Abstract

Replacing photographic recording by electronic processing has some obvious advantages. An algorithm used for electronic speckle pattern photography is presented, and the reliability and accuracy is analyzed by using computer-generated speckle patterns. The algorithm is based on a two-dimensional discrete cross correlation between subimages from different images. Subpixel accuracy is obtained by a Fourier series expansion of the discrete correlation surface. The accuracy of the algorithm was found to vary in proportion to σ/n(1 − δ)2, where σ is the speckle size, n is the subimage size, and δ is the amount of decorrelation, with negligible systematic errors. For typical values the uncertainty in the displacement is approximately 0.05 pixels. The uncertainty is found to increase with increased displacement gradients.

© 1993 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. G. H. Kaufmann, A. E. Ennos, B. Gale, D. J. Pugh, “An electro-optical read-out system for analysis of speckle photographs,” J. Phys. E 13, 579–584 (1980).
    [CrossRef]
  4. J. M. Huntley, “An image processing system for the analysis of speckle photographs,” J. Phys. E 19, 43–48 (1986).
    [CrossRef]
  5. J. M. Huntley, “Speckle photography fringe analysis: assessment of current algorithms,” Appl. Opt. 28, 4316–4322 (1989).
    [CrossRef] [PubMed]
  6. 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]
  7. 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]
  8. 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]
  9. D. J. Chen, F. P. Chiang, “Computer speckle interferometry (CSI),” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 49–58.
  10. D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Computer-aided speckle interferometry (CASI). Part II. An alternate approach using spectral amplitude and phase information,” in Second International Conference on Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1554A, 706–717 (1991).
  11. H. H. Bailey, F. W. Blackwell, C. L. Lowery, J. A. Ratkovic, “Image correlation. Part I. Simulation and analysis,” Rand Corporation Rep. R-2057/1-PR (Rand Corporation, Santa Monica, Calif., 1976).
  12. R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).
  13. V. Cizek, Discrete Fourier Transforms and Their Applications (Hilger, London, 1986).
  14. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in fortran, (Cambridge U. Press, Cambridge, UK, 1986).

1989

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]

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

1986

J. M. Huntley, “An image processing system for the analysis of speckle photographs,” J. Phys. E 19, 43–48 (1986).
[CrossRef]

1985

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

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]

1980

G. H. Kaufmann, A. E. Ennos, B. Gale, D. J. Pugh, “An electro-optical read-out system for analysis of speckle photographs,” J. Phys. E 13, 579–584 (1980).
[CrossRef]

Bailey, H. H.

H. H. Bailey, F. W. Blackwell, C. L. Lowery, J. A. Ratkovic, “Image correlation. Part I. Simulation and analysis,” Rand Corporation Rep. R-2057/1-PR (Rand Corporation, Santa Monica, Calif., 1976).

Blackwell, F. W.

H. H. Bailey, F. W. Blackwell, C. L. Lowery, J. A. Ratkovic, “Image correlation. Part I. Simulation and analysis,” Rand Corporation Rep. R-2057/1-PR (Rand Corporation, Santa Monica, Calif., 1976).

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]

Chen, D. J.

D. J. Chen, F. P. Chiang, “Computer speckle interferometry (CSI),” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 49–58.

D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Computer-aided speckle interferometry (CASI). Part II. An alternate approach using spectral amplitude and phase information,” in Second International Conference on Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1554A, 706–717 (1991).

Chiang, F. P.

D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Computer-aided speckle interferometry (CASI). Part II. An alternate approach using spectral amplitude and phase information,” in Second International Conference on Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1554A, 706–717 (1991).

D. J. Chen, F. P. Chiang, “Computer speckle interferometry (CSI),” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 49–58.

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]

Cizek, V.

V. Cizek, Discrete Fourier Transforms and Their Applications (Hilger, London, 1986).

Don, H. S.

D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Computer-aided speckle interferometry (CASI). Part II. An alternate approach using spectral amplitude and phase information,” in Second International Conference on Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1554A, 706–717 (1991).

Ennos, A. E.

G. H. Kaufmann, A. E. Ennos, B. Gale, D. J. Pugh, “An electro-optical read-out system for analysis of speckle photographs,” J. Phys. E 13, 579–584 (1980).
[CrossRef]

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

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in fortran, (Cambridge U. Press, Cambridge, UK, 1986).

Gale, B.

G. H. Kaufmann, A. E. Ennos, B. Gale, D. J. Pugh, “An electro-optical read-out system for analysis of speckle photographs,” J. Phys. E 13, 579–584 (1980).
[CrossRef]

Gonzalez, R. C.

R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).

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]

Huntley, J. M.

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

J. M. Huntley, “An image processing system for the analysis of speckle photographs,” J. Phys. E 19, 43–48 (1986).
[CrossRef]

Kaufmann, G. H.

G. H. Kaufmann, A. E. Ennos, B. Gale, D. J. Pugh, “An electro-optical read-out system for analysis of speckle photographs,” J. Phys. E 13, 579–584 (1980).
[CrossRef]

Lowery, C. L.

H. H. Bailey, F. W. Blackwell, C. L. Lowery, J. A. Ratkovic, “Image correlation. Part I. Simulation and analysis,” Rand Corporation Rep. R-2057/1-PR (Rand Corporation, Santa Monica, Calif., 1976).

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]

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]

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]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in fortran, (Cambridge U. Press, Cambridge, UK, 1986).

Pugh, D. J.

G. H. Kaufmann, A. E. Ennos, B. Gale, D. J. Pugh, “An electro-optical read-out system for analysis of speckle photographs,” J. Phys. E 13, 579–584 (1980).
[CrossRef]

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]

Ratkovic, J. A.

H. H. Bailey, F. W. Blackwell, C. L. Lowery, J. A. Ratkovic, “Image correlation. Part I. Simulation and analysis,” Rand Corporation Rep. R-2057/1-PR (Rand Corporation, Santa Monica, Calif., 1976).

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]

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]

Tan, Y. S.

D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Computer-aided speckle interferometry (CASI). Part II. An alternate approach using spectral amplitude and phase information,” in Second International Conference on Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1554A, 706–717 (1991).

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in fortran, (Cambridge U. Press, Cambridge, UK, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in fortran, (Cambridge U. Press, Cambridge, UK, 1986).

Wintz, P.

R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).

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]

Appl. Opt.

Exp. Mech.

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–Raphson method of partial differential correction,” Exp. Mech. 29, 261–267 (1989).
[CrossRef]

Image Vision Comput.

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. Phys. E

G. H. Kaufmann, A. E. Ennos, B. Gale, D. J. Pugh, “An electro-optical read-out system for analysis of speckle photographs,” J. Phys. E 13, 579–584 (1980).
[CrossRef]

J. M. Huntley, “An image processing system for the analysis of speckle photographs,” J. Phys. E 19, 43–48 (1986).
[CrossRef]

Other

D. J. Chen, F. P. Chiang, “Computer speckle interferometry (CSI),” in Proceedings of the International Conference on Hologram Interferometry and Speckle Metrology, (Society for Experimental Mechanics, Bethel, Conn., 1990), pp. 49–58.

D. J. Chen, F. P. Chiang, Y. S. Tan, H. S. Don, “Computer-aided speckle interferometry (CASI). Part II. An alternate approach using spectral amplitude and phase information,” in Second International Conference on Photomechanics and Speckle Metrology, F. Chiang, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1554A, 706–717 (1991).

H. H. Bailey, F. W. Blackwell, C. L. Lowery, J. A. Ratkovic, “Image correlation. Part I. Simulation and analysis,” Rand Corporation Rep. R-2057/1-PR (Rand Corporation, Santa Monica, Calif., 1976).

R. C. Gonzalez, P. Wintz, Digital Image Processing, 2nd ed. (Addison-Wesley, Reading, Mass., 1987).

V. Cizek, Discrete Fourier Transforms and Their Applications (Hilger, London, 1986).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in fortran, (Cambridge U. Press, Cambridge, UK, 1986).

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]

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Figures (5)

Fig. 1
Fig. 1

Computer-generated 128 × 128 pixel speckle pattern as seen by a video camera.

Fig. 2
Fig. 2

Contour map of a discrete correlation, using Eq. (3), between two subimages. The true displacement between the two subimages was kx = ky = 7.5 pixels. The first subimage has been shifted in order to obtain maximum correlation.

Fig. 3
Fig. 3

Correlation intensity versus displacement in pixels; estimations of the continuous correlation function, along one row close to the peak position, given by Eq. (6) (the continuous curve), cubic splines (dashed curve), and a least-squares parabolic fit of the five largest values (dash–dot curve). The true displacement between the two subimages was kx = ky = 7.5 pixels.

Fig. 4
Fig. 4

Relative orientation of planes 1 and 2 that was used for simulating speckle patterns.

Fig. 5
Fig. 5

Probability density function P(I) of the registered intensity in plane 2 versus intensity over mean intensity. Results from one 256 × 256 pixel computer-generated speckle pattern. The continuous curve is the expected negative exponential curve for an ideal detector.

Tables (2)

Tables Icon

Table 1 Parameters Used to Generate Twelve Datasets of Speckle Patterns

Tables Icon

Table 2 Results of Analysis of Twelve Datasets

Equations (16)

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c ( k , l ) = i = 0 m 1 j = 0 m 1 [ h s 2 * ( i , j ) h s 1 ( k + i , l + j ) ] , k , l = 0 , 1 , , m 1 ,
H s 1 ( r , s ) = ( h s 1 ) = 1 m 2 k = 0 m 1 l = 0 m 1 { h s 1 ( k , l ) exp [ 2 π i ( r k + s l ) / m ] } , H s 2 ( r , s ) = ( h s 2 ) = 1 m 2 k = 0 m 1 l = 0 m 1 { h s 2 ( k , l ) exp [ 2 π i ( r k + s l ) / m ] } , r , s = 0 , 1 , , m 1
c ( k , l ) = 1 ( H s 2 * H s 1 ) ,
u ( x , y ) = r = M M s = M M { C ( r , s ) exp [ 2 π i ( r x + s y ) / P ] } ,
C ( r , s ) = 1 P 2 k = 0 P 1 l = 0 P 1 { c ( k , l ) exp [ 2 π i ( r k + s l ) / P ] } , r , s = 0 , ± 1 , ± 2 , , ± M ,
u ( x , y ) = 1 P 2 k = 0 P 1 l = 0 P 1 c ( k , l ) × sin [ π ( x k ) ] sin [ π ( y l ) ] sin [ π ( x k ) / P ] sin [ π ( y l ) / P ] .
I 2 ( m 2 , n 2 ) = | 1 { W 1 ( m 1 , n 1 ) [ A 2 ( m 2 , n 2 ) ] } | 2 ,
W 1 ( m 1 , n 1 ) = 1 , m 1 2 + n 1 2 ( D / 2 ) 2 , = 0 m 1 2 + n 1 2 > ( D / 2 ) 2 , m 1 , n 1 = N / 2 , N / 2 + 1 , , N / 2 2 , N / 2 1 .
A 1 d ( m 1 , n 1 ) = A 1 ( m 1 , n 1 ) exp [ 2 π i ( k m m 1 + k n n 1 ) / N ] .
I 2 d ( m 2 , n 2 ) = | ( 1 δ ) A 2 d ( m 2 , n 2 ) + δ A 2 i ( m 2 , n 2 ) | 2 .
I c ( m 2 , n 2 ) = round { α [ I 2 ( m 2 , n 2 ) I t ] / ( I max I t ) } , I 2 ( m 2 , n 2 ) I t , = 0 , I 2 ( m 2 , n 2 ) < I t ,
e 0.66 σ / n ( 1 δ ) 2 .
k x ¯
k y ¯
Δ k x ¯
k y ¯

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