Abstract

Electronic speckle photography is an accurate, easy-to-use, video-based technique for the analysis of two- and three-dimensional deformation fields and in-plane strain fields, based on numerical cross correlation. Through the use of statistical optics, simulated speckle patterns, and experiments the accuracy in electronic speckle photography was found to depend on correlation, speckle size, window size, and correlation filter. The estimated correlation was found to be the combined effect of three mutually competing factors because of classical speckle correlation, subimage overlap, and displacement gradients. In many applications white-light speckle patterns provide a more accurate estimate of the displacement field than do laser speckle patterns.

© 1997 Optical Society of America

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  1. D. J. Chen, F. P. Chiang, “Optimal sampling and range of measurement in displacement-only laser-speckle correlation,” Exp. Mech. 32, 145–153 (1992).
    [CrossRef]
  2. D. R. Matthys, J. A. Gilbert, P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng. 30, 1455–1460 (1990).
    [CrossRef]
  3. S. Noh, I. Yamaguchi, “Two-dimensional measurement of strain distribution by speckle correlation,” Jpn. J. Appl. Phys. 31, L1299–L1301 (1992).
    [CrossRef]
  4. C. E. Willert, M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10, 181–193 (1991).
    [CrossRef]
  5. M. Sjödahl, L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993).
    [CrossRef] [PubMed]
  6. M. Sjödahl, L. R. Benckert, “Systematic and random errors in electronic speckle photography,” Appl. Opt. 33, 7461–7471 (1994).
    [CrossRef] [PubMed]
  7. M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
    [CrossRef] [PubMed]
  8. M. R. James, W. L. Morris, B. N. Cox, “A high accuracy strain-field mapper,” Exp. Mech. 30, 60–67 (1990).
    [CrossRef]
  9. D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Digital speckle-displacement measurement using a complex spectrum method,” Appl. Opt. 32, 1839–1849 (1993).
    [CrossRef] [PubMed]
  10. P. Synnergren, “Measurement of 3-D displacement fields and shape using electronic speckle photography,” to be published in Opt. Eng.
  11. M. Sjödahl, “Electronic speckle photography: measurement of in-plane strain fields through the use of defocused laser speckle,” Appl. Opt. 34, 5799–5808 (1995).
    [CrossRef] [PubMed]
  12. J. P. Fillard, H. M’timet, J. M. Lussert, M. Castagné, “Computer simulation of super-resolution point source image detection,” Opt. Eng. 32, 2936–2944 (1993).
    [CrossRef]
  13. H. T. Huang, H. E. Fiedler, J. J. Wang, “Limitation and improvement of PIV; Part II: particle image distortion, a novel technique,” Exp. Fluids 15, 263–273 (1993).
  14. M. A. Sutton, W. J. Wolters, W. H. Peters, W. F. Ranson, S. R. McNeill, “Determination of displacements using an improved digital correlation method,” Image Vision Comput. 1, 133–139 (1983).
    [CrossRef]
  15. H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton-Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
    [CrossRef]
  16. D. W. Li, J. B. Chen, F. P. Chiang, “Statistical analysis of one-beam subjective laser-speckle interferometry,” J. Opt. Soc. Am. A 2, 657–666 (1985).
    [CrossRef]
  17. J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.
  18. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  19. I. Yamaguchi, “Fringe formation in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1985), Vol. 22, pp. 272–340.
  20. M. Sjödahl, “Calculation of speckle displacement, decorrelation, and object-point location in imaging systems,” Appl. Opt. 34, 7998–8010 (1995).
    [CrossRef] [PubMed]
  21. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  22. M. P. Wernet, A. Pline, “Particle displacement technique and Cramer–Rao lower bound error in centroid estimates from CCD imagery,” Exp. Fluids 15, 295–307 (1993).
  23. B. V. K. V. Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
    [CrossRef] [PubMed]
  24. A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).
  25. J. M. Huntley, “Speckle photography fringe analysis: assessment of current algorithms,” Appl. Opt. 28, 4316–4322 (1989).
    [CrossRef] [PubMed]
  26. D. W. Li, F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. A 3, 1023–1031 (1986).
    [CrossRef]

1995 (2)

1994 (2)

1993 (5)

M. P. Wernet, A. Pline, “Particle displacement technique and Cramer–Rao lower bound error in centroid estimates from CCD imagery,” Exp. Fluids 15, 295–307 (1993).

J. P. Fillard, H. M’timet, J. M. Lussert, M. Castagné, “Computer simulation of super-resolution point source image detection,” Opt. Eng. 32, 2936–2944 (1993).
[CrossRef]

H. T. Huang, H. E. Fiedler, J. J. Wang, “Limitation and improvement of PIV; Part II: particle image distortion, a novel technique,” Exp. Fluids 15, 263–273 (1993).

M. Sjödahl, L. R. Benckert, “Electronic speckle photography: analysis of an algorithm giving the displacement with subpixel accuracy,” Appl. Opt. 32, 2278–2284 (1993).
[CrossRef] [PubMed]

D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Digital speckle-displacement measurement using a complex spectrum method,” Appl. Opt. 32, 1839–1849 (1993).
[CrossRef] [PubMed]

1992 (2)

D. J. Chen, F. P. Chiang, “Optimal sampling and range of measurement in displacement-only laser-speckle correlation,” Exp. Mech. 32, 145–153 (1992).
[CrossRef]

S. Noh, I. Yamaguchi, “Two-dimensional measurement of strain distribution by speckle correlation,” Jpn. J. Appl. Phys. 31, L1299–L1301 (1992).
[CrossRef]

1991 (1)

C. E. Willert, M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10, 181–193 (1991).
[CrossRef]

1990 (3)

M. R. James, W. L. Morris, B. N. Cox, “A high accuracy strain-field mapper,” Exp. Mech. 30, 60–67 (1990).
[CrossRef]

D. R. Matthys, J. A. Gilbert, P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng. 30, 1455–1460 (1990).
[CrossRef]

B. V. K. V. Kumar, L. Hassebrook, “Performance measures for correlation filters,” Appl. Opt. 29, 2997–3006 (1990).
[CrossRef] [PubMed]

1989 (2)

J. M. Huntley, “Speckle photography fringe analysis: assessment of current algorithms,” Appl. Opt. 28, 4316–4322 (1989).
[CrossRef] [PubMed]

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

1986 (1)

1985 (1)

1983 (1)

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

1981 (1)

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

1964 (1)

A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Benckert, L. R.

Bruck, H. A.

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

Castagné, M.

J. P. Fillard, H. M’timet, J. M. Lussert, M. Castagné, “Computer simulation of super-resolution point source image detection,” Opt. Eng. 32, 2936–2944 (1993).
[CrossRef]

Chen, D. J.

D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Digital speckle-displacement measurement using a complex spectrum method,” Appl. Opt. 32, 1839–1849 (1993).
[CrossRef] [PubMed]

D. J. Chen, F. P. Chiang, “Optimal sampling and range of measurement in displacement-only laser-speckle correlation,” Exp. Mech. 32, 145–153 (1992).
[CrossRef]

Chen, J. B.

Chiang, F. P.

Cox, B. N.

M. R. James, W. L. Morris, B. N. Cox, “A high accuracy strain-field mapper,” Exp. Mech. 30, 60–67 (1990).
[CrossRef]

Don, H. S.

Fiedler, H. E.

H. T. Huang, H. E. Fiedler, J. J. Wang, “Limitation and improvement of PIV; Part II: particle image distortion, a novel technique,” Exp. Fluids 15, 263–273 (1993).

Fillard, J. P.

J. P. Fillard, H. M’timet, J. M. Lussert, M. Castagné, “Computer simulation of super-resolution point source image detection,” Opt. Eng. 32, 2936–2944 (1993).
[CrossRef]

Gharib, M.

C. E. Willert, M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10, 181–193 (1991).
[CrossRef]

Gilbert, J. A.

D. R. Matthys, J. A. Gilbert, P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng. 30, 1455–1460 (1990).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

Greguss, P.

D. R. Matthys, J. A. Gilbert, P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng. 30, 1455–1460 (1990).
[CrossRef]

Hassebrook, L.

Huang, H. T.

H. T. Huang, H. E. Fiedler, J. J. Wang, “Limitation and improvement of PIV; Part II: particle image distortion, a novel technique,” Exp. Fluids 15, 263–273 (1993).

Huntley, J. M.

James, M. R.

M. R. James, W. L. Morris, B. N. Cox, “A high accuracy strain-field mapper,” Exp. Mech. 30, 60–67 (1990).
[CrossRef]

Kumar, B. V. K. V.

Li, D. W.

Lussert, J. M.

J. P. Fillard, H. M’timet, J. M. Lussert, M. Castagné, “Computer simulation of super-resolution point source image detection,” Opt. Eng. 32, 2936–2944 (1993).
[CrossRef]

M’timet, H.

J. P. Fillard, H. M’timet, J. M. Lussert, M. Castagné, “Computer simulation of super-resolution point source image detection,” Opt. Eng. 32, 2936–2944 (1993).
[CrossRef]

Matthys, D. R.

D. R. Matthys, J. A. Gilbert, P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng. 30, 1455–1460 (1990).
[CrossRef]

McNeill, S. R.

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

Morris, W. L.

M. R. James, W. L. Morris, B. N. Cox, “A high accuracy strain-field mapper,” Exp. Mech. 30, 60–67 (1990).
[CrossRef]

Noh, S.

S. Noh, I. Yamaguchi, “Two-dimensional measurement of strain distribution by speckle correlation,” Jpn. J. Appl. Phys. 31, L1299–L1301 (1992).
[CrossRef]

Peters, W. H.

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

Pline, A.

M. P. Wernet, A. Pline, “Particle displacement technique and Cramer–Rao lower bound error in centroid estimates from CCD imagery,” Exp. Fluids 15, 295–307 (1993).

Ranson, W. F.

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

Sjödahl, M.

Sutton, M. A.

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

Synnergren, P.

P. Synnergren, “Measurement of 3-D displacement fields and shape using electronic speckle photography,” to be published in Opt. Eng.

Tan, Y. S.

VanderLugt, A.

A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Wang, J. J.

H. T. Huang, H. E. Fiedler, J. J. Wang, “Limitation and improvement of PIV; Part II: particle image distortion, a novel technique,” Exp. Fluids 15, 263–273 (1993).

Wernet, M. P.

M. P. Wernet, A. Pline, “Particle displacement technique and Cramer–Rao lower bound error in centroid estimates from CCD imagery,” Exp. Fluids 15, 295–307 (1993).

Willert, C. E.

C. E. Willert, M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10, 181–193 (1991).
[CrossRef]

Wolters, W. J.

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

Yamaguchi, I.

S. Noh, I. Yamaguchi, “Two-dimensional measurement of strain distribution by speckle correlation,” Jpn. J. Appl. Phys. 31, L1299–L1301 (1992).
[CrossRef]

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

I. Yamaguchi, “Fringe formation in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1985), Vol. 22, pp. 272–340.

Appl. Opt. (8)

Exp. Fluids (3)

M. P. Wernet, A. Pline, “Particle displacement technique and Cramer–Rao lower bound error in centroid estimates from CCD imagery,” Exp. Fluids 15, 295–307 (1993).

H. T. Huang, H. E. Fiedler, J. J. Wang, “Limitation and improvement of PIV; Part II: particle image distortion, a novel technique,” Exp. Fluids 15, 263–273 (1993).

C. E. Willert, M. Gharib, “Digital particle image velocimetry,” Exp. Fluids 10, 181–193 (1991).
[CrossRef]

Exp. Mech. (3)

D. J. Chen, F. P. Chiang, “Optimal sampling and range of measurement in displacement-only laser-speckle correlation,” Exp. Mech. 32, 145–153 (1992).
[CrossRef]

M. R. James, W. L. Morris, B. N. Cox, “A high accuracy strain-field mapper,” Exp. Mech. 30, 60–67 (1990).
[CrossRef]

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

IEEE Trans. Inf. Theory (1)

A. VanderLugt, “Signal Detection by Complex Spatial Filtering,” IEEE Trans. Inf. Theory IT-10, 139–145 (1964).

Image Vision Comput. (1)

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

J. Opt. Soc. Am. A (2)

Jpn. J. Appl. Phys. (1)

S. Noh, I. Yamaguchi, “Two-dimensional measurement of strain distribution by speckle correlation,” Jpn. J. Appl. Phys. 31, L1299–L1301 (1992).
[CrossRef]

Opt. Acta (1)

I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
[CrossRef]

Opt. Eng. (2)

D. R. Matthys, J. A. Gilbert, P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng. 30, 1455–1460 (1990).
[CrossRef]

J. P. Fillard, H. M’timet, J. M. Lussert, M. Castagné, “Computer simulation of super-resolution point source image detection,” Opt. Eng. 32, 2936–2944 (1993).
[CrossRef]

Other (4)

P. Synnergren, “Measurement of 3-D displacement fields and shape using electronic speckle photography,” to be published in Opt. Eng.

J. W. Goodman, “Statistical Properties of Laser Speckle Patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 9–75.

I. Yamaguchi, “Fringe formation in deformation and vibration measurements using laser light,” in Progress in Optics, E. Wolf, ed. (Elsevier Science, New York, 1985), Vol. 22, pp. 272–340.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

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Figures (8)

Fig. 1
Fig. 1

Geometry of the recording system.

Fig. 2
Fig. 2

Area of correlated speckle: S, before deformation of the object; S′, after deformation of the object.

Fig. 3
Fig. 3

Shape of the correlation peaks (heights are normalized by the CMF peak height): (a) and their corresponding power spectra; (b) obtained when analyzing a speckle pattern recorded with a quadratic aperture with the CMF (continuous black curve), the POF (continuous grey curve), and the IF (dashed curve), respectively.

Fig. 4
Fig. 4

Simulated random error e for the CMF (continuous curve), and the POF (dotted curve) correlation filters as a function of speckle correlation for three subimage sizes and for a given speckle size, σ, of 2.5 pixels. The fitted curves follow the relation e = k [(1 - δ)/δ]1/2 where k equals 0.251 (CMF, B = 16 pixels), 0.178 (POF, B = 16 pixels), 0.142 (CMF, B = 32 pixels), 0.105 (POF, B = 32 pixels), 0.076 (CMF, B = 64 pixels), and 0.059 (POF, B = 64 pixels), respectively.

Fig. 5
Fig. 5

Simulated random error e for the CMF (continuous curve) and the POF (dashed curve) correlation filters as a function of speckle size for three subimage sizes and for a given speckle correlation, δ, of 90%. The results follow the relations e = 0.022σ1.47 (CMF, B = 16 pixels), e = 0.010σ1.67 (CMF, B = 32 pixels), e = 0.0047σ1.87 (CMF, B = 64 pixels), e = 0.040σ0.47 (POF, B = 16 pixels), e = 0.019σ 0.67 (POF, B = 32 pixels), and e = 0.009σ0.87 (POF, B = 64 pixels), respectively.

Fig. 6
Fig. 6

Measured (dots) and analytical (continuous curve) correlation as a function of rotation angle for the white-light speckle pattern.

Fig. 7
Fig. 7

Measured and predicted random errors from the CMF (black dots, continuous curve) and the POF (grey dots, dashed curve) algorithms as a function of rotation angle for the laser speckle pattern (0–3.5 mrad) and the white-light speckle pattern (0–15 mrad), respectively.

Fig. 8
Fig. 8

Measured (dots) and analytical (continuous line) correlation as a function of rotation angle for the laser speckle pattern.

Equations (42)

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

cΔx=1SH0xH0x+Δx×I1xI2x+Δxdx,
I1xI2x+Δx=I21+γμΔx-A2,
γ=ΣPυP*υ+dexpjΘdυΣPυ2dυ2
μΔx=ΣPυ2 exp-2πjΔx·υdυΣPυ2dυ
cΔx=ccΔx+cfΔx,
ccΔx=I2SH0xH0x+Δxdx,
cfΔx=I2γSH0xH0x+ΔxμΔx-A2dx;
cfΔx=I2γΩS1S1μΔx-A2dx,
Ax=A0+xgradA|x=0,
cfΔx=I2γΩS1 exp-2πjxgradA|x=0·υdxS1×μΔx-A02.
Φυx, υy=1S1S1 exp-2πjxrx+yrydxdy,
rx=Axxx=0υx+Ayxx=0υy,
ry=Axyx=0υx+Ayyx=0υy.
Φυx, υy=sincsxAxxυx+Ayxυy×sincsyAxyυx+Ayyυy,
Axxx=0, Ayxx=0, Axyx=0, Ayyx=0,
ΓΔx, Δy=1AxxAyy-AxyAyx rectkxsxrectkysy,
kx=ΔxAxx+AxyAxxAyy-AxyAyxΔxAyxAxx-Δy,
ky=ΔyAxx-ΔxAyxAxxAyy-AxyAyx,
cfΔx, Δy=I2γΩΓΔx, ΔyμΔx-Ax, Δy-Ay2,
Ψ= rectkx/sxrectky/syμΔx, Δy2dΔx dΔy rectkx/sxrectky/sydΔx dΔy.
cfΔx, Δy=I2γΩΨμΔx-Ax, Δy-Ay2.
SAc1γ,
σI2=I1I21-γ2AcS.
pθθ=12π1/2σθ exp-θ22σθ2,
σθ=σIJ12=σ1.2B21-γπγ1/2.
e=kσ2B1-γγ1/2,
S1υ=S1υexp-jθυ.
HFPFpυ=S1υp exp-jθυ,
cΔx=-1HFPFpυS2υ,
δ=γΩΨ,
δm=i=0m-1j=0m-1I1i, j-I1I2i, j-I2i=0m-1j=0m-1I1i, j-I12i=0m-1j=0m-1I2i, j-I221/2,
δa=ΩΨ,
Ω=1-1/21-tanθ/22 tan θ,
Ψ=1-124πBθσ2+76912πBθσ4+OπBθσ6,
γ=ρ-sinρπ2,
ρ=2 cos-1f#1+Mθ sin α.
B=16: e=0.066σ1.471-δ/δ1/2,
B=32: e=0.030σ1.671-δ/δ1/2,
B=64: e=0.014σ1.871-δ/δ1/2,
B=16: e=0.120σ0.471-δ/δ1/2,
B=32: e=0.057σ0.671-δ/δ1/2,
B=64: e=0.027σ0.871-δ/δ1/2.

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