Abstract

Digital Image Correlation (DIC) is a well-established non-contact optical metrology method. It employs digital image analysis to extract the full-field displacements and strains that occur in objects subjected to external stresses. Despite recent DIC progress, many problematic areas which greatly affect accuracy and that can seldomly be avoided, received very little attention. Problems posed by the presence of sharp displacement discontinuities, reflections, object borders or edges can be linked to the analysed object’s properties and deformation. Other problematic areas, such as image noise, localized reflections or shadows are related more to the image acquisition process. This paper proposes a new subset-based pixel-level robust DIC method for in-plane displacement measurement which addresses all of these problems in a straightforward and unified approach, significantly improving DIC measurement accuracy compared to classic approaches. The proposed approach minimizes a robust energy functional which adaptively weighs pixel differences in the motion estimation process. The aim is to limit the negative influence of pixels that present erroneous or inconsistent motions by enforcing local motion consistency. The proposed method is compared to the classic Newton-Raphson DIC method in terms of displacement accuracy in three experiments. The first experiment is numerical and presents three combined problems: sharp displacement discontinuities, missing image information and image noise. The second experiment is a real experiment in which a plastic specimen is developing a lateral crack due to the application of uniaxial stress. The region around the crack presents both reflections that saturate the image intensity levels leading to missing image information, as well as sharp motion discontinuities due to the plastic film rupturing. The third experiment compares the proposed and classic DIC approaches with generic computer vision optical flow methods using images from the popular Middlebury optical flow evaluation dataset. Results in all experiments clearly show the proposed method’s improved measurement accuracy with respect to the classic approach considering the challenging conditions. Furthermore, in image areas where the classic approach completely fails to recover motion due to severe image de-correlation, the proposed method provides reliable results.

© 2013 OSA

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. W. H. Peters and W. F. Ranson, “Digital imaging techniques in experimental stress-analysis,” Opt. Eng.21, 427–431 (1982).
    [CrossRef]
  2. M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
    [CrossRef]
  3. W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
    [CrossRef]
  4. B. Pan, A. Asundi, H-M. Xie, and J. X. Gao, “Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements,” Opt. Lasers Eng.47, 865–874 (2009).
    [CrossRef]
  5. J. Zhang, Y. Cai, W. Ye, and T.X. Yu, “On the use of the digital image correlation method for heterogeneous deformation measurement of porous solids,” Opt. Lasers Eng.49, 200–209 (2011).
    [CrossRef]
  6. L.B. Meng, G.C. Jin, and X.F. Yao, “Application of iteration and finite element smoothing technique for displacement and strain measurement of digital speckle correlation,” Opt. Lasers Eng.45, 56–73 (2007).
    [CrossRef]
  7. C. Tang, L. Wang, S. Yan, J. Wu, L. Cheng, and C. Li, “Displacement field analysis based on the combination digital speckle correlation method with radial basis function interpolation,” Appl. Opt.49, 4545–4553 (2010).
    [CrossRef] [PubMed]
  8. Y. Sun, J.H.L. Pang, C.K. Wong, and F. Su, “Finite element formulation for a digital image correlation method,” Appl. Opt.44, 7357–7363 (2005).
    [CrossRef] [PubMed]
  9. Y.N. Chen, W.Q. Jin, L. Zhao, and F.W. Li, “A subpixel motion estimation algorithm based on digital correlation for illumination variant and noise image sequences” Optik120, 835–844 (2009).
    [CrossRef]
  10. B. Pan, Z. Wang, and Z. Lu, “Genuine full-field deformation measurement of an object with complex shape using reliability-guided digital image correlation,” Opt. Express18, 1011–1023 (2010).
    [CrossRef] [PubMed]
  11. J. Poissant and F. Barthelat, “A novel subset splitting procedure for digital image correlation on discontinuous displacement fields,” Exp. Mech.50, 353–364 (2010).
    [CrossRef]
  12. C. Cofaru, W. Philips, and W. Van Paepegem, “Improved Newton-Raphson digital image correlation method for full-field displacement and strain calculation,” Appl. Opt.49, 6472–6484 (2010).
    [CrossRef] [PubMed]
  13. C. Cofaru, W. Philips, and W. Van Paepegem, “A three-frame digital image correlation (DIC) method for the measurement of small displacements and strains,” Meas. Sci. Technol.23, 105406 (14 pp.) (2012).
    [CrossRef]
  14. G. Besnard, F. Hild, and S. Roux, “Finite-Element’ displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands,” Exp. Mech.46, 789–803 (2006).
    [CrossRef]
  15. J. Réthoré, S. Roux, and F. Hild, “From pictures to extended finite elements: extended digital image correlation (X-DIC),” C.R. Mécanique335, 131–137 (2007).
    [CrossRef]
  16. J. Réthoré, F. Hild, and S. Roux, “Extended digital image correlation with crack shape optimization,” Int. J. Numer. Meth. Eng.73, 248–272 (2008).
    [CrossRef]
  17. M. A. Sutton, J-J. Orteu, and H. W. Schreier, Image correlation for shape, motion and deformation measurements (Springer, 2009).
  18. B. Pan, H. Xie, and Z. Wang, “Equivalence of digital image correlation criteria for pattern matching,” Appl. Opt.49, 5501–5509 (2010).
    [CrossRef] [PubMed]
  19. B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
    [CrossRef]
  20. C. Cofaru, W. Philips, and W. Van Paepegem, “A novel speckle pattern - adaptive Digital Image Correlation approach with robust strain calculation,” Opt. Lasers Eng.50, 187–198 (2012).
    [CrossRef]
  21. P.J. Huber and E.M. Ronchetti, Robust Statistics, 2nd Edition (John Wiley & Sons, New York (NY), 2009).
    [CrossRef]
  22. P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection (John Wiley & Sons, 1987)
    [CrossRef]
  23. F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions (John Wiley & Sons, 1986).
  24. 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, 261–267 (1989).
    [CrossRef]
  25. G. Vendroux and W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech.38, 86–92 (1998).
    [CrossRef]
  26. M. J. Black and P. Anandan, “The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields,” Comput. Vis. Image Underst.63, 75–104 (1996).
    [CrossRef]
  27. M. Ye, R. M. Haralick, and L. G. Shapiro, “Estimating piecewise-smooth optical flow with global matching and graduated optimization,” IEEE Trans. Pattern Anal. Mach. Intell.25, 1625–1630 (2003).
    [CrossRef]
  28. C-H. Teng, S-H. Lai, Y-S. Chen, and W-H. Hsu, “Accurate optical flow computation under non-uniform brightness variations,” Comput. Vis. Image Underst.97, 315–346 (2005).
    [CrossRef]
  29. Y-H. Kim, A. M. Martìnez, and A. C. Kak, “Robust motion estimation under varying illumination,” Image Vis. Comput.23, 365–375 (2005).
    [CrossRef]
  30. S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
    [CrossRef]
  31. X. Ren, “Local grouping for optical flow,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2008) pp. 1–8.
  32. D. Sun, E. B. Sudderth, and M. J. Black, “Layered image motion with explicit occlusions, temporal consistency, and depth ordering,” Adv. Neural Inf. Process. Syst.23, 2226–2234 (2010).
  33. D. Sun, E. B. Sudderth, and M. J. Black, “Layered segmentation and optical flow estimation over time,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2012) pp. 1768–1775.
  34. Y. Weiss, “Smoothness in layers: Motion segmentation using nonparametric mixture estimation,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997) pp. 520–526.
    [CrossRef]
  35. M. Werlberger, T. Pock, and H. Bischof, “Motion estimation with non-local total variation regularization,” Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2010) pp. 2464–2471.
  36. M.J. Black, Robust incremental optical flow(PhD. Thesis, Yale University, 1992).
  37. C. Cofaru, W. Philips, and W. Van Paepegem, “Evaluation of digital image correlation techniques using realistic ground truth speckle images,” Meas. Sci. Technol.21, 055102 (17 pp.) (2010).
    [CrossRef]

2012

C. Cofaru, W. Philips, and W. Van Paepegem, “A three-frame digital image correlation (DIC) method for the measurement of small displacements and strains,” Meas. Sci. Technol.23, 105406 (14 pp.) (2012).
[CrossRef]

B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “A novel speckle pattern - adaptive Digital Image Correlation approach with robust strain calculation,” Opt. Lasers Eng.50, 187–198 (2012).
[CrossRef]

2011

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

J. Zhang, Y. Cai, W. Ye, and T.X. Yu, “On the use of the digital image correlation method for heterogeneous deformation measurement of porous solids,” Opt. Lasers Eng.49, 200–209 (2011).
[CrossRef]

2010

2009

Y.N. Chen, W.Q. Jin, L. Zhao, and F.W. Li, “A subpixel motion estimation algorithm based on digital correlation for illumination variant and noise image sequences” Optik120, 835–844 (2009).
[CrossRef]

B. Pan, A. Asundi, H-M. Xie, and J. X. Gao, “Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements,” Opt. Lasers Eng.47, 865–874 (2009).
[CrossRef]

2008

J. Réthoré, F. Hild, and S. Roux, “Extended digital image correlation with crack shape optimization,” Int. J. Numer. Meth. Eng.73, 248–272 (2008).
[CrossRef]

2007

J. Réthoré, S. Roux, and F. Hild, “From pictures to extended finite elements: extended digital image correlation (X-DIC),” C.R. Mécanique335, 131–137 (2007).
[CrossRef]

L.B. Meng, G.C. Jin, and X.F. Yao, “Application of iteration and finite element smoothing technique for displacement and strain measurement of digital speckle correlation,” Opt. Lasers Eng.45, 56–73 (2007).
[CrossRef]

2006

G. Besnard, F. Hild, and S. Roux, “Finite-Element’ displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands,” Exp. Mech.46, 789–803 (2006).
[CrossRef]

2005

Y. Sun, J.H.L. Pang, C.K. Wong, and F. Su, “Finite element formulation for a digital image correlation method,” Appl. Opt.44, 7357–7363 (2005).
[CrossRef] [PubMed]

C-H. Teng, S-H. Lai, Y-S. Chen, and W-H. Hsu, “Accurate optical flow computation under non-uniform brightness variations,” Comput. Vis. Image Underst.97, 315–346 (2005).
[CrossRef]

Y-H. Kim, A. M. Martìnez, and A. C. Kak, “Robust motion estimation under varying illumination,” Image Vis. Comput.23, 365–375 (2005).
[CrossRef]

2003

M. Ye, R. M. Haralick, and L. G. Shapiro, “Estimating piecewise-smooth optical flow with global matching and graduated optimization,” IEEE Trans. Pattern Anal. Mach. Intell.25, 1625–1630 (2003).
[CrossRef]

1998

G. Vendroux and W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech.38, 86–92 (1998).
[CrossRef]

1996

M. J. Black and P. Anandan, “The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields,” Comput. Vis. Image Underst.63, 75–104 (1996).
[CrossRef]

1989

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, 261–267 (1989).
[CrossRef]

1983

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
[CrossRef]

W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
[CrossRef]

1982

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

Anandan, P.

M. J. Black and P. Anandan, “The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields,” Comput. Vis. Image Underst.63, 75–104 (1996).
[CrossRef]

Anderson, J.

W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
[CrossRef]

Asundi, A.

B. Pan, A. Asundi, H-M. Xie, and J. X. Gao, “Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements,” Opt. Lasers Eng.47, 865–874 (2009).
[CrossRef]

Baker, S.

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

Barthelat, F.

J. Poissant and F. Barthelat, “A novel subset splitting procedure for digital image correlation on discontinuous displacement fields,” Exp. Mech.50, 353–364 (2010).
[CrossRef]

Besnard, G.

G. Besnard, F. Hild, and S. Roux, “Finite-Element’ displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands,” Exp. Mech.46, 789–803 (2006).
[CrossRef]

Bischof, H.

M. Werlberger, T. Pock, and H. Bischof, “Motion estimation with non-local total variation regularization,” Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2010) pp. 2464–2471.

Black, M. J.

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

D. Sun, E. B. Sudderth, and M. J. Black, “Layered image motion with explicit occlusions, temporal consistency, and depth ordering,” Adv. Neural Inf. Process. Syst.23, 2226–2234 (2010).

M. J. Black and P. Anandan, “The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields,” Comput. Vis. Image Underst.63, 75–104 (1996).
[CrossRef]

D. Sun, E. B. Sudderth, and M. J. Black, “Layered segmentation and optical flow estimation over time,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2012) pp. 1768–1775.

Black, M.J.

M.J. Black, Robust incremental optical flow(PhD. Thesis, Yale University, 1992).

Bruck, H. A.

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, 261–267 (1989).
[CrossRef]

Cai, Y.

J. Zhang, Y. Cai, W. Ye, and T.X. Yu, “On the use of the digital image correlation method for heterogeneous deformation measurement of porous solids,” Opt. Lasers Eng.49, 200–209 (2011).
[CrossRef]

Chen, Y.N.

Y.N. Chen, W.Q. Jin, L. Zhao, and F.W. Li, “A subpixel motion estimation algorithm based on digital correlation for illumination variant and noise image sequences” Optik120, 835–844 (2009).
[CrossRef]

Chen, Y-S.

C-H. Teng, S-H. Lai, Y-S. Chen, and W-H. Hsu, “Accurate optical flow computation under non-uniform brightness variations,” Comput. Vis. Image Underst.97, 315–346 (2005).
[CrossRef]

Cheng, L.

Chu, T.C.

W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
[CrossRef]

Cofaru, C.

C. Cofaru, W. Philips, and W. Van Paepegem, “A three-frame digital image correlation (DIC) method for the measurement of small displacements and strains,” Meas. Sci. Technol.23, 105406 (14 pp.) (2012).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “A novel speckle pattern - adaptive Digital Image Correlation approach with robust strain calculation,” Opt. Lasers Eng.50, 187–198 (2012).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “Improved Newton-Raphson digital image correlation method for full-field displacement and strain calculation,” Appl. Opt.49, 6472–6484 (2010).
[CrossRef] [PubMed]

C. Cofaru, W. Philips, and W. Van Paepegem, “Evaluation of digital image correlation techniques using realistic ground truth speckle images,” Meas. Sci. Technol.21, 055102 (17 pp.) (2010).
[CrossRef]

Ding, L.

B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
[CrossRef]

Gao, J. X.

B. Pan, A. Asundi, H-M. Xie, and J. X. Gao, “Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements,” Opt. Lasers Eng.47, 865–874 (2009).
[CrossRef]

Hampel, F.R.

F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions (John Wiley & Sons, 1986).

Hao, X.

B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
[CrossRef]

Haralick, R. M.

M. Ye, R. M. Haralick, and L. G. Shapiro, “Estimating piecewise-smooth optical flow with global matching and graduated optimization,” IEEE Trans. Pattern Anal. Mach. Intell.25, 1625–1630 (2003).
[CrossRef]

Hild, F.

J. Réthoré, F. Hild, and S. Roux, “Extended digital image correlation with crack shape optimization,” Int. J. Numer. Meth. Eng.73, 248–272 (2008).
[CrossRef]

J. Réthoré, S. Roux, and F. Hild, “From pictures to extended finite elements: extended digital image correlation (X-DIC),” C.R. Mécanique335, 131–137 (2007).
[CrossRef]

G. Besnard, F. Hild, and S. Roux, “Finite-Element’ displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands,” Exp. Mech.46, 789–803 (2006).
[CrossRef]

Hsu, W-H.

C-H. Teng, S-H. Lai, Y-S. Chen, and W-H. Hsu, “Accurate optical flow computation under non-uniform brightness variations,” Comput. Vis. Image Underst.97, 315–346 (2005).
[CrossRef]

Huber, P.J.

P.J. Huber and E.M. Ronchetti, Robust Statistics, 2nd Edition (John Wiley & Sons, New York (NY), 2009).
[CrossRef]

Jin, G.C.

L.B. Meng, G.C. Jin, and X.F. Yao, “Application of iteration and finite element smoothing technique for displacement and strain measurement of digital speckle correlation,” Opt. Lasers Eng.45, 56–73 (2007).
[CrossRef]

Jin, W.Q.

Y.N. Chen, W.Q. Jin, L. Zhao, and F.W. Li, “A subpixel motion estimation algorithm based on digital correlation for illumination variant and noise image sequences” Optik120, 835–844 (2009).
[CrossRef]

Kak, A. C.

Y-H. Kim, A. M. Martìnez, and A. C. Kak, “Robust motion estimation under varying illumination,” Image Vis. Comput.23, 365–375 (2005).
[CrossRef]

Kim, Y-H.

Y-H. Kim, A. M. Martìnez, and A. C. Kak, “Robust motion estimation under varying illumination,” Image Vis. Comput.23, 365–375 (2005).
[CrossRef]

Knauss, W. G.

G. Vendroux and W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech.38, 86–92 (1998).
[CrossRef]

Lai, S-H.

C-H. Teng, S-H. Lai, Y-S. Chen, and W-H. Hsu, “Accurate optical flow computation under non-uniform brightness variations,” Comput. Vis. Image Underst.97, 315–346 (2005).
[CrossRef]

Leroy, A.M.

P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection (John Wiley & Sons, 1987)
[CrossRef]

Lewis, J.P

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

Li, C.

Li, F.W.

Y.N. Chen, W.Q. Jin, L. Zhao, and F.W. Li, “A subpixel motion estimation algorithm based on digital correlation for illumination variant and noise image sequences” Optik120, 835–844 (2009).
[CrossRef]

Lu, Z.

Martìnez, A. M.

Y-H. Kim, A. M. Martìnez, and A. C. Kak, “Robust motion estimation under varying illumination,” Image Vis. Comput.23, 365–375 (2005).
[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, 261–267 (1989).
[CrossRef]

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
[CrossRef]

Meng, L.B.

L.B. Meng, G.C. Jin, and X.F. Yao, “Application of iteration and finite element smoothing technique for displacement and strain measurement of digital speckle correlation,” Opt. Lasers Eng.45, 56–73 (2007).
[CrossRef]

Orteu, J-J.

M. A. Sutton, J-J. Orteu, and H. W. Schreier, Image correlation for shape, motion and deformation measurements (Springer, 2009).

Pan, B.

Pang, J.H.L.

Peng, B.

B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
[CrossRef]

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, 261–267 (1989).
[CrossRef]

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
[CrossRef]

W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
[CrossRef]

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

Philips, W.

C. Cofaru, W. Philips, and W. Van Paepegem, “A novel speckle pattern - adaptive Digital Image Correlation approach with robust strain calculation,” Opt. Lasers Eng.50, 187–198 (2012).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “A three-frame digital image correlation (DIC) method for the measurement of small displacements and strains,” Meas. Sci. Technol.23, 105406 (14 pp.) (2012).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “Improved Newton-Raphson digital image correlation method for full-field displacement and strain calculation,” Appl. Opt.49, 6472–6484 (2010).
[CrossRef] [PubMed]

C. Cofaru, W. Philips, and W. Van Paepegem, “Evaluation of digital image correlation techniques using realistic ground truth speckle images,” Meas. Sci. Technol.21, 055102 (17 pp.) (2010).
[CrossRef]

Pock, T.

M. Werlberger, T. Pock, and H. Bischof, “Motion estimation with non-local total variation regularization,” Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2010) pp. 2464–2471.

Poissant, J.

J. Poissant and F. Barthelat, “A novel subset splitting procedure for digital image correlation on discontinuous displacement fields,” Exp. Mech.50, 353–364 (2010).
[CrossRef]

Ranson, W. F.

W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
[CrossRef]

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
[CrossRef]

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

Ren, X.

X. Ren, “Local grouping for optical flow,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2008) pp. 1–8.

Réthoré, J.

J. Réthoré, F. Hild, and S. Roux, “Extended digital image correlation with crack shape optimization,” Int. J. Numer. Meth. Eng.73, 248–272 (2008).
[CrossRef]

J. Réthoré, S. Roux, and F. Hild, “From pictures to extended finite elements: extended digital image correlation (X-DIC),” C.R. Mécanique335, 131–137 (2007).
[CrossRef]

Ronchetti, E.M.

P.J. Huber and E.M. Ronchetti, Robust Statistics, 2nd Edition (John Wiley & Sons, New York (NY), 2009).
[CrossRef]

F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions (John Wiley & Sons, 1986).

Roth, S.

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

Rousseeuw, P.J.

F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions (John Wiley & Sons, 1986).

P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection (John Wiley & Sons, 1987)
[CrossRef]

Roux, S.

J. Réthoré, F. Hild, and S. Roux, “Extended digital image correlation with crack shape optimization,” Int. J. Numer. Meth. Eng.73, 248–272 (2008).
[CrossRef]

J. Réthoré, S. Roux, and F. Hild, “From pictures to extended finite elements: extended digital image correlation (X-DIC),” C.R. Mécanique335, 131–137 (2007).
[CrossRef]

G. Besnard, F. Hild, and S. Roux, “Finite-Element’ displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands,” Exp. Mech.46, 789–803 (2006).
[CrossRef]

Scharstein, D.

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

Schreier, H. W.

M. A. Sutton, J-J. Orteu, and H. W. Schreier, Image correlation for shape, motion and deformation measurements (Springer, 2009).

Shapiro, L. G.

M. Ye, R. M. Haralick, and L. G. Shapiro, “Estimating piecewise-smooth optical flow with global matching and graduated optimization,” IEEE Trans. Pattern Anal. Mach. Intell.25, 1625–1630 (2003).
[CrossRef]

Stahel, W.A.

F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions (John Wiley & Sons, 1986).

Su, F.

Sudderth, E. B.

D. Sun, E. B. Sudderth, and M. J. Black, “Layered image motion with explicit occlusions, temporal consistency, and depth ordering,” Adv. Neural Inf. Process. Syst.23, 2226–2234 (2010).

D. Sun, E. B. Sudderth, and M. J. Black, “Layered segmentation and optical flow estimation over time,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2012) pp. 1768–1775.

Sun, D.

D. Sun, E. B. Sudderth, and M. J. Black, “Layered image motion with explicit occlusions, temporal consistency, and depth ordering,” Adv. Neural Inf. Process. Syst.23, 2226–2234 (2010).

D. Sun, E. B. Sudderth, and M. J. Black, “Layered segmentation and optical flow estimation over time,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2012) pp. 1768–1775.

Sun, Y.

Sutton, M. A.

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, 261–267 (1989).
[CrossRef]

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
[CrossRef]

W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
[CrossRef]

M. A. Sutton, J-J. Orteu, and H. W. Schreier, Image correlation for shape, motion and deformation measurements (Springer, 2009).

Szeliski, R.

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

Tang, C.

Teng, C-H.

C-H. Teng, S-H. Lai, Y-S. Chen, and W-H. Hsu, “Accurate optical flow computation under non-uniform brightness variations,” Comput. Vis. Image Underst.97, 315–346 (2005).
[CrossRef]

Van Paepegem, W.

C. Cofaru, W. Philips, and W. Van Paepegem, “A novel speckle pattern - adaptive Digital Image Correlation approach with robust strain calculation,” Opt. Lasers Eng.50, 187–198 (2012).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “A three-frame digital image correlation (DIC) method for the measurement of small displacements and strains,” Meas. Sci. Technol.23, 105406 (14 pp.) (2012).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “Improved Newton-Raphson digital image correlation method for full-field displacement and strain calculation,” Appl. Opt.49, 6472–6484 (2010).
[CrossRef] [PubMed]

C. Cofaru, W. Philips, and W. Van Paepegem, “Evaluation of digital image correlation techniques using realistic ground truth speckle images,” Meas. Sci. Technol.21, 055102 (17 pp.) (2010).
[CrossRef]

Vendroux, G.

G. Vendroux and W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech.38, 86–92 (1998).
[CrossRef]

Wang, L.

Wang, Z.

Weiss, Y.

Y. Weiss, “Smoothness in layers: Motion segmentation using nonparametric mixture estimation,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997) pp. 520–526.
[CrossRef]

Werlberger, M.

M. Werlberger, T. Pock, and H. Bischof, “Motion estimation with non-local total variation regularization,” Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2010) pp. 2464–2471.

Wolters, W. J.

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
[CrossRef]

Wong, C.K.

Wu, J.

Xie, H.

Xie, H-M.

B. Pan, A. Asundi, H-M. Xie, and J. X. Gao, “Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements,” Opt. Lasers Eng.47, 865–874 (2009).
[CrossRef]

Yan, S.

Yao, X.F.

L.B. Meng, G.C. Jin, and X.F. Yao, “Application of iteration and finite element smoothing technique for displacement and strain measurement of digital speckle correlation,” Opt. Lasers Eng.45, 56–73 (2007).
[CrossRef]

Ye, M.

M. Ye, R. M. Haralick, and L. G. Shapiro, “Estimating piecewise-smooth optical flow with global matching and graduated optimization,” IEEE Trans. Pattern Anal. Mach. Intell.25, 1625–1630 (2003).
[CrossRef]

Ye, W.

J. Zhang, Y. Cai, W. Ye, and T.X. Yu, “On the use of the digital image correlation method for heterogeneous deformation measurement of porous solids,” Opt. Lasers Eng.49, 200–209 (2011).
[CrossRef]

Yu, T.X.

J. Zhang, Y. Cai, W. Ye, and T.X. Yu, “On the use of the digital image correlation method for heterogeneous deformation measurement of porous solids,” Opt. Lasers Eng.49, 200–209 (2011).
[CrossRef]

Zhang, J.

J. Zhang, Y. Cai, W. Ye, and T.X. Yu, “On the use of the digital image correlation method for heterogeneous deformation measurement of porous solids,” Opt. Lasers Eng.49, 200–209 (2011).
[CrossRef]

Zhang, Q.

B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
[CrossRef]

Zhao, L.

Y.N. Chen, W.Q. Jin, L. Zhao, and F.W. Li, “A subpixel motion estimation algorithm based on digital correlation for illumination variant and noise image sequences” Optik120, 835–844 (2009).
[CrossRef]

Zhou, W.

B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
[CrossRef]

Adv. Neural Inf. Process. Syst.

D. Sun, E. B. Sudderth, and M. J. Black, “Layered image motion with explicit occlusions, temporal consistency, and depth ordering,” Adv. Neural Inf. Process. Syst.23, 2226–2234 (2010).

Appl. Opt.

C.R. Mécanique

J. Réthoré, S. Roux, and F. Hild, “From pictures to extended finite elements: extended digital image correlation (X-DIC),” C.R. Mécanique335, 131–137 (2007).
[CrossRef]

Comput. Vis. Image Underst.

C-H. Teng, S-H. Lai, Y-S. Chen, and W-H. Hsu, “Accurate optical flow computation under non-uniform brightness variations,” Comput. Vis. Image Underst.97, 315–346 (2005).
[CrossRef]

M. J. Black and P. Anandan, “The robust estimation of multiple motions: Parametric and piecewise-smooth flow fields,” Comput. Vis. Image Underst.63, 75–104 (1996).
[CrossRef]

Exp. Mech.

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, 261–267 (1989).
[CrossRef]

G. Vendroux and W. G. Knauss, “Submicron deformation field measurements: Part 2. Improved digital image correlation,” Exp. Mech.38, 86–92 (1998).
[CrossRef]

J. Poissant and F. Barthelat, “A novel subset splitting procedure for digital image correlation on discontinuous displacement fields,” Exp. Mech.50, 353–364 (2010).
[CrossRef]

G. Besnard, F. Hild, and S. Roux, “Finite-Element’ displacement fields analysis from digital images: Application to Portevin-Le Châtelier bands,” Exp. Mech.46, 789–803 (2006).
[CrossRef]

IEEE Trans. Pattern Anal. Mach. Intell.

M. Ye, R. M. Haralick, and L. G. Shapiro, “Estimating piecewise-smooth optical flow with global matching and graduated optimization,” IEEE Trans. Pattern Anal. Mach. Intell.25, 1625–1630 (2003).
[CrossRef]

Image Vis. Comput.

Y-H. Kim, A. M. Martìnez, and A. C. Kak, “Robust motion estimation under varying illumination,” Image Vis. Comput.23, 365–375 (2005).
[CrossRef]

Image Vision Comput.

M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, and S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput.1, 133–139 (1983).
[CrossRef]

Int. J. Comput. Vis.

S. Baker, D. Scharstein, J.P Lewis, S. Roth, M. J. Black, and R. Szeliski, “A database and evaluation methodology for optical flow,” Int. J. Comput. Vis.92, 1–31 (2011).
[CrossRef]

Int. J. Numer. Meth. Eng.

J. Réthoré, F. Hild, and S. Roux, “Extended digital image correlation with crack shape optimization,” Int. J. Numer. Meth. Eng.73, 248–272 (2008).
[CrossRef]

Meas. Sci. Technol.

C. Cofaru, W. Philips, and W. Van Paepegem, “Evaluation of digital image correlation techniques using realistic ground truth speckle images,” Meas. Sci. Technol.21, 055102 (17 pp.) (2010).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “A three-frame digital image correlation (DIC) method for the measurement of small displacements and strains,” Meas. Sci. Technol.23, 105406 (14 pp.) (2012).
[CrossRef]

Opt. Eng.

B. Peng, Q. Zhang, W. Zhou, X. Hao, and L. Ding, “Modified correlation criterion for digital image correlation considering the effect of lighting variations in deformation measurements,” Opt. Eng.51, 017004 (2012).
[CrossRef]

W. H. Peters, W. F. Ranson, M. A. Sutton, T.C. Chu, and J. Anderson, “Application of digital correlation methods to rigid body mechanics,” Opt. Eng.22, 738–742 (1983).
[CrossRef]

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

Opt. Express

Opt. Lasers Eng.

B. Pan, A. Asundi, H-M. Xie, and J. X. Gao, “Digital image correlation using iterative least squares and pointwise least squares for displacement field and strain field measurements,” Opt. Lasers Eng.47, 865–874 (2009).
[CrossRef]

J. Zhang, Y. Cai, W. Ye, and T.X. Yu, “On the use of the digital image correlation method for heterogeneous deformation measurement of porous solids,” Opt. Lasers Eng.49, 200–209 (2011).
[CrossRef]

L.B. Meng, G.C. Jin, and X.F. Yao, “Application of iteration and finite element smoothing technique for displacement and strain measurement of digital speckle correlation,” Opt. Lasers Eng.45, 56–73 (2007).
[CrossRef]

C. Cofaru, W. Philips, and W. Van Paepegem, “A novel speckle pattern - adaptive Digital Image Correlation approach with robust strain calculation,” Opt. Lasers Eng.50, 187–198 (2012).
[CrossRef]

Optik

Y.N. Chen, W.Q. Jin, L. Zhao, and F.W. Li, “A subpixel motion estimation algorithm based on digital correlation for illumination variant and noise image sequences” Optik120, 835–844 (2009).
[CrossRef]

Other

P.J. Huber and E.M. Ronchetti, Robust Statistics, 2nd Edition (John Wiley & Sons, New York (NY), 2009).
[CrossRef]

P.J. Rousseeuw and A.M. Leroy, Robust Regression and Outlier Detection (John Wiley & Sons, 1987)
[CrossRef]

F.R. Hampel, E.M. Ronchetti, P.J. Rousseeuw, and W.A. Stahel, Robust Statistics: The Approach Based on Influence Functions (John Wiley & Sons, 1986).

D. Sun, E. B. Sudderth, and M. J. Black, “Layered segmentation and optical flow estimation over time,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2012) pp. 1768–1775.

Y. Weiss, “Smoothness in layers: Motion segmentation using nonparametric mixture estimation,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 1997) pp. 520–526.
[CrossRef]

M. Werlberger, T. Pock, and H. Bischof, “Motion estimation with non-local total variation regularization,” Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2010) pp. 2464–2471.

M.J. Black, Robust incremental optical flow(PhD. Thesis, Yale University, 1992).

M. A. Sutton, J-J. Orteu, and H. W. Schreier, Image correlation for shape, motion and deformation measurements (Springer, 2009).

X. Ren, “Local grouping for optical flow,” in Proceedings of the IEEE conference on Computer Vision and Pattern Recognition (Institute of Electrical and Electronics Engineers, New York, 2008) pp. 1–8.

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 (15)

Fig. 1:
Fig. 1:

The quadratic estimator (left) and its influence function (right).

Fig. 2:
Fig. 2:

The Welsch estimator (left) and its influence function (right). The shape parameter was considered constant with σ = 0.3

Fig. 3:
Fig. 3:

The Geman-McClure estimator (left) and its influence function (right). The shape parameter was considered constant with σ = 0.3

Fig. 4:
Fig. 4:

The reference (left) and deformed image (right) used in the first experiment.

Fig. 5:
Fig. 5:

The numerical displacement schematic (left) and image difference (right) corresponding to the first experiment.

Fig. 6:
Fig. 6:

The mean absolute horizontal displacement errors for the evaluated DIC methods in the first experiment.

Fig. 7:
Fig. 7:

The mean absolute vertical displacement errors for the evaluated DIC methods in the first experiment.

Fig. 8:
Fig. 8:

The variation with subset size of the number of non-converging displacement solutions in the first experiment.

Fig. 9:
Fig. 9:

The horizontal (left) and vertical (right) displacements calculated with (a) the classic and (b) proposed robust DIC approaches (μS = 0) using a subset size of 15 × 15 pixels, in the first experiment.

Fig. 10:
Fig. 10:

The horizontal (left) and vertical (right) displacements calculated with (a) the classic and (b) proposed robust DIC approaches (μS = 0) using a subset size of 33 × 33 pixels, in the first experiment.

Fig. 11:
Fig. 11:

The reference (left) and deformed image (right) used in the second experiment.

Fig. 12:
Fig. 12:

The horizontal (left) and vertical (right) displacements calculated with (a) the classic and (b) proposed robust DIC approaches (μS = 3000) using a subset size of 15 × 15 pixels, in the second experiment.

Fig. 13:
Fig. 13:

Vertical displacements (36×31 displacement points detail) from the second experiment calculated with the classic (left) and proposed robust (right) DIC approaches.

Fig. 14:
Fig. 14:

Motion vectors (36×31 displacement points detail) from the second experiment calculated with the classic (left) and proposed robust (right) DIC approaches. The locations marked in red represent displacement solutions that did not converge.

Fig. 15:
Fig. 15:

First image of the each sequence (left column) and pixelwise optic flow fields recovered with the classic (middle column) and proposed robust (right column) DIC approaches in the third experiment. Both DIC methods employed a 21 × 21 subset size with a step size of 5 pixels.

Tables (1)

Tables Icon

Table 1: Average endpoint errors (AEE) for the classic and proposed robust DIC methods for the Middlebury training dataset. Both DIC methods employed a 21 × 21 pixel subset size and a 5 pixel step size. The errors corresponding to other state-of-the-art optical flow methods are presented for reference. The results marked with * are taken from [31] and [32] respectively.

Equations (19)

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

ρ ( x ) = x 2 , ψ ( x ) = 2 x ,
ρ ( x , σ ) = σ 2 2 [ 1 exp ( ( x / σ ) 2 ) ] , ψ ( x , σ ) = x exp ( ( x / σ ) 2 ) ,
ρ ( x , σ ) = x 2 σ + x 2 , ψ ( x , σ ) = 2 x σ ( σ + x 2 ) 2 .
τ = ± σ 3 ,
τ = ± 1 2 σ .
u ( x , y ) = p 1 + p 3 ( x x 0 ) + p 5 ( y y 0 ) ,
v ( x , y ) = p 2 + p 4 ( x x 0 ) + p 6 ( y y 0 ) .
C ( p ) = E D ( p ) + μ S E S ( p ) ,
E D ( p ) = x = 1 M y = 1 N σ D 2 2 [ 1 exp ( ( f ( x , y ) g ( x , y ; p ) ) 2 / σ D 2 ) ] ,
E S ( p ) = i = 1 6 k = 1 8 ( p i p i k ) 2 σ i + ( p i p i k ) 2
p ( t ) = p ( t 1 ) H C 1 ( p ( t 1 ) ) J C ( p ( t 1 ) ) ,
J C ( p ( t 1 ) ) = ( p 1 C ( p ( t 1 ) ) p 6 C ( p ( t 1 ) ) )
H C ( p ) = ( 2 p 1 2 C ( p ( t 1 ) ) 2 p 1 p 6 C ( p ( t 1 ) ) 2 p 6 p 1 C ( p ( t 1 ) ) 2 p 6 2 C ( p ( t 1 ) ) ) .
p i C ( p ( t 1 ) ) = x = 1 M y = 1 N [ p i g ( x , y ; p ( t 1 ) ) ( f ( x , y ) g ( x , y ; p ( t 1 ) ) ) exp ( ( f ( x , y ) g ( x , y ; p ( t 1 ) ) ) 2 / ( σ D ( t 1 ) ) 2 ) ] + μ S k = 1 8 2 σ i ( t 1 ) ( p i ( t 1 ) p i k ( t 1 ) ) ( σ i ( t 1 ) + ( p i ( t 1 ) p i k ( t 1 ) ) 2 ) 2 ,
2 p i p j C ( p ( t 1 ) ) = x = 1 M y = 1 N [ p i g ( x , y ; p ( t 1 ) ) p j g ( x , y ; p ( t 1 ) ) exp ( ( f ( x , y ) g ( x , y ; p ( t 1 ) ) ) 2 / ( σ D ( t 1 ) ) 2 ) ] ,
2 p i 2 C ( p ( t 1 ) ) = x = 1 M y = 1 N [ ( p i g ( x , y ; p ( t 1 ) ) ) 2 exp ( ( f ( x , y ) g ( x , y ; p ( t 1 ) ) ) 2 / ( σ D ( t 1 ) ) 2 ) ] + μ S k = 1 8 ( 2 σ i ( t 1 ) ) 2 6 σ i ( t 1 ) ( p i ( t 1 ) p i k ( t 1 ) ) 2 ( σ i ( t 1 ) + ( p i ( t 1 ) p i k ( t 1 ) ) 2 ) 3 ,
σ D ( t 1 ) = 2 median ( | f ( x , y ) g ( x , y ; p ( t 1 ) ) | ) .
r i k = p i p i k , for i = 1 , , 6 ; k = 1 , , 8
σ ˜ i = 1 8 1 k = 1 8 ( r i k r i ¯ ) 2 , r i ¯ = 1 8 k = 1 8 r i k

Metrics