] 2 .
E( u e ) n= [ f( n )g( n+ u 0 ) ] g ( n+ u 0 ) N σ 2 k= φ int ( u 0 k ) φ int ( u 0 k ) n= g 2 ( n+ u 0 ) n= [ f( n )g( n+ u 0 ) ] g ( n+ u 0 ) +N σ 2 k= [ φ int 2 ( u 0 k )+ φ int ( u 0 k ) φ int ( u 0 k ) ] .
E( u e ) u ib + u nb ; u ib = n= [ f( n )g( n+ u 0 ) ] g ( n+ u 0 ) n= g 2 ( n+ u 0 ) , u nb = N σ 2 k= φ int ( u 0 k ) φ int ( u 0 k ) n= g 2 ( n+ u 0 ) .
u nb N σ 2 n= f 2 ( n ) Φ( u 0 ),Φ( u 0 )= k= φ int ( u 0 k ) φ int ( u 0 k ) .
Φ( x )= k= Θ( xk ) =comb( x )Θ( x ),
Φ ^ ( ν )=comb( ν ) Θ ^ ( ν ),
Θ ^ ( ν )= j2πξ φ ^ int ( ξ ) φ ^ int ( νξ )dξ .
Φ( x )= m=1 a m sin( 2πmx ) , a m =4π ν φ ^ int ( ν ) φ ^ int ( mν )dν .
u nb N σ 2 n= f 2 ( n ) m=1 M a m sin( 2πm u 0 ) ,
f( x )= k=1 K I 0 exp[ ( x x k ) 2 R 2 ] ,
e u = 1 M i=1 M ( u i u 0 ) , σ u = i=1 M ( u i u 0 e u ) 2 /( M1 ) ,
Keys: φ ^ int ( ν )=sin c 3 ( ν )( 3sinν2cosπν ); CubicBSpline: φ ^ int ( ν )= 3sin c 4 ( ν ) 2+cos2πν ; QuinticBSpline: φ ^ int ( ν )= 60sin c 6 ( ν ) 33+26cos2πν+cos4πν ; SepticBSpline: φ ^ int ( ν )= 2520sin c 8 ( ν ) 1208+1191cos2πν+120cos4πν+cos6πν .
u nb Φ( u 0 ) m=1 M n=1 N σ 2 ( m,n ) m=1 M n=1 N f x 2 ( m,n ) =Φ( u 0 ) σ 2 ¯ f x 2 ¯ ; σ 2 ¯ = m=1 M n=1 N σ 2 ( m,n ) /( MN ) , f x 2 ¯ = m=1 M n=1 N f x 2 ( m,n ) /( MN ) ,
φ Keys ( x )={ 3 2 | x | 3 5 2 | x | 2 +1, 0<| x |<1 1 2 | x | 3 + 5 2 | x | 2 4| x |+2, 1<| x |<2 0, 2<| x |
Φ Keys ( x )= 9 2 x6 x 2 21 x 3 + 75 2 x 4 15 x 5 .
φ linear ( x )={ 1| x | 1x1 0 otherwise
Φ linear ( x )=[ ( 1x )+x ]=12x,when0x1.
σ 2 ( u )= σ 0 2 0 1 h ^ ( k,u ) 2 dk,
f noise ( u )= k= [ f( k )+ε( k ) ] φ int ( uk ) = f T ( u )+ k= ε( k ) φ int ( uk ) ,
σ 2 ( u )=VA( k= ε( k ) φ int ( uk ) )= σ 0 2 k= φ int 2 ( uk ) .
E( C SSD ( u ) )=E( m= [ f( m )+ ε f ( m )g( m+u ) k= ε g ( k ) φ int ( m+uk ) ] 2 ) = m= [ f( m )g( m+u ) ] 2 +N σ f 2 +N σ g 2 k= φ int 2 ( uk ) .
Γ( u )= n= [ f( n )g( n+u ) ] 2 .
1 2 Γ ( u )= n= [ f( n )g( n+u ) ] g ( n+u ) , 1 2 Γ ( u )= n= g 2 ( n+u ) n= [ f( n )g( n+u ) ] g ( n+u ) .
Λ( u )=2 n= [ f( n )g( n+u ) ][ ε f ( n ) ε g ( n+u ) ] + n= [ ε f 2 ( n )2 ε f ( n ) ε g ( n+u )+ ε g 2 ( n+u ) ] .
1 2 Λ ( u )= n= g ( n+u )[ ε f ( n ) ε g ( n+u ) ] n= [ f( n )g( n+u ) ] ε g ( n+u ) n= ε f ( n ) ε g ( n+u ) + n= ε g ( n+u ) ε g ( n+u ) .
E( 1 2 Λ ( u ) )=E( n= ε g ( n+u ) ε g ( n+u ) )=E( n= k 1 = ε g ( k 1 ) φ int ( n+u k 1 ) k 2 = ε g ( k 2 ) φ int ( n+u k 2 ) ) =E( n= k 1 = k 2 = ε g ( k 1 ) ε g ( k 2 ) φ int ( n+u k 1 ) φ int ( n+u k 2 ) ) = k= σ 2 ( k ) n= φ int ( n+uk ) φ int ( n+uk ) = k= σ 2 ( k ) n= φ int ( n+u ) φ int ( n+u ) .
E( 1 2 Λ ( u ) )=N σ 2 k= φ int ( uk ) φ int ( uk ) .
1 2 Λ ( u )= n= g ( n+u )[ ε f ( n ) ε g ( n+u ) ] +2 n= g ( n+u ) ε g ( n+u ) n= [ f( n )g( n+u ) ] ε g ( n+u ) n= ε f ( n ) ε g ( n+u ) + n= ε g 2 ( n+u ) + n= ε g ( n+u ) ε g ( n+u ).
E( 1 2 Λ ( u ) )=E( n= ε g 2 ( n+u ) + n= ε g ( n+u ) ε g ( n+u ) ) =E( n= k 1 = ε g ( k 1 ) φ int ( n+u k 1 ) k 2 = ε g ( k 2 ) φ int ( n+u k 2 ) + n= k 1 = ε g ( k 1 ) φ int ( n+u k 1 ) k 2 = ε g ( k 2 ) φ int ( n+u k 2 ) ) =E( k 1 = k 2 = ε g ( k 1 ) ε g ( k 2 ) n= [ φ int ( n+u k 1 ) φ int ( n+u k 2 )+ φ int ( n+u k 1 ) φ int ( n+u k 2 ) ] ) = k= σ 2 ( k ) n= [ φ int 2 ( n+uk )+ φ int ( n+uk ) φ int ( n+uk ) ] = k= σ 2 ( k ) n= [ φ int 2 ( n+u )+ φ int ( n+u ) φ int ( n+u ) ] =N σ 2 k= [ φ int 2 ( uk )+ φ int ( uk ) φ int ( uk ) ] .
f ^ ( ν )= f( x ) e j2πνx dx .
{ g ( x+ u 0 ) }=j2πν φ ^ int ( ν ) k= e j2πk u 0 f ^ ( νk ) ,
n= g ( n+ u 0 ) e j2πνn = m= j2π( νm ) φ ^ int ( νm ) k= e j2πk u 0 f ^ ( νkm ) .
n= g 2 ( n+ u 0 ) =4 π 2 1/2 1/2 | m= e j2πm u 0 ( νm ) φ ^ int ( νm ) | 2 | f ^ ( ν ) | 2 dν .
n= g 2 ( n+ u 0 ) 4 π 2 1/2 1/2 [ E DC ( ν )+ E COS ( ν )cos2π u 0 ] | f ^ ( ν ) | 2 dν ; E DC ( ν )= ( ν1 ) 2 φ ^ int 2 ( ν1 )+ ν 2 φ ^ int 2 ( ν )+ ( ν+1 ) 2 φ ^ int 2 ( ν+1 ), E COS ( ν )=2ν φ ^ int ( ν )[ ( ν1 ) φ ^ int ( ν1 )+( ν+1 ) φ ^ int ( ν+1 ) ].

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