Abstract

In digital image correlation (DIC), to obtain the displacements of each point of interest, a correlation criterion must be predefined to evaluate the similarity between the reference subset and the target subset. The correlation criterion is of fundamental importance in DIC, and various correlation criteria have been designed and used in literature. However, little research has been carried out to investigate their relations. In this paper, we first provide a comprehensive overview of various correlation criteria used in DIC. Then we focus on three robust and most widely used correlation criteria, i.e., a zero-mean normalized cross-correlation (ZNCC) criterion, a zero-mean normalized sum of squared difference (ZNSSD) criterion, and a parametric sum of squared difference (PSSDab) criterion with two additional unknown parameters, since they are insensitive to the scale and offset changes of the target subset intensity and have been highly recommended for practical use in literature. The three correlation criteria are analyzed to establish their transversal relationships, and the theoretical analyses clearly indicate that the three correlation criteria are actually equivalent, which elegantly unifies these correlation criteria for pattern matching. Finally, the equivalence of these correlation criteria is further validated by numerical simulation and actual experiment.

© 2010 Optical Society of America

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).
  2. T. C. Chu, W. F. Ranson, M. A. Sutton, and W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
    [CrossRef]
  3. R. S. Sirohi, Optical Methods of Measurement: Wholefield Techniques, 2nd ed. (CRC, 2009).
    [CrossRef]
  4. M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, Advances in Two-Dimensional and Three-Dimensional Computer Vision, P.K.Rastogi, ed., Topics in Applied Physics (Springer–Verlag, 2000), Vol. 77, pp. 323–372.
  5. M. A. Sutton, J. J. Orteu, and H. W. Schreier, Image Correlation for Shape, Motion, and Deformation Measurements (Springer, 2009).
  6. B. Pan, K. M. Qian, H. M. Xie, and A. Asundi, “Two-dimensional digital image correlation for in-plane displacement and strain measurement: a review,” Meas. Sci. Technol. 20, 062001 (2009).
    [CrossRef]
  7. H. A. Bruck, S. R. McNeil, 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]
  8. B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of subpixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
    [CrossRef]
  9. A. Giachetti, “Matching techniques to compute image motion,” Image Vis. Comput. 18, 247–260 (2000).
    [CrossRef]
  10. S. P. Ma and G. C. Jin, “New correlation coefficients designed for digital speckle correlation method (DSCM),” Proc. SPIE 5058, 25–33 (2003).
    [CrossRef]
  11. W. Tong, “An evaluation of digital image correlation criteria for strain mapping applications,” Strain 41, 167–175 (2005).
    [CrossRef]
  12. P. Zhou and K. E. Goodson, “Subpixel displacement and deformation gradient measurement using digital image/speckle correlation,” Opt. Eng. 40, 1613–1620 (2001).
    [CrossRef]
  13. M. Z. Brown, D. Burschka, and G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
    [CrossRef]
  14. A. W. Gruen, “Adaptive least squares correlation: a powerful image matching technique,” S. Afr. J. Photogr. Rem. Sensing Cartogr. 14(3), 175–187 (1985).
  15. Y. Altunbasak, R. M. Mersereau, and A. J. Patti, “A fast parametric motion estimation algorithm with illumination and lens distortion correction,” IEEE Trans. Image Process. 12, 395–408 (2003).
    [CrossRef]
  16. 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]
  17. B. Pan, Z. Y. Wang, and H. M. Xie, “Generalized spatial-gradient based digital image correlation for displacement and shape measurement with subpixel accuracy,” J. Strain Anal. Eng. Des. 44, 659–669 (2009).
    [CrossRef]
  18. B. Pan, H. M. Xie, J. X. Gao, and A. Asundi, “Improved speckle projection profilometry for out-of-plane shape measurement,” Appl. Opt. 47, 5527–5533 (2008).
    [CrossRef] [PubMed]
  19. D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. V. Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44, 1132–1145 (2006).
    [CrossRef]
  20. B. Pan, H. M. Xie, Z. Y. Wang, K. M. Qian, and Z. Y. Wang, “Study of subset size selection in digital image correlation for speckle patterns,” Opt. Express 16, 7037–7048 (2008).
    [CrossRef] [PubMed]
  21. B. Pan, Z. X. Lu, and H. M. Xie, “Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation,” Opt. Lasers Eng. 48, 469–477 (2010).
    [CrossRef]
  22. D. J. Chen, F. P. Chiang, Y. S. Tan, and H. S. Don, “Digital speckle-displacement measurement using a complex spectrum method,” Appl. Opt. 32, 1839–1849 (1993).
    [CrossRef] [PubMed]
  23. M. Sjodahl and L. R. Benckert, “Electronic speckle photography: Analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993).
    [CrossRef] [PubMed]
  24. F. Hild, B. Raka, M. Baudequin, S. Roux, and F. Cantelaube, “Multiscale displacement field measurements of compressed mineral-wool samples by digital image correlation,” Appl. Opt. 41, 6815–6828 (2002).
    [CrossRef] [PubMed]
  25. J. Y. Liu and M. Iskander, “Adaptive cross correlation for imaging displacements in soils,” J. Comput. Civ. Eng. 18, 46–57 (2004).
    [CrossRef]
  26. J. P. Lewis, “Fast normalized cross correlation,” 2003, available at http://www.idiom.com/˜zilla/Papers/nvisionInterface/nip.html.
  27. J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
    [CrossRef]
  28. B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation,” Opt. Eng. 46, 033601 (2007).
    [CrossRef]
  29. H. Lu and P. D. Cary, “Deformation measurement by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
    [CrossRef]
  30. Y. Wang and A. M. Cuitiño, “Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation,” Int. J. Solids Struct. 39, 3777–3796 (2002).
    [CrossRef]
  31. G. Vendroux and W. G. Knauss, “Submicron deformation field measurements. Part 2. Improved digital image correlation,” Exp. Mech. 38, 86–92 (1998).
    [CrossRef]
  32. H. Q. Jin and H. A. Bruck, “Theoretical development for pointwise digital image correlation,” Opt. Eng. 44, 067003(2005).
    [CrossRef]
  33. B. Pan, “Reliability-guided digital image correlation for image deformation measurement,” Appl. Opt. 48, 1535–1542(2009).
    [CrossRef] [PubMed]
  34. B. Pan, Z. Y. Wang, and Z. X. Lu, “Genuine full-field deformation measurement of an object with complex shape using reliability-guided digital image correlation,” Opt. Express 18, 1011–1023 (2010).
    [CrossRef] [PubMed]
  35. B. Pan, D. F. Wu, and Y. Xia, “High-temperature field measurement by combing transient aerodynamic heating system and reliability-guided digital image correlation,” Opt. Lasers Eng. 48, 841–848 (2010).
    [CrossRef]
  36. W. H. Press, C++ Numerical Algorithms (Publishing House of Electronics Industry, 2003).
  37. H. W. Schreier, J. R. Braasch, and M. A. Sutton, “Systematic errors in digital image correlation caused by intensity interpolation,” Opt. Eng. 39, 2915–2921 (2000).
    [CrossRef]
  38. Y. Q. Wang, M. A. Sutton, H. A. Bruch, and H. W. Schreier, “Quantitative error assessment in pattern matching: effects of intensity pattern noise, interpolation, strain and image contrast on motion measurement,” Strain 45, 160–178(2009).
    [CrossRef]

2010 (4)

B. Pan, Z. X. Lu, and H. M. Xie, “Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation,” Opt. Lasers Eng. 48, 469–477 (2010).
[CrossRef]

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

B. Pan, Z. Y. Wang, and Z. X. Lu, “Genuine full-field deformation measurement of an object with complex shape using reliability-guided digital image correlation,” Opt. Express 18, 1011–1023 (2010).
[CrossRef] [PubMed]

B. Pan, D. F. Wu, and Y. Xia, “High-temperature field measurement by combing transient aerodynamic heating system and reliability-guided digital image correlation,” Opt. Lasers Eng. 48, 841–848 (2010).
[CrossRef]

2009 (5)

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

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

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

B. Pan, Z. Y. Wang, and H. M. Xie, “Generalized spatial-gradient based digital image correlation for displacement and shape measurement with subpixel accuracy,” J. Strain Anal. Eng. Des. 44, 659–669 (2009).
[CrossRef]

2008 (2)

2007 (1)

B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation,” Opt. Eng. 46, 033601 (2007).
[CrossRef]

2006 (2)

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

B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of subpixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

2005 (2)

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

H. Q. Jin and H. A. Bruck, “Theoretical development for pointwise digital image correlation,” Opt. Eng. 44, 067003(2005).
[CrossRef]

2004 (1)

J. Y. Liu and M. Iskander, “Adaptive cross correlation for imaging displacements in soils,” J. Comput. Civ. Eng. 18, 46–57 (2004).
[CrossRef]

2003 (3)

M. Z. Brown, D. Burschka, and G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

S. P. Ma and G. C. Jin, “New correlation coefficients designed for digital speckle correlation method (DSCM),” Proc. SPIE 5058, 25–33 (2003).
[CrossRef]

Y. Altunbasak, R. M. Mersereau, and A. J. Patti, “A fast parametric motion estimation algorithm with illumination and lens distortion correction,” IEEE Trans. Image Process. 12, 395–408 (2003).
[CrossRef]

2002 (2)

Y. Wang and A. M. Cuitiño, “Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation,” Int. J. Solids Struct. 39, 3777–3796 (2002).
[CrossRef]

F. Hild, B. Raka, M. Baudequin, S. Roux, and F. Cantelaube, “Multiscale displacement field measurements of compressed mineral-wool samples by digital image correlation,” Appl. Opt. 41, 6815–6828 (2002).
[CrossRef] [PubMed]

2001 (1)

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

2000 (3)

A. Giachetti, “Matching techniques to compute image motion,” Image Vis. Comput. 18, 247–260 (2000).
[CrossRef]

H. Lu and P. D. Cary, “Deformation measurement by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

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

1998 (1)

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

1993 (2)

1989 (1)

H. A. Bruck, S. R. McNeil, 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]

1985 (2)

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

A. W. Gruen, “Adaptive least squares correlation: a powerful image matching technique,” S. Afr. J. Photogr. Rem. Sensing Cartogr. 14(3), 175–187 (1985).

1982 (1)

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

Altunbasak, Y.

Y. Altunbasak, R. M. Mersereau, and A. J. Patti, “A fast parametric motion estimation algorithm with illumination and lens distortion correction,” IEEE Trans. Image Process. 12, 395–408 (2003).
[CrossRef]

Asundi, A.

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

B. Pan, H. M. Xie, J. X. Gao, and A. Asundi, “Improved speckle projection profilometry for out-of-plane shape measurement,” Appl. Opt. 47, 5527–5533 (2008).
[CrossRef] [PubMed]

Baudequin, M.

Benckert, L. R.

Bossuyt, S.

D. Lecompte, A. Smits, S. Bossuyt, H. Sol, J. Vantomme, D. V. Hemelrijck, and A. M. Habraken, “Quality assessment of speckle patterns for digital image correlation,” Opt. Lasers Eng. 44, 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, 2915–2921 (2000).
[CrossRef]

Brown, M. Z.

M. Z. Brown, D. Burschka, and G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

Bruch, H. A.

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

Bruck, H. A.

H. Q. Jin and H. A. Bruck, “Theoretical development for pointwise digital image correlation,” Opt. Eng. 44, 067003(2005).
[CrossRef]

H. A. Bruck, S. R. McNeil, 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]

Burschka, D.

M. Z. Brown, D. Burschka, and G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

Cantelaube, F.

Cary, P. D.

H. Lu and P. D. Cary, “Deformation measurement by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

Chao, Y. J.

M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, Advances in Two-Dimensional and Three-Dimensional Computer Vision, P.K.Rastogi, ed., Topics in Applied Physics (Springer–Verlag, 2000), Vol. 77, pp. 323–372.

Chen, D. J.

Chiang, F. P.

Chu, T. C.

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

Cuitiño, A. M.

Y. Wang and A. M. Cuitiño, “Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation,” Int. J. Solids Struct. 39, 3777–3796 (2002).
[CrossRef]

Dai, F. L.

B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of subpixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Don, H. S.

Fang, J.

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[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]

B. Pan, H. M. Xie, J. X. Gao, and A. Asundi, “Improved speckle projection profilometry for out-of-plane shape measurement,” Appl. Opt. 47, 5527–5533 (2008).
[CrossRef] [PubMed]

Giachetti, A.

A. Giachetti, “Matching techniques to compute image motion,” Image Vis. Comput. 18, 247–260 (2000).
[CrossRef]

Goodson, K. E.

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

Gruen, A. W.

A. W. Gruen, “Adaptive least squares correlation: a powerful image matching technique,” S. Afr. J. Photogr. Rem. Sensing Cartogr. 14(3), 175–187 (1985).

Guo, Z. Q.

B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation,” Opt. Eng. 46, 033601 (2007).
[CrossRef]

Habraken, A. M.

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

Hager, G. D.

M. Z. Brown, D. Burschka, and G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

Helm, J. D.

M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, Advances in Two-Dimensional and Three-Dimensional Computer Vision, P.K.Rastogi, ed., Topics in Applied Physics (Springer–Verlag, 2000), Vol. 77, pp. 323–372.

Hemelrijck, D. V.

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

Hild, F.

Hua, T.

B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation,” Opt. Eng. 46, 033601 (2007).
[CrossRef]

Huang, J. Y.

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

Iskander, M.

J. Y. Liu and M. Iskander, “Adaptive cross correlation for imaging displacements in soils,” J. Comput. Civ. Eng. 18, 46–57 (2004).
[CrossRef]

Jin, G. C.

S. P. Ma and G. C. Jin, “New correlation coefficients designed for digital speckle correlation method (DSCM),” Proc. SPIE 5058, 25–33 (2003).
[CrossRef]

Jin, H. Q.

H. Q. Jin and H. A. Bruck, “Theoretical development for pointwise digital image correlation,” Opt. Eng. 44, 067003(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]

Lecompte, D.

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

Lewis, J. P.

J. P. Lewis, “Fast normalized cross correlation,” 2003, available at http://www.idiom.com/˜zilla/Papers/nvisionInterface/nip.html.

Liu, J. Y.

J. Y. Liu and M. Iskander, “Adaptive cross correlation for imaging displacements in soils,” J. Comput. Civ. Eng. 18, 46–57 (2004).
[CrossRef]

Lu, H.

H. Lu and P. D. Cary, “Deformation measurement by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

Lu, Z. X.

B. Pan, Z. X. Lu, and H. M. Xie, “Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation,” Opt. Lasers Eng. 48, 469–477 (2010).
[CrossRef]

B. Pan, Z. Y. Wang, and Z. X. Lu, “Genuine full-field deformation measurement of an object with complex shape using reliability-guided digital image correlation,” Opt. Express 18, 1011–1023 (2010).
[CrossRef] [PubMed]

Ma, S. P.

S. P. Ma and G. C. Jin, “New correlation coefficients designed for digital speckle correlation method (DSCM),” Proc. SPIE 5058, 25–33 (2003).
[CrossRef]

McNeil, S. R.

H. A. Bruck, S. R. McNeil, 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]

McNeill, S. R.

M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, Advances in Two-Dimensional and Three-Dimensional Computer Vision, P.K.Rastogi, ed., Topics in Applied Physics (Springer–Verlag, 2000), Vol. 77, pp. 323–372.

Mersereau, R. M.

Y. Altunbasak, R. M. Mersereau, and A. J. Patti, “A fast parametric motion estimation algorithm with illumination and lens distortion correction,” IEEE Trans. Image Process. 12, 395–408 (2003).
[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.

B. Pan, Z. X. Lu, and H. M. Xie, “Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation,” Opt. Lasers Eng. 48, 469–477 (2010).
[CrossRef]

B. Pan, D. F. Wu, and Y. Xia, “High-temperature field measurement by combing transient aerodynamic heating system and reliability-guided digital image correlation,” Opt. Lasers Eng. 48, 841–848 (2010).
[CrossRef]

B. Pan, Z. Y. Wang, and Z. X. Lu, “Genuine full-field deformation measurement of an object with complex shape using reliability-guided digital image correlation,” Opt. Express 18, 1011–1023 (2010).
[CrossRef] [PubMed]

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

B. Pan, Z. Y. Wang, and H. M. Xie, “Generalized spatial-gradient based digital image correlation for displacement and shape measurement with subpixel accuracy,” J. Strain Anal. Eng. Des. 44, 659–669 (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]

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

B. Pan, H. M. Xie, J. X. Gao, and A. Asundi, “Improved speckle projection profilometry for out-of-plane shape measurement,” Appl. Opt. 47, 5527–5533 (2008).
[CrossRef] [PubMed]

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

B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation,” Opt. Eng. 46, 033601 (2007).
[CrossRef]

B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of subpixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Pan, X. Y.

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

Patti, A. J.

Y. Altunbasak, R. M. Mersereau, and A. J. Patti, “A fast parametric motion estimation algorithm with illumination and lens distortion correction,” IEEE Trans. Image Process. 12, 395–408 (2003).
[CrossRef]

Peng, X. L.

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

Peters, W. H.

H. A. Bruck, S. R. McNeil, 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]

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

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

Press, W. H.

W. H. Press, C++ Numerical Algorithms (Publishing House of Electronics Industry, 2003).

Qian, K. M.

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

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

Qin, L.

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

Raka, B.

Ranson, W. F.

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

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

Roux, S.

Schreier, H. W.

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

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

M. A. Sutton, J. J. Orteu, and H. W. Schreier, Image Correlation for Shape, Motion, and Deformation Measurements (Springer, 2009).

Sirohi, R. S.

R. S. Sirohi, Optical Methods of Measurement: Wholefield Techniques, 2nd ed. (CRC, 2009).
[CrossRef]

Sjodahl, M.

Smits, A.

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

Sol, H.

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

Sutton, M. A.

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

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

H. A. Bruck, S. R. McNeil, 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]

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

M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, Advances in Two-Dimensional and Three-Dimensional Computer Vision, P.K.Rastogi, ed., Topics in Applied Physics (Springer–Verlag, 2000), Vol. 77, pp. 323–372.

M. A. Sutton, J. J. Orteu, and H. W. Schreier, Image Correlation for Shape, Motion, and Deformation Measurements (Springer, 2009).

Tan, Y. S.

Tong, W.

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

Vantomme, J.

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

Y. Wang and A. M. Cuitiño, “Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation,” Int. J. Solids Struct. 39, 3777–3796 (2002).
[CrossRef]

Wang, Y. Q.

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

Wang, Z. Y.

Wu, D. F.

B. Pan, D. F. Wu, and Y. Xia, “High-temperature field measurement by combing transient aerodynamic heating system and reliability-guided digital image correlation,” Opt. Lasers Eng. 48, 841–848 (2010).
[CrossRef]

Xia, Y.

B. Pan, D. F. Wu, and Y. Xia, “High-temperature field measurement by combing transient aerodynamic heating system and reliability-guided digital image correlation,” Opt. Lasers Eng. 48, 841–848 (2010).
[CrossRef]

Xie, H. M.

B. Pan, Z. X. Lu, and H. M. Xie, “Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation,” Opt. Lasers Eng. 48, 469–477 (2010).
[CrossRef]

B. Pan, Z. Y. Wang, and H. M. Xie, “Generalized spatial-gradient based digital image correlation for displacement and shape measurement with subpixel accuracy,” J. Strain Anal. Eng. Des. 44, 659–669 (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]

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

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

B. Pan, H. M. Xie, J. X. Gao, and A. Asundi, “Improved speckle projection profilometry for out-of-plane shape measurement,” Appl. Opt. 47, 5527–5533 (2008).
[CrossRef] [PubMed]

B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation,” Opt. Eng. 46, 033601 (2007).
[CrossRef]

B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of subpixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Xiong, C. Y.

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

Xu, B. Q.

B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of subpixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

Zhou, P.

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

Zhu, T.

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

Appl. Opt. (5)

Exp. Mech. (4)

H. Lu and P. D. Cary, “Deformation measurement by digital image correlation: implementation of a second-order displacement gradient,” Exp. Mech. 40, 393–400 (2000).
[CrossRef]

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

H. A. Bruck, S. R. McNeil, 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]

IEEE Trans. Image Process. (1)

Y. Altunbasak, R. M. Mersereau, and A. J. Patti, “A fast parametric motion estimation algorithm with illumination and lens distortion correction,” IEEE Trans. Image Process. 12, 395–408 (2003).
[CrossRef]

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

M. Z. Brown, D. Burschka, and G. D. Hager, “Advances in computational stereo,” IEEE Trans. Pattern Anal. Mach. Intell. 25, 993–1008 (2003).
[CrossRef]

Image Vis. Comput. (1)

A. Giachetti, “Matching techniques to compute image motion,” Image Vis. Comput. 18, 247–260 (2000).
[CrossRef]

Int. J. Solids Struct. (1)

Y. Wang and A. M. Cuitiño, “Full-field measurements of heterogeneous deformation patterns on polymeric foams using digital image correlation,” Int. J. Solids Struct. 39, 3777–3796 (2002).
[CrossRef]

J. Comput. Civ. Eng. (1)

J. Y. Liu and M. Iskander, “Adaptive cross correlation for imaging displacements in soils,” J. Comput. Civ. Eng. 18, 46–57 (2004).
[CrossRef]

J. Strain Anal. Eng. Des. (1)

B. Pan, Z. Y. Wang, and H. M. Xie, “Generalized spatial-gradient based digital image correlation for displacement and shape measurement with subpixel accuracy,” J. Strain Anal. Eng. Des. 44, 659–669 (2009).
[CrossRef]

Meas. Sci. Technol. (3)

B. Pan, H. M. Xie, B. Q. Xu, and F. L. Dai, “Performance of subpixel registration algorithms in digital image correlation,” Meas. Sci. Technol. 17, 1615–1621 (2006).
[CrossRef]

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

J. Y. Huang, T. Zhu, X. Y. Pan, L. Qin, X. L. Peng, C. Y. Xiong, and J. Fang, “A high-efficiency digital image correlation method based on a fast recursive scheme,” Meas. Sci. Technol. 21, 035101 (2010).
[CrossRef]

Opt. Eng. (5)

B. Pan, H. M. Xie, Z. Q. Guo, and T. Hua, “Full-field strain measurement using a two-dimensional Savitzky–Golay digital differentiator in digital image correlation,” Opt. Eng. 46, 033601 (2007).
[CrossRef]

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

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

H. Q. Jin and H. A. Bruck, “Theoretical development for pointwise digital image correlation,” Opt. Eng. 44, 067003(2005).
[CrossRef]

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

Opt. Express (2)

Opt. Lasers Eng. (4)

B. Pan, Z. X. Lu, and H. M. Xie, “Mean intensity gradient: an effective global parameter for quality assessment of the speckle patterns used in digital image correlation,” Opt. Lasers Eng. 48, 469–477 (2010).
[CrossRef]

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

B. Pan, D. F. Wu, and Y. Xia, “High-temperature field measurement by combing transient aerodynamic heating system and reliability-guided digital image correlation,” Opt. Lasers Eng. 48, 841–848 (2010).
[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]

Proc. SPIE (1)

S. P. Ma and G. C. Jin, “New correlation coefficients designed for digital speckle correlation method (DSCM),” Proc. SPIE 5058, 25–33 (2003).
[CrossRef]

S. Afr. J. Photogr. Rem. Sensing Cartogr. (1)

A. W. Gruen, “Adaptive least squares correlation: a powerful image matching technique,” S. Afr. J. Photogr. Rem. Sensing Cartogr. 14(3), 175–187 (1985).

Strain (2)

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

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

Other (5)

R. S. Sirohi, Optical Methods of Measurement: Wholefield Techniques, 2nd ed. (CRC, 2009).
[CrossRef]

M. A. Sutton, S. R. McNeill, J. D. Helm, and Y. J. Chao, Advances in Two-Dimensional and Three-Dimensional Computer Vision, P.K.Rastogi, ed., Topics in Applied Physics (Springer–Verlag, 2000), Vol. 77, pp. 323–372.

M. A. Sutton, J. J. Orteu, and H. W. Schreier, Image Correlation for Shape, Motion, and Deformation Measurements (Springer, 2009).

W. H. Press, C++ Numerical Algorithms (Publishing House of Electronics Industry, 2003).

J. P. Lewis, “Fast normalized cross correlation,” 2003, available at http://www.idiom.com/˜zilla/Papers/nvisionInterface/nip.html.

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 (7)

Fig. 1
Fig. 1

Example of tracking the reference subset in the deformed image using DIC. The center position of the target subset is obtained through searching the peak position of the distribution of the correlation coefficient.

Fig. 2
Fig. 2

Reference image (left), deformed image (middle), and deformed image after artificially adjusted with 20% increase in brightness and 20% decreases in contrast (right) and their histograms. The rectangle of the reference image indicates the specified region of interest, and the two small squares denote the subset used for calculation.

Fig. 3
Fig. 3

Computed u-field displacements by using the ZNSSD criterion (a) and the PSSD a b criterion (b) for a simulated uniaxial tensile test.

Fig. 4
Fig. 4

Computed ZNCC coefficients by using the ZNSSD criterion (a) and the PSSD a b criterion (b) for a simulated uniaxial tensile test.

Fig. 5
Fig. 5

Schematic of a three-point bending experiment (top) and the captured reference and deformed images (bottom).

Fig. 6
Fig. 6

Computed u-displacement (left), v-displacement (middle), and ZNCC coefficient distribution (right) by optimizing the ZNSSD criterion for a three-point bending test.

Fig. 7
Fig. 7

Computed u-displacement (left), v-displacement (middle), and the ZNCC coefficient distribution (right) by optimizing the PSSD a b criterion for a three-point bending test.

Equations (20)

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

C CC = f i g i .
C ZCC = [ ( f i f ¯ ) ( g i g ¯ ) ] ,
C NCC = f i g i f i 2 g i 2 .
C ZNCC = f ¯ i g ¯ i f ¯ i 2 g ¯ i 2 ,
C SSD = ( f i g i ) 2 .
C ZSSD = [ ( f i f ¯ ) ( g i g ¯ ) ] 2 .
C NSSD = ( f i f i 2 g i g i 2 ) 2 .
C ZNSSD = ( f ¯ i f ¯ i 2 g ¯ i g ¯ i 2 ) 2 .
C PSSD b = ( f i + b g i ) 2 .
C PSSD a = ( a f i g i ) 2 .
C PSSD a b = ( a f i + b g i ) 2 .
C ZNSSD = ( f ¯ i f ¯ i 2 g ¯ i g ¯ i 2 ) 2 = ( f ¯ i 2 f ¯ i 2 2 f ¯ i g ¯ i f ¯ i 2 g ¯ i 2 + g ¯ i 2 g ¯ i 2 ) = 2 2 f ¯ i g ¯ i f ¯ i 2 g ¯ i 2 = 2 ( 1 C ZNCC ) .
{ C PSSD a b a = 0 C PSSD a b b = 0 { 2 [ ( a f i + b g i ) f i ] = 0 2 ( a f i + b g i ) = 0 .
a = [ ( g i b ) f i ] f i 2 ,
b = ( g i a f i ) 1 = ( g i a f i ) n = g ¯ a f ¯ .
a = [ ( g i g ¯ + a f ¯ ) f i ] f i 2 a f i 2 = [ ( g i g ¯ + a f ¯ ) f i ] a = [ ( g i g ¯ ) f i ] [ ( f i f ¯ ) f i ] = g ¯ i f i f ¯ i f i .
a = g ¯ i f ¯ i f ¯ i 2 ,
b = g ¯ g ¯ i f ¯ i f ¯ i 2 f ¯ .
C PSSD a b = ( a f i + b g i ) 2 = ( a f i + g ¯ a f ¯ g i ) 2 = ( a f ¯ i g ¯ i ) 2 = ( g ¯ i f ¯ i f ¯ i 2 f ¯ i g ¯ i ) 2 = [ ( g ¯ i f ¯ i f ¯ i 2 ) 2 f ¯ i 2 2 g ¯ i f ¯ i f ¯ i 2 f ¯ i g ¯ i + g ¯ i 2 ] = ( g ¯ i f ¯ i ) 2 f ¯ i 2 + g ¯ i 2 = g ¯ i 2 [ 1 ( g ¯ i f ¯ i ) 2 g ¯ i 2 f ¯ i 2 ] = g ¯ i 2 ( 1 C ZNCC 2 ) .
C ZNCC = 1 C PSSD a b g ¯ i 2 .

Metrics