Abstract

Image registration is the key technique of optical metrologies such as digital image correlation (DIC), particle image velocimetry (PIV), and speckle metrology. Its performance depends critically on the quality of image pattern, and thus pattern optimization attracts extensive attention. In this article, a statistical model is built to optimize speckle patterns that are composed of randomly positioned speckles. It is found that the process of speckle pattern generation is essentially a filtered Poisson process. The dependence of measurement errors (including systematic errors, random errors, and overall errors) upon speckle pattern generation parameters is characterized analytically. By minimizing the errors, formulas of the optimal speckle radius are presented. Although the primary motivation is from the field of DIC, we believed that scholars in other optical measurement communities, such as PIV and speckle metrology, will benefit from these discussions.

© 2017 Optical Society of America under the terms of the OSA Open Access Publishing Agreement

Full Article  |  PDF Article
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References

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2018 (1)

Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018).
[Crossref]

2017 (3)

W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017).
[Crossref]

G. Bomarito, J. Hochhalter, T. Ruggles, and A. Cannon, “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017).
[Crossref]

A. Lavatelli and E. Zappa, “A displacement uncertainty model for 2-D DIC measurement under motion blur conditions,” IEEE Trans. Instrum. Meas. 66(3), 451–459 (2017).
[Crossref]

2016 (5)

Z. Gao, X. Xu, Y. Su, and Q. Zhang, “Experimental analysis of image noise and interpolation bias in digital image correlation,” Opt. Lasers Eng. 81, 46–53 (2016).
[Crossref]

B. Blaysat, M. Grédiac, and F. Sur, “On the propagation of camera sensor noise to displacement maps obtained by DIC - an experimental study,” Exp. Mech. 56(6), 919–944 (2016).
[Crossref]

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

Y. Su, Q. Zhang, Z. Gao, and X. Xu, “Noise-induced bias for convolution-based interpolation in digital image correlation,” Opt. Express 24(2), 1175–1195 (2016).
[Crossref] [PubMed]

Z. Hu, T. Xu, H. Luo, R. Z. Gan, and H. Lu, “Measurement of thickness and profile of a transparent material using fluorescent stereo microscopy,” Opt. Express 24(26), 29822–29829 (2016).
[Crossref]

2015 (7)

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

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

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

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

M. Chakrabarti, M. L. Jakobsen, and S. G. Hanson, “Speckle-based spectrometer,” Opt. Lett. 40(14), 3264–3267 (2015).
[Crossref] [PubMed]

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]

Y. Wang, P. Lava, P. Reu, and D. Debruyne, “Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements,” Strain 52(2), 110–128 (2015).

2013 (1)

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

2011 (3)

W. Tong, “Subpixel image registration with reduced bias,” Opt. Lett. 36(5), 763–765 (2011).
[Crossref] [PubMed]

L. Luu, Z. Wang, M. Vo, T. Hoang, and J. Ma, “Accuracy enhancement of digital image correlation with B-spline interpolation,” Opt. Lett. 36(16), 3070–3072 (2011).
[Crossref] [PubMed]

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

2009 (2)

Y. Wang, M. Sutton, H. Bruck, and H. 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]

2008 (1)

2006 (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]

D. Lecompte, H. Sol, J. Vantomme, and A. Habraken, “Analysis of speckle patterns for deformation measurements by digital image correlation,” Proc. SPIE 6341, 63410 (2006).
[Crossref]

2004 (1)

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

2003 (2)

T. Roesgen, “Optimal subpixel interpolation in particle image velocimetry,” Exp. Fluids 35(3), 252–256 (2003).
[Crossref]

B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[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)

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]

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

1997 (1)

J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8(12), 1379 (1997).
[Crossref]

1994 (1)

1989 (1)

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

1988 (1)

B. Kumar and S. C. D. Roy, “Design of digital differentiators for low frequencies,” Proc. IEEE 76(3), 287–289 (1988).
[Crossref]

1944 (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23(3), 282–332 (1944).
[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]

Baker, S.

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

Benckert, L. R.

Blaysat, B.

B. Blaysat, M. Grédiac, and F. Sur, “On the propagation of camera sensor noise to displacement maps obtained by DIC - an experimental study,” Exp. Mech. 56(6), 919–944 (2016).
[Crossref]

Blu, T.

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

Bomarito, G.

G. Bomarito, J. Hochhalter, T. Ruggles, and A. Cannon, “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017).
[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]

Bruck, H.

Y. Wang, M. Sutton, H. Bruck, and H. 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]

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(3), 261–267 (1989).
[Crossref]

Cannon, A.

G. Bomarito, J. Hochhalter, T. Ruggles, and A. Cannon, “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017).
[Crossref]

Chakrabarti, M.

Chen, P.

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

Chen, W.

W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017).
[Crossref]

Chen, Z.

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

Cheng, T.

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

Cigada, A.

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

Dai, X.

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

Debruyne, D.

Y. Wang, P. Lava, P. Reu, and D. Debruyne, “Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements,” Strain 52(2), 110–128 (2015).

Flannery, B. P.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University, 2007).

Flusser, J.

B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[Crossref]

Gan, R. Z.

Gao, Y.

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

Gao, Z.

Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018).
[Crossref]

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

Y. Su, Q. Zhang, Z. Gao, and X. Xu, “Noise-induced bias for convolution-based interpolation in digital image correlation,” Opt. Express 24(2), 1175–1195 (2016).
[Crossref] [PubMed]

Z. Gao, X. Xu, Y. Su, and Q. Zhang, “Experimental analysis of image noise and interpolation bias in digital image correlation,” Opt. Lasers Eng. 81, 46–53 (2016).
[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]

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]

Grédiac, M.

B. Blaysat, M. Grédiac, and F. Sur, “On the propagation of camera sensor noise to displacement maps obtained by DIC - an experimental study,” Exp. Mech. 56(6), 919–944 (2016).
[Crossref]

Habraken, A.

D. Lecompte, H. Sol, J. Vantomme, and A. Habraken, “Analysis of speckle patterns for deformation measurements by digital image correlation,” Proc. SPIE 6341, 63410 (2006).
[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]

Hanson, S. G.

He, X.

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

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

Hoang, T.

Hochhalter, J.

G. Bomarito, J. Hochhalter, T. Ruggles, and A. Cannon, “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017).
[Crossref]

Hu, Z.

Z. Hu, T. Xu, H. Luo, R. Z. Gan, and H. Lu, “Measurement of thickness and profile of a transparent material using fluorescent stereo microscopy,” Opt. Express 24(26), 29822–29829 (2016).
[Crossref]

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

Hua, T.

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

Jakobsen, M. L.

Jiang, Z.

W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017).
[Crossref]

Kompenhans, J.

M. Raffel, C. E. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: a Practical Guide (Springer, 2013).

Kumar, B.

B. Kumar and S. C. D. Roy, “Design of digital differentiators for low frequencies,” Proc. IEEE 76(3), 287–289 (1988).
[Crossref]

Lava, P.

Y. Wang, P. Lava, P. Reu, and D. Debruyne, “Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements,” Strain 52(2), 110–128 (2015).

Lavatelli, A.

A. Lavatelli and E. Zappa, “A displacement uncertainty model for 2-D DIC measurement under motion blur conditions,” IEEE Trans. Instrum. Meas. 66(3), 451–459 (2017).
[Crossref]

Lecompte, D.

D. Lecompte, H. Sol, J. Vantomme, and A. Habraken, “Analysis of speckle patterns for deformation measurements by digital image correlation,” Proc. SPIE 6341, 63410 (2006).
[Crossref]

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

Li, K.

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

Liu, Y.

W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017).
[Crossref]

Liu, Z.

W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017).
[Crossref]

Lu, H.

Luo, H.

Luu, L.

Ma, J.

Matta, F.

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

Matthews, I.

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

Mazzoleni, P.

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

McNeill, S. R.

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

Orteu, J. J.

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).

Pan, B.

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

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

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

Papoulis, A.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (Tata McGraw-Hill Education, 2002).

Peters, W. H.

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

Pillai, S. U.

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (Tata McGraw-Hill Education, 2002).

Press, W. H.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University, 2007).

Qian, K.

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

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

Quan, C.

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

Raffel, M.

M. Raffel, C. E. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: a Practical Guide (Springer, 2013).

Reu, P.

Y. Wang, P. Lava, P. Reu, and D. Debruyne, “Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements,” Strain 52(2), 110–128 (2015).

Rice, S. O.

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23(3), 282–332 (1944).
[Crossref]

Roesgen, T.

T. Roesgen, “Optimal subpixel interpolation in particle image velocimetry,” Exp. Fluids 35(3), 252–256 (2003).
[Crossref]

Roy, S. C. D.

B. Kumar and S. C. D. Roy, “Design of digital differentiators for low frequencies,” Proc. IEEE 76(3), 287–289 (1988).
[Crossref]

Ruggles, T.

G. Bomarito, J. Hochhalter, T. Ruggles, and A. Cannon, “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017).
[Crossref]

Schreier, H.

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

Schreier, H. W.

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]

Shao, X.

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

Sjödahl, M.

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]

D. Lecompte, H. Sol, J. Vantomme, and A. Habraken, “Analysis of speckle patterns for deformation measurements by digital image correlation,” Proc. SPIE 6341, 63410 (2006).
[Crossref]

Su, Y.

Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018).
[Crossref]

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

Z. Gao, X. Xu, Y. Su, and Q. Zhang, “Experimental analysis of image noise and interpolation bias in digital image correlation,” Opt. Lasers Eng. 81, 46–53 (2016).
[Crossref]

Y. Su, Q. Zhang, Z. Gao, and X. Xu, “Noise-induced bias for convolution-based interpolation in digital image correlation,” Opt. Express 24(2), 1175–1195 (2016).
[Crossref] [PubMed]

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

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]

Sur, F.

B. Blaysat, M. Grédiac, and F. Sur, “On the propagation of camera sensor noise to displacement maps obtained by DIC - an experimental study,” Exp. Mech. 56(6), 919–944 (2016).
[Crossref]

Sutton, M.

Y. Wang, M. Sutton, H. Bruck, and H. 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]

Sutton, M. A.

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

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]

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

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

Szeliski, R.

R. Szeliski, Computer Vision: Algorithms and Applications (Springer Science & Business Media, 2010).

Tang, L.

W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017).
[Crossref]

Teukolsky, S. A.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University, 2007).

Thevenaz, P.

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

Tong, W.

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

W. Tong, “Subpixel image registration with reduced bias,” Opt. Lett. 36(5), 763–765 (2011).
[Crossref] [PubMed]

Unser, M.

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

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]

D. Lecompte, H. Sol, J. Vantomme, and A. Habraken, “Analysis of speckle patterns for deformation measurements by digital image correlation,” Proc. SPIE 6341, 63410 (2006).
[Crossref]

Vetterling, W. T.

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University, 2007).

Vo, M.

Wang, S.

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

Wang, Y.

Y. Wang, P. Lava, P. Reu, and D. Debruyne, “Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements,” Strain 52(2), 110–128 (2015).

Y. Wang, M. Sutton, H. Bruck, and H. 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.

Wereley, S.

M. Raffel, C. E. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: a Practical Guide (Springer, 2013).

Westerweel, J.

J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8(12), 1379 (1997).
[Crossref]

Willert, C. E.

M. Raffel, C. E. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: a Practical Guide (Springer, 2013).

Wu, S.

Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018).
[Crossref]

Wu, X.

Xie, H.

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

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

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

Xu, T.

Xu, X.

Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018).
[Crossref]

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

Y. Su, Q. Zhang, Z. Gao, and X. Xu, “Noise-induced bias for convolution-based interpolation in digital image correlation,” Opt. Express 24(2), 1175–1195 (2016).
[Crossref] [PubMed]

Z. Gao, X. Xu, Y. Su, and Q. Zhang, “Experimental analysis of image noise and interpolation bias in digital image correlation,” Opt. Lasers Eng. 81, 46–53 (2016).
[Crossref]

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

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]

Zappa, E.

A. Lavatelli and E. Zappa, “A displacement uncertainty model for 2-D DIC measurement under motion blur conditions,” IEEE Trans. Instrum. Meas. 66(3), 451–459 (2017).
[Crossref]

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

Zhang, Q.

Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018).
[Crossref]

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

Z. Gao, X. Xu, Y. Su, and Q. Zhang, “Experimental analysis of image noise and interpolation bias in digital image correlation,” Opt. Lasers Eng. 81, 46–53 (2016).
[Crossref]

Y. Su, Q. Zhang, Z. Gao, and X. Xu, “Noise-induced bias for convolution-based interpolation in digital image correlation,” Opt. Express 24(2), 1175–1195 (2016).
[Crossref] [PubMed]

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

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]

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

Zhang, Y.

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

Zhou, P.

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

Zhu, F.

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

Zitová, B.

B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[Crossref]

Appl. Opt. (1)

Bell Syst. Tech. J. (1)

S. O. Rice, “Mathematical analysis of random noise,” Bell Syst. Tech. J. 23(3), 282–332 (1944).
[Crossref]

Exp. Fluids (1)

T. Roesgen, “Optimal subpixel interpolation in particle image velocimetry,” Exp. Fluids 35(3), 252–256 (2003).
[Crossref]

Exp. Mech. (4)

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

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

B. Blaysat, M. Grédiac, and F. Sur, “On the propagation of camera sensor noise to displacement maps obtained by DIC - an experimental study,” Exp. Mech. 56(6), 919–944 (2016).
[Crossref]

W. Chen, Z. Jiang, L. Tang, Y. Liu, and Z. Liu, “Equal noise resistance of two mainstream iterative sub-pixel registration algorithms in digital image correlation,” Exp. Mech. 57(6), 979–996 (2017).
[Crossref]

IEEE Trans. Instrum. Meas. (1)

A. Lavatelli and E. Zappa, “A displacement uncertainty model for 2-D DIC measurement under motion blur conditions,” IEEE Trans. Instrum. Meas. 66(3), 451–459 (2017).
[Crossref]

IEEE Trans. Med. Imaging (1)

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

Image Vis. Comput. (1)

B. Zitová and J. Flusser, “Image registration methods: a survey,” Image Vis. Comput. 21(11), 977–1000 (2003).
[Crossref]

Int. J. Comput. Vision (1)

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

Meas. Sci. Technol. (3)

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

J. Westerweel, “Fundamentals of digital particle image velocimetry,” Meas. Sci. Technol. 8(12), 1379 (1997).
[Crossref]

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

Opt. Eng. (2)

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

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]

Opt. Express (4)

Opt. Laser Technol. (1)

T. Hua, H. Xie, S. Wang, Z. Hu, P. Chen, and Q. Zhang, “Evaluation of the quality of a speckle pattern in the digital image correlation method by mean subset fluctuation,” Opt. Laser Technol. 43(1), 9–13 (2011).
[Crossref]

Opt. Lasers Eng. (8)

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

G. Bomarito, J. Hochhalter, T. Ruggles, and A. Cannon, “Increasing accuracy and precision of digital image correlation through pattern optimization,” Opt. Lasers Eng. 91, 73–85 (2017).
[Crossref]

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

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

Y. Su, Q. Zhang, X. Xu, Z. Gao, and S. Wu, “Interpolation bias for the inverse compositional Gauss-Newton algorithm in digital image correlation,” Opt. Lasers Eng. 100, 267–278 (2018).
[Crossref]

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

Z. Gao, X. Xu, Y. Su, and Q. Zhang, “Experimental analysis of image noise and interpolation bias in digital image correlation,” Opt. Lasers Eng. 81, 46–53 (2016).
[Crossref]

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

Opt. Lett. (3)

Proc. IEEE (1)

B. Kumar and S. C. D. Roy, “Design of digital differentiators for low frequencies,” Proc. IEEE 76(3), 287–289 (1988).
[Crossref]

Proc. SPIE (1)

D. Lecompte, H. Sol, J. Vantomme, and A. Habraken, “Analysis of speckle patterns for deformation measurements by digital image correlation,” Proc. SPIE 6341, 63410 (2006).
[Crossref]

Strain (2)

Y. Wang, P. Lava, P. Reu, and D. Debruyne, “Theoretical analysis on the measurement errors of local 2D DIC: part I temporal and spatial uncertainty quantification of displacement measurements,” Strain 52(2), 110–128 (2015).

Y. Wang, M. Sutton, H. Bruck, and H. 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]

Other (6)

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).

M. Raffel, C. E. Willert, S. Wereley, and J. Kompenhans, Particle Image Velocimetry: a Practical Guide (Springer, 2013).

R. Szeliski, Computer Vision: Algorithms and Applications (Springer Science & Business Media, 2010).

A. Papoulis and S. U. Pillai, Probability, Random Variables, and Stochastic Processes (Tata McGraw-Hill Education, 2002).

Y. Su, “Optimal speckle radius for image registration,” figshare (2017) [retrieved 01 Nov 2017], http://dx.doi.org/10.6084/m9.figshare.5558542.v1 .

W. H. Press, S. A. Teukolsky, W. T. Vetterling, and B. P. Flannery, Numerical Recipes: The Art of Scientific Computing (Cambridge University, 2007).

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Figures (7)

Fig. 1
Fig. 1

A schematic of speckle patterns which are composed of random positioned speckles. These speckles are randomly scattered over a region, and thus the speckle positions are random variables. Due to this spatial randomness, patterns shown in (a) and (b) are different, though the generation parameters are identical.

Fig. 2
Fig. 2

Systematic errors eb, random errors en, and root mean squared errors er of Gaussian speckle patterns with speckle radii (a) R = 1.3 pixels, (b) R = 1.5 pixels, (c) R = 2.0 pixels, and (d) R = 3.0 pixels. The theoretical estimates are evaluated using Eq. (12), Eq. (21), and Eq. (27). The variability of the numerical results is due to the spatial randomness illustrated in Fig. 1.

Fig. 3
Fig. 3

Influence of subset size on measurement errors: systematic errors for speckle radius (a1) R = 2.0 pixels and (a2) R = 5.0 pixels; random errors for speckle radius (b1) R = 2.0 pixels and (b2) R = 5.0 pixels. The actual displacement is u0 = 0.25 pixels. The theoretical estimates are evaluated using Eq. (12) and Eq. (21).

Fig. 4
Fig. 4

Influence of speckle coverage on measurement errors: systematic errors for speckle radius (a1) R = 2.0 pixels and (a2) R = 5.0 pixels; random errors for speckle radius (b1) R = 2.0 pixels and (b2) R = 5.0 pixels. The actual displacement is u0 = 0.25 pixels. The theoretical estimates are evaluated using Eq. (12) and Eq. (21).

Fig. 5
Fig. 5

Influence of speckle radius on measurement errors. (a1) Systematic errors; (a2) the difference between systematic errors by theoretical estimations and by numerical experiments. Random errors for subset size (b1) M = 31 pixels and (b2) M = 91 pixels. The actual displacement u0 = 0.25 pixels. The theoretical estimates are evaluated using Eq. (12) and Eq. (21).

Fig. 6
Fig. 6

The normalized random errors are approximately proportional to the speckle radius.

Fig. 7
Fig. 7

Relationship between pattern quality measure and speckle radius: (a) subset size M = 31 pixels, speckle coverage 50%; (b) subset size M = 31 pixels, speckle coverage 80%; (c) subset size M = 91 pixels, speckle coverage 50%; (d) subset size M = 91 pixels, speckle coverage 80%. The theoretical estimations of the pattern quality measure are evaluated using Eq. (29) and the optimal speckle radii Ropt are evaluated using Eq. (31).

Equations (43)

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

f ( x , y ) = i = 1 N ψ ( x x i , y y i ) ,
f ( x , y ) = i = ψ ( x x i , y y i ) .
z ( x , y ) = i = δ ( x x i , y y i ) ,
f ( x , y ) = ψ z ( x , y ) = i = ψ ( x x i , y y i ) .
𝔼 { f ( x , y ) } = λ ψ ( ξ , ζ ) d ξ d ζ , Var ( f ( x , y ) ) = λ ψ 2 ( ξ , ζ ) d ξ d ζ .
S f f ( ω x , ω y ) = 4 π 2 λ 2 ψ ^ 2 ( 0 , 0 ) δ ( ω x , ω y ) + λ | ψ ^ ( ω x , ω y ) | 2 ,
f [ m , n ] = f ( m , n ) + w f [ m , n ] , g [ m , n ] = f ( m u 0 , n ) + w g [ m , n ] .
u = arg min m = S S n = S S { f [ m , n ] g ( m + u , n ) } 2 ,
e b = 𝔼 { Δ u | f ( x , y ) | } , e n = Var { Δ u | f ( x , y ) | } .
e b C b sin 2 π u 0 , C b = π π π π E b ( ω x , ω y ) | f ^ ( ω x , ω y ) | 2 d ω x d ω y π π π π ω x d ^ ( ω x , ω y ) φ ^ ( ω x , ω y ) | f ^ ( ω x , ω y ) | 2 d ω x d ω y , E b ( ω x , ω y ) = d ^ ( ω x , ω y ) [ φ ^ ( ω x 2 π , ω y ) φ ^ ( ω x + 2 π , ω y ) ] .
𝔼 { C b } π π π π E b ( ω x , ω y ) S f f ( ω x , ω y ) d ω x d ω y π π π π ω x d ^ ( ω x , ω y ) φ ^ ( ω x , ω y ) S f f ( ω x , ω y ) d ω x d ω y ,
𝔼 { e b } 𝔼 { C b } sin 2 π u 0 , 𝔼 { C b } π π π π E b ( ω x , ω y ) | ψ ^ ( ω x , ω y ) | 2 d ω x d ω y π π π π ω x d ^ ( ω x , ω y ) φ ^ ( ω x , ω y ) | ψ ^ ( ω x , ω y ) | 2 d ω x d ω y , E b ( ω x , ω y ) = d ^ ( ω x , ω y ) [ φ ^ ( ω x 2 π , ω y ) φ ^ ( ω x + 2 π , ω y ) ] .
e n = σ f 2 + σ g 2 m = S S n = S S f x 2 [ m , n ] ,
e n = σ Q ,
𝔼 { Φ ( x ) } Φ ( 𝔼 { x } ) + 1 2 Var ( x ) Φ ( 𝔼 { x } ) .
𝔼 { e n } σ 𝔼 { Q } [ 1 + 3 8 Var ( Q ) 𝔼 { Q } 2 ] .
f x ( x , y ) = ( d f ) ( x , y ) = ( d ψ z ) ( x , y ) .
q = 𝔼 { f x 2 ( x , y ) } = λ 4 π 2 | d ^ ( ω x , ω y ) | 2 | ψ ^ ( ω x , ω y ) | 2 d ω x d ω y .
𝔼 { Q } = m = S S n = S S 𝔼 { f x 2 [ m , n ] } = M 2 q ,
Var ( Q ) 2 M 2 q .
𝔼 { e n } σ M q ( 1 + 3 4 M 2 q ) .
e m = e b 2 + e n 2 ,
𝔼 { e m } = 𝔼 { e b 2 } + 𝔼 𝔼 { e n 2 } .
𝔼 { e b 2 } = 𝔼 { C b 2 } sin 2 2 π u 0 𝔼 { C b } 2 sin 2 2 π u 0 ,
𝔼 { e n 2 } σ 2 𝔼 { Q } [ 1 + Var ( Q ) 𝔼 { Q } 2 ] = σ 2 M 2 q ( 1 + 2 M 2 q ) .
𝔼 { e m } 𝔼 { C b } 2 sin 2 2 π u 0 + σ 2 M 2 q ( 1 + 2 M 2 q ) ,
𝔼 { e r } 𝔼 { e m } 1 / 2 .
V = 0 1 e m ( u 0 ) d u 0 .
𝔼 { V } 1 2 𝔼 { C b } 2 + σ 2 M 2 q ( 1 + 2 M 2 q ) .
d 𝔼 { V } d R = 0 .
𝔼 { C b } d 𝔼 { C b } d R σ 2 M 2 q 2 ( 1 + 4 M 2 q ) d q d R = 0 ,
ψ g ( x , y ; R ) = exp ( x 2 + y 2 R 2 ) ,
d ^ ( ω x , ω y ) ω x and φ ^ ( ω x , ω y ) 1 , for π < ω x < π , π < ω y < π .
π π π π ω x 2 ψ ^ g 2 ( ω x , ω y ; R ) d ω x d ω y ω x 2 ψ ^ g 2 ( ω x , ω y ; R ) d ω x d ω y = 2 π 3 .
𝔼 { C b } 1 2 π 3 π π π π E b ( ω x , ω y ) ψ g 2 ( ω x , ω y ; R ) d ω x d ω y ,
𝔼 { e n 2 } σ 2 M 2 q ( 1 + 2 M 2 q ) σ 2 M 2 q 2 σ 2 R 2 ρ M 2 .
π π π π E b ( ω x , ω y ) ψ ^ g 2 ( ω x , ω y ; R ) d ω x d ω y × π π π π E b ( ω x , ω y ) ψ ^ g ( ω x , ω y ; R ) d ψ ^ g ( ω x , ω y ; R ) d R d ω x d ω y + 8 π 6 σ 2 R ρ M 2 = 0 ,
f ( k ) ( x , y ) = i = 1 N exp { [ x x i ( k ) ] 2 + [ y y i ( k ) ] 2 R 2 } ,
f ( k ) [ m , n ] = f ( k ) ( m , n ) , g ( k , i ) [ m , n ] = f ( k ) ( m u 0 , n ) + w ( k , i ) [ m , n ] ,
e b ( k ) = 1 I i = 1 I [ u ( k , i ) u 0 ] , e n ( k ) = 1 I 1 i = 1 I [ u ( k , i ) u 0 e b ( k ) ] 2 ,
𝔼 { e b } = 1 K k = 1 K e b ( k ) , Var { e b } = 1 K 1 k = 1 K ( e b ( k ) 𝔼 { e b } ) 2 .
𝔼 { e n } = 1 K k = 1 K e n ( k ) , Var { e n } = 1 K 1 k = 1 K ( e n ( k ) 𝔼 { e n } ) 2 .
𝔼 { e n } 2 σ ρ M R ,