Abstract

Electronic speckle photography offers a simple and fast technique for measuring in-plane displacement fields in solid and fluid mechanics. Errors from undersampling, illumination divergence, and displacement magnitude have been analyzed and measured. The nature of the systematic error is such that a drift toward the closest integral pixel value is introduced. Because of the finite extent of the sensor area, considerable undersampling is tolerable before systematic errors occur. The random errors are mainly dependent on the effective f-number of the imaging system and speckle decorrelation introduced by object displacement. When sampling at a rate of ~70% of the Nyquist frequency, we avoided systematic errors and minimized random errors.

© 1994 Optical Society of America

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References

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  1. A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 203–253.
    [CrossRef]
  2. 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.
    [CrossRef]
  3. 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]
  4. T. C. Chu, W. F. Ranson, M. A. Sutton, W. H. Peters, “Applications of digital-image-correlation techniques to experimental mechanics,” Exp. Mech. 25, 232–244 (1985).
    [CrossRef]
  5. H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphon method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
    [CrossRef]
  6. D. J. Chen, F. P. Chiang, “Computer-aided speckle interferometry using spectral amplitude fringes,” Appl. Opt. 32, 225–236 (1993).
    [CrossRef] [PubMed]
  7. 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]
  8. 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]
  9. W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).
  10. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  11. D. W. Li, F. P. Chiang, “Decorrelation functions in laser speckle photography,” J. Opt. Soc. Am. A 3, 1023–1031 (1986).
    [CrossRef]
  12. G. E. P. Box, W. G. Hunter, J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978).
  13. M. Sjödahl, “Electronic speckle photography: some applications,” in Proceedings of Lasers 93, Laser in Engineering, W. Waidelich, ed. (Springer-Verlag, Berlin, 1994).

1993 (2)

1992 (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]

1989 (1)

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

1986 (1)

1985 (1)

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

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]

Benckert, L. R.

Box, G. E. P.

G. E. P. Box, W. G. Hunter, J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978).

Bruck, H. A.

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

Chen, D. J.

D. J. Chen, F. P. Chiang, “Computer-aided speckle interferometry using spectral amplitude fringes,” Appl. Opt. 32, 225–236 (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]

Chiang, F. P.

Chu, T. C.

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

Ennos, A. E.

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 203–253.
[CrossRef]

Goodman, J. W.

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.
[CrossRef]

Hunter, J. S.

G. E. P. Box, W. G. Hunter, J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978).

Hunter, W. G.

G. E. P. Box, W. G. Hunter, J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978).

Li, D. W.

McNeill, S. R.

H. A. Bruck, S. R. McNeill, M. A. Sutton, W. H. Peters, “Digital image correlation using Newton–Raphon 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]

Peters, W. H.

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

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

Pratt, W. K.

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

Ranson, W. F.

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

Sjödahl, M.

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]

M. Sjödahl, “Electronic speckle photography: some applications,” in Proceedings of Lasers 93, Laser in Engineering, W. Waidelich, ed. (Springer-Verlag, Berlin, 1994).

Sutton, M. A.

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

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

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.

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

Appl. Opt. (2)

Exp. Mech. (3)

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

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

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]

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 (1)

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]

Other (5)

G. E. P. Box, W. G. Hunter, J. S. Hunter, Statistics for Experimenters (Wiley, New York, 1978).

M. Sjödahl, “Electronic speckle photography: some applications,” in Proceedings of Lasers 93, Laser in Engineering, W. Waidelich, ed. (Springer-Verlag, Berlin, 1994).

A. E. Ennos, “Speckle interferometry,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), pp. 203–253.
[CrossRef]

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.
[CrossRef]

W. K. Pratt, Digital Image Processing (Wiley, New York, 1978).

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Figures (9)

Fig. 1
Fig. 1

Relative orientation of the planes used in the analysis of the speckle patterns: y1, x1, coordinates on the object; ψ, ξ, pupil plane coordinates; y, x, detector plane coordinates; D, aperture diameter; z, pupil to detector plane distance; L 0, object to pupil plane distance.

Fig. 2
Fig. 2

Process of sampling a speckle pattern imaged through a circular aperture. (a) Power spectral density of a continuous speckle pattern. (b) Power spectral density of the pixel sensitivity profile. (c) Sampling function. (d) Power spectral density of the digitized speckle pattern.

Fig. 3
Fig. 3

Effect of undersampling in the cross-correlation function. The sampling frequency is 60% of the Nyquist frequency. (a) The triangle part of Eq. (10). (b) The reconstructed cross-correlation function [response in R II x, 0)/〈 I2], where the translation is 0.3 pixels, versus displacement. (c) The reconstructed cross-correlation function [response in R II x, 0)/〈 I2], where the translation is 0.7 pixels, versus displacement.

Fig. 4
Fig. 4

Probability density functions for a speckle pattern recorded by a detector with a finite size of 11 × 11 μm for f # 36 (continuous black curve), f # 26 (continuous light curve), f # 18 (dashed curve), andf # 13 (dash–dotted curve).

Fig. 5
Fig. 5

Schematic of the experimental setup.

Fig. 6
Fig. 6

Rigid-body deformation fields obtained in the experiment. (a) Rigid-body translation. (b) Rigid-body rotation. Tho measured area is 9.1 × 9.1 mm subdivided into 16 × 16 data points.

Fig. 7
Fig. 7

Rotation field from Fig. 6(b) replotted as (a) the predicted rotation angle versus position vector magnitude and as (b) the predicted displacement magnitude versus the position vector magnitude.

Fig. 8
Fig. 8

Displacement residuals relative to f # 36. The mean values for f # 36 in Table 1 have been subtracted from the rest of the mean values with the same translation and illumination divergence: +, translation 5.25 pixels; *, translation 5.75 pixels; ○, translation 10.25 pixels; ×, translation 10.75 pixels.

Fig. 9
Fig. 9

Comparison between the experiments (dashed curves) and an expression based on decorrelation (continuous curves) for (a) pure translation and (b) pure rotation. The experimental fluctuations are indicated by bars.

Tables (2)

Tables Icon

Table 1 Pure Translation

Equations (22)

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μ A ( Δ x , Δ y ) = F [ P ( ξ , ψ ) 2 ] - P ( ξ , ψ ) 2 d ξ d ψ ,
R I ( Δ x , Δ y ) = I 2 [ 1 + | μ A ( Δ x , Δ y ) | 2 ] .
R I r = I 2 [ 1 + | 2 J 1 ( π D r λ z ) π D r λ z | 2 ] ,
σ = 1.22 λ z D .
ζ I ( ν ) = I 2 ( δ ( ν x , ν y ) + ( λ z D ) 2 4 π × { cos - 1 ( λ z D ν ) - λ z D ν [ 1 - ( λ z D ν ) 2 ] 1 / 2 } )
F ( x , y ) = [ I ( x , y ) * p ( x , y ) ] i = - j = - δ ( x - i δ x , y - j δ y ) ,
ζ F ( ν x , ν y ) = F [ ( x , y ) ] 2 F [ p ( x , y ) ] 2 * i = - j = - δ ( ν x - i / δ x , ν y - j / δ y ) .
max δ x , δ y λ z 2 D
max δ x , δ y 0.41 σ ,
Φ F ( ν x , ν y ) = I 2 [ δ ( ν x , ν y ) + ( λ z L ) 2 Λ ( λ z L ν x ) Λ ( λ z L ν y ) ] × exp [ - i 2 π ( ν x d x + ν y d y ) ] * i = - j = - δ ( ν x - i / δ , ν y - j / δ ) ,
R I I ( Δ x , Δ y ) = - 1 / 2 δ 1 / 2 δ Φ F ( ν x , ν y ) × exp [ 2 π i ( ν x Δ x + ν y Δ y ) ] d ν x d ν y .
p ( I ) ( κ I ) κ I κ - 1 exp ( - κ I I ) Γ ( κ )
κ = [ - R s ( Δ x , Δ y ) μ A ( Δ x , Δ y ) 2 d Δ x d Δ y ] - 1 ,
κ = { 16 0 1 0 1 ( 1 - Δ x ) ( 1 - Δ y ) [ J 1 ( a π D r λ z ) a π D r λ z ] 2 d Δ x d Δ y } - 1 r = [ ( Δ x ) 2 + ( Δ y ) 2 ] 1 / 2 ,
κ = [ 1 - 1 12 c 2 + 17 3456 c 4 - 29 138240 c 6 + 187 27648000 c 8 - 239 1393459200 c 10 + O ( c 12 ) ] - 1 .
f ( x 0 , y 0 ) = i = 0 n - 1 j = 0 m - 1 [ ( x 0 + j Δ x ) u ( i , j ) + ( y 0 + i Δ y ) v ( i , j ) ] 2 ,
s 0.092 - 0.0073 f # + 0.000087 τ f # + 0.00017 f # 2 ,
s 0.55 - 0.037 f # - 0.39 θ + 0.021 f # θ + 0.00063 f # 2 ,
e f # 711 ( 1 - δ ) 2 .
γ = [ ( θ - sin θ ) / π ] 2 ,
θ = 2 cos - 1 ( d r D ) ,
A ξ = a x - L 0 [ ɛ x x l s x + ɛ x y l s y + Ω z l s y - Ω y ( l s z + 1 ) ] , A ψ = a y - L 0 [ ɛ y y l s y + ɛ x y l s x + Ω z l s x - Ω x ( l s z + 1 ) ] ,

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