Abstract

Digital speckle correlation is based on a detailed analysis of changes in speckle images that are recorded from laser-illuminated rough surfaces. The two in-plane components are obtained by cross-correlation of corresponding subimages, a method also known as digital speckle photography. The local gradient of the hitherto inaccessible out-of-plane component is determined from the characteristic dependence of the speckle correlation on the spatial frequency. A detailed experimental study is carried out to analyze the new technique for systematic and random measuring errors. For moderate decorrelation the accuracy of the out-of-plane measurement is better than λ/10 and thus comparable with interferometric techniques. Yet the extremely simple and robust optical setup is suited to nondestructive-testing applications in harsh environments. The quality of the deformation maps is demonstrated in a practical application.

© 2003 Optical Society of America

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References

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  1. P. K. Rastogi, ed., Optical Measurement Techniques and Applications (Artech House, Norwood, Mass., 1997).
  2. R. S. Sirohi, F. S. Chau, Optical Methods of Measurement: Wholefield Techniques (Marcel Dekker, New York, 1999).
  3. P. K. Rastogi, D. Inaudi, eds., Trends in Optical Nondestructive Testing and Inspection (Elsevier, Amsterdam, 2000).
  4. P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, Chichester, UK, 2001).
  5. D. R. Matthys, J. A. Gilbert, P. Greguss, “Endoscopic measurement using radial metrology with digital correlation,” Opt. Eng. 30, 1455–1460 (1990).
    [CrossRef]
  6. S. Noh, I. Yamaguchi, “Two-dimensional measurement of strain distribution by speckle correlation,” Jpn. J. Appl. Phys. 31, L1299–L1301 (1992).
    [CrossRef]
  7. 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]
  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. E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
    [CrossRef]
  10. D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
    [CrossRef]
  11. B. Gombköto, J. Kornis, “Success rate and speckle correlation in electronic speckle photography,” Opt. Commun. 201, 289–292 (2002).
    [CrossRef]
  12. H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, London, 1978), pp. 73–109.
    [CrossRef]
  13. I. D. C. Tullis, N. A. Halliwell, S. J. Rothberg, “Spatially integrated speckle intensity: maximum resistance to decorrelation caused by in-plane target displacement,” Appl. Opt. 37, 7062–7069 (1998).
    [CrossRef]
  14. B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “A laser speckle sensor to measure the distribution of static torsion angles of twisted targets,” Meas. Sci. Technol. 9, 42–49 (1998).
    [CrossRef]
  15. P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers. Eng. 31, 425–443 (1999).
    [CrossRef]
  16. M. A. Sutton, S. R. McNeill, J. D. Helm, H. W. Schreier, “Computer vision applied to shape and deformation measurement,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 571–591.
  17. G. Schirripa Spagnolo, D. Ambrosini, D. Paoletti, “Image decorrelation for in situ diagnostics of wooden artifacts,” Appl. Opt. 36, 8358–8362 (1997).
    [CrossRef]
  18. L. Bruno, L. Pagnotta, A. Poggialini, “Laser speckle decorrelation in NDT,” in Proceedings of the International Conference on Trends in Optical Nondestructive Testing, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 187–195.
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).
  20. A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).
  21. M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
    [CrossRef]
  22. T. Fricke-Begemann, “Optical measurement of deformation fields and surface processes with digital speckle correlation,” Ph.D. dissertation (Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany, 2002).
  23. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  24. T. Fricke-Begemann, J. Burke, “Speckle interferometry: three-dimensional deformation field measurement with a single interferogram,” Appl. Opt. 40, 5011–5022 (2001).
    [CrossRef]
  25. J. W. Goodman, “Statistical properties of laser speckle patterns,” in Laser Speckle and Related Phenomena, J. C. Dainty, ed. (Springer-Verlag, Berlin, 1975), Chap. 2, pp. 9–75.
    [CrossRef]
  26. I. Yamaguchi, “Speckle displacement and decorrelation in the diffraction and image fields for small object deformation,” Opt. Acta 28, 1359–1376 (1981).
    [CrossRef]
  27. M. Sjödahl, “Calculation of speckle displacement, decorrelation, and object-point location in imaging systems,” Appl. Opt. 34, 7998–8010 (1995).
    [CrossRef] [PubMed]
  28. M. Owner-Petersen, “Decorrelation and fringe visibility: on the limiting behavior of various electronic speckle-pattern interferometers,” J. Opt. Soc. Am. A 8, 1082–1089 (1991).
    [CrossRef]
  29. L. Leushacke, M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (1990).
    [CrossRef]
  30. R. Feiel, P. Wilksch, “High-resolution laser speckle correlation for displacement and strain measurement,” Appl. Opt. 39, 54–60 (2000).
    [CrossRef]
  31. J. M. Huntley, “Speckle photography fringe analysis: assessment of current algorithms,” Appl. Opt. 28, 4316–4322 (1989).
    [CrossRef] [PubMed]
  32. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, Cambridge, England, 1988).
  33. M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
    [CrossRef] [PubMed]
  34. T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, London, 2000).
  35. T. Fricke-Begemann, K. D. Hinsch, “The measurement of random processes at rough surfaces with digital speckle correlation,” J. Opt. Soc. Am. A. (to be published).

2002 (1)

B. Gombköto, J. Kornis, “Success rate and speckle correlation in electronic speckle photography,” Opt. Commun. 201, 289–292 (2002).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers. Eng. 31, 425–443 (1999).
[CrossRef]

1998 (3)

M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
[CrossRef]

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “A laser speckle sensor to measure the distribution of static torsion angles of twisted targets,” Meas. Sci. Technol. 9, 42–49 (1998).
[CrossRef]

I. D. C. Tullis, N. A. Halliwell, S. J. Rothberg, “Spatially integrated speckle intensity: maximum resistance to decorrelation caused by in-plane target displacement,” Appl. Opt. 37, 7062–7069 (1998).
[CrossRef]

1997 (1)

1995 (1)

1994 (1)

1993 (2)

1992 (1)

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

1991 (1)

1990 (2)

L. Leushacke, M. Kirchner, “Three-dimensional correlation coefficient of speckle intensity for rectangular and circular apertures,” J. Opt. Soc. Am. A 7, 827–832 (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]

1989 (1)

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]

1976 (1)

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

1972 (1)

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Ambrosini, D.

Archbold, E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Benckert, L. R.

Bruno, L.

L. Bruno, L. Pagnotta, A. Poggialini, “Laser speckle decorrelation in NDT,” in Proceedings of the International Conference on Trends in Optical Nondestructive Testing, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 187–195.

Burke, J.

Chau, F. S.

R. S. Sirohi, F. S. Chau, Optical Methods of Measurement: Wholefield Techniques (Marcel Dekker, New York, 1999).

Chen, D. J.

Chiang, F. P.

Don, H. S.

Ennos, A. E.

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

Feiel, R.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, Cambridge, England, 1988).

Fricke-Begemann, T.

T. Fricke-Begemann, J. Burke, “Speckle interferometry: three-dimensional deformation field measurement with a single interferogram,” Appl. Opt. 40, 5011–5022 (2001).
[CrossRef]

T. Fricke-Begemann, “Optical measurement of deformation fields and surface processes with digital speckle correlation,” Ph.D. dissertation (Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany, 2002).

T. Fricke-Begemann, K. D. Hinsch, “The measurement of random processes at rough surfaces with digital speckle correlation,” J. Opt. Soc. Am. A. (to be published).

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]

Gombköto, B.

B. Gombköto, J. Kornis, “Success rate and speckle correlation in electronic speckle photography,” Opt. Commun. 201, 289–292 (2002).
[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), Chap. 2, pp. 9–75.
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

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

Gregory, D. A.

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

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]

Halliwell, N. A.

Hanson, S. G.

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “A laser speckle sensor to measure the distribution of static torsion angles of twisted targets,” Meas. Sci. Technol. 9, 42–49 (1998).
[CrossRef]

Hassibi, B.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, London, 2000).

Helm, J. D.

M. A. Sutton, S. R. McNeill, J. D. Helm, H. W. Schreier, “Computer vision applied to shape and deformation measurement,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 571–591.

Hinsch, K. D.

T. Fricke-Begemann, K. D. Hinsch, “The measurement of random processes at rough surfaces with digital speckle correlation,” J. Opt. Soc. Am. A. (to be published).

Huntley, J. M.

Imam, H.

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “A laser speckle sensor to measure the distribution of static torsion angles of twisted targets,” Meas. Sci. Technol. 9, 42–49 (1998).
[CrossRef]

Kailath, T.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, London, 2000).

Kirchner, M.

Kornis, J.

B. Gombköto, J. Kornis, “Success rate and speckle correlation in electronic speckle photography,” Opt. Commun. 201, 289–292 (2002).
[CrossRef]

Leushacke, L.

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.

M. A. Sutton, S. R. McNeill, J. D. Helm, H. W. Schreier, “Computer vision applied to shape and deformation measurement,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 571–591.

Noh, S.

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

Owner-Petersen, M.

Pagnotta, L.

L. Bruno, L. Pagnotta, A. Poggialini, “Laser speckle decorrelation in NDT,” in Proceedings of the International Conference on Trends in Optical Nondestructive Testing, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 187–195.

Paoletti, D.

Papoulis, A.

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

Poggialini, A.

L. Bruno, L. Pagnotta, A. Poggialini, “Laser speckle decorrelation in NDT,” in Proceedings of the International Conference on Trends in Optical Nondestructive Testing, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 187–195.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, Cambridge, England, 1988).

Rose, B.

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “A laser speckle sensor to measure the distribution of static torsion angles of twisted targets,” Meas. Sci. Technol. 9, 42–49 (1998).
[CrossRef]

Rothberg, S. J.

Sayed, A. H.

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, London, 2000).

Schirripa Spagnolo, G.

Schreier, H. W.

M. A. Sutton, S. R. McNeill, J. D. Helm, H. W. Schreier, “Computer vision applied to shape and deformation measurement,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 571–591.

Sirohi, R. S.

R. S. Sirohi, F. S. Chau, Optical Methods of Measurement: Wholefield Techniques (Marcel Dekker, New York, 1999).

Sjödahl, M.

Sutton, M. A.

M. A. Sutton, S. R. McNeill, J. D. Helm, H. W. Schreier, “Computer vision applied to shape and deformation measurement,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 571–591.

Synnergren, P.

P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers. Eng. 31, 425–443 (1999).
[CrossRef]

Tan, Y. S.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, Cambridge, England, 1988).

Tiziani, H. J.

H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, London, 1978), pp. 73–109.
[CrossRef]

Tullis, I. D. C.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, Cambridge, England, 1988).

Wilksch, P.

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]

Yura, H. T.

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “A laser speckle sensor to measure the distribution of static torsion angles of twisted targets,” Meas. Sci. Technol. 9, 42–49 (1998).
[CrossRef]

Appl. Opt. (9)

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

M. Sjödahl, “Electronic speckle photography: increased accuracy by nonintegral pixel shifting,” Appl. Opt. 33, 6667–6673 (1994).
[CrossRef] [PubMed]

G. Schirripa Spagnolo, D. Ambrosini, D. Paoletti, “Image decorrelation for in situ diagnostics of wooden artifacts,” Appl. Opt. 36, 8358–8362 (1997).
[CrossRef]

I. D. C. Tullis, N. A. Halliwell, S. J. Rothberg, “Spatially integrated speckle intensity: maximum resistance to decorrelation caused by in-plane target displacement,” Appl. Opt. 37, 7062–7069 (1998).
[CrossRef]

M. Sjödahl, “Calculation of speckle displacement, decorrelation, and object-point location in imaging systems,” Appl. Opt. 34, 7998–8010 (1995).
[CrossRef] [PubMed]

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]

R. Feiel, P. Wilksch, “High-resolution laser speckle correlation for displacement and strain measurement,” Appl. Opt. 39, 54–60 (2000).
[CrossRef]

T. Fricke-Begemann, J. Burke, “Speckle interferometry: three-dimensional deformation field measurement with a single interferogram,” Appl. Opt. 40, 5011–5022 (2001).
[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]

Meas. Sci. Technol. (1)

B. Rose, H. Imam, S. G. Hanson, H. T. Yura, “A laser speckle sensor to measure the distribution of static torsion angles of twisted targets,” Meas. Sci. Technol. 9, 42–49 (1998).
[CrossRef]

Opt. Acta (2)

E. Archbold, A. E. Ennos, “Displacement measurement from double-exposure laser photographs,” Opt. Acta 19, 253–271 (1972).
[CrossRef]

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

Opt. Commun. (1)

B. Gombköto, J. Kornis, “Success rate and speckle correlation in electronic speckle photography,” Opt. Commun. 201, 289–292 (2002).
[CrossRef]

Opt. Eng. (1)

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

Opt. Laser Technol. (1)

D. A. Gregory, “Basic physical principles of defocused speckle photography: a tilt topology inspection technique,” Opt. Laser Technol. 8, 201–213 (1976).
[CrossRef]

Opt. Lasers Eng. (1)

M. Sjödahl, “Some recent advances in electronic speckle photography,” Opt. Lasers Eng. 29, 125–144 (1998).
[CrossRef]

Opt. Lasers. Eng. (1)

P. Synnergren, M. Sjödahl, “A stereoscopic digital speckle photography system for 3-D displacement field measurements,” Opt. Lasers. Eng. 31, 425–443 (1999).
[CrossRef]

Other (15)

M. A. Sutton, S. R. McNeill, J. D. Helm, H. W. Schreier, “Computer vision applied to shape and deformation measurement,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 571–591.

L. Bruno, L. Pagnotta, A. Poggialini, “Laser speckle decorrelation in NDT,” in Proceedings of the International Conference on Trends in Optical Nondestructive Testing, P. K. Rastogi, D. Inaudi, eds. (Elsevier, Amsterdam, 2000), pp. 187–195.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, San Francisco, Calif., 1968).

A. Papoulis, Probability, Random Variables, and Stochastic Processes, 3rd ed. (McGraw-Hill, New York, 1991).

T. Fricke-Begemann, “Optical measurement of deformation fields and surface processes with digital speckle correlation,” Ph.D. dissertation (Carl von Ossietzky Universität Oldenburg, Oldenburg, Germany, 2002).

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), Chap. 2, pp. 9–75.
[CrossRef]

H. J. Tiziani, “Vibration analysis and deformation measurement,” in Speckle Metrology, R. K. Erf, ed. (Academic, London, 1978), pp. 73–109.
[CrossRef]

P. K. Rastogi, ed., Optical Measurement Techniques and Applications (Artech House, Norwood, Mass., 1997).

R. S. Sirohi, F. S. Chau, Optical Methods of Measurement: Wholefield Techniques (Marcel Dekker, New York, 1999).

P. K. Rastogi, D. Inaudi, eds., Trends in Optical Nondestructive Testing and Inspection (Elsevier, Amsterdam, 2000).

P. K. Rastogi, ed., Digital Speckle Pattern Interferometry and Related Techniques (Wiley, Chichester, UK, 2001).

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in C (Cambridge University, Cambridge, England, 1988).

T. Kailath, A. H. Sayed, B. Hassibi, Linear Estimation (Prentice-Hall, London, 2000).

T. Fricke-Begemann, K. D. Hinsch, “The measurement of random processes at rough surfaces with digital speckle correlation,” J. Opt. Soc. Am. A. (to be published).

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Figures (11)

Fig. 1
Fig. 1

Experimental setup for digital speckle correlation and illustration of object tilt.

Fig. 2
Fig. 2

Geometric relations for a small object tilt γ x about the y′ axis.

Fig. 3
Fig. 3

Normalized cross power spectrum for (a) surface microstructure changes and (b) object tilt: white, complete correlation; black, zero correlation. The tilt angle in (b) is ∼2 mrad and generates 50 fringes over an image size of 1024 pixels in an interferometric measurement.

Fig. 4
Fig. 4

Calculation of the cross PSD for a circular aperture (see text).

Fig. 5
Fig. 5

Out-of-plane displacement error and bias versus tilt angle for determination of the tilt parameter Δ ν from the width of the correlation peak.

Fig. 6
Fig. 6

Out-of-plane displacement error and bias versus tilt angle for determination of the tilt parameter Δ ν from a maximum likelihood estimation.

Fig. 7
Fig. 7

Out-of-plane displacement error versus correlation coefficient for a circular aperture. The random errors from Figs. 5 and 6 (symbols) are now drawn versus cc ̃ and compared with Eq. (34) (curves).

Fig. 8
Fig. 8

Out-of-plane displacement error versus total correlation coefficient for increasing tilt and with fixed primary decorrelation to cc μ ≈ 0.89. The experimental data (symbols) are compared with the expected error from Eq. (34) (curves).

Fig. 9
Fig. 9

Out-of-plane displacement error versus aperture size. The experimental data (symbols) are compared with the expected error from Eq. (34) (curves).

Fig. 10
Fig. 10

Out-of-plane deformation measurement with DSC: (a) differential deformation field Δz; (b) integration by a Gaussian filter; (c) the isolines of the integral deformation superimposed with the phase map of a simultaneous interferometric measurement.

Fig. 11
Fig. 11

Three-dimensional deformation measurement with DSC: left, demonstration object with an observed region measuring 14 mm × 10 mm indicated by the dashed rectangle; center, out-of-plane component of the deformation in a surface plot; right, out-of-plane component coded in a gray scale (isolines distance, 0.1λ) with superimposed arrows indicating in-plane displacement.

Equations (36)

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

ar=ur * pr,
ir=ir * hr+nr,
ur=u0r+Δur, ũr=u0r-Δr+Δũrexp-2πiΔνr.
Δν=1+cos θ/λMγ.
Δzx=BM γx=λ1+cos θ BΔνx=λ1+cos θN2ΔνxνNy.
ciı˜s=irı˜r+s-iı˜/σiσı˜,
cciı˜=|a*ã|2/σa2σã2=|ccaã|2,
a*rãr+s=F-1A*Ã,
Aν=U0ν+ΔUνPν, Ãν=U0νexp-2πiνΔr+ΔŨνPν+Δν,
a*rãr+s=σu02 P*νPν+Δν×exp2πiνs-Δrd2ν,
cciı˜=ccμccγ=σu04σu02+σΔu22 P*νPν+Δνd2ν |Pν|2d2ν2.
ccγ=4π-2arccos|Δν|-|Δν|1-|Δν|21/22,
Iν=ACFAνHν+Nν,
I*Ĩ=ACF*AACFÃ|H|2.
ACF*AACFà=- UνU*ν+νŨ*νŨν+ν×PνP*ν+νP*ν+Δν×Pν+ν+Δνd2νd2ν,
UνU*ν+νŨ*νŨν+ν=σu04 exp-2πiνΔrδν-ν.
I*νĨν=σu04 exp-2πiνΔr×ACFP*νPν+Δν|Hν|2.
ccĨ=σu04 exp-2πiνΔrACFP*νPν+Δν|Hν[|2σu02+σΔu22ACF|Pν|2|Hν|2+|Nν|2 =ccμ exp-2πiνΔrccTν; ΔνccNν,
ccTν; Δν=ACFP*νPν+ΔνACF|Pν|2.
ccNν=1+|Nν|2/σu4ACF|Pν|2|Hν|2-1.
ACFPνPν+Δν=DνxDνy1-|Δνx|Dνx-|νx|Dνx×1-|Δνy|Dνy-|νy|Dνy
ccTν; Δν=1-|Δνx|/Dνx-|νx|×1-|Δνy|/Dνy-|νy|
iı˜  sinc2Dνx-|Δνx|xsinc2Dνy-|Δνy|y,
|Δνx|=Dνx1-xFWHM0/xFWHM
νR2=|ν|2+|Δν|2+2|νxΔνx+νyΔνy|.
ACFPνPν+Δν=Dν22arccosνR-νR1-νR21/2if Q > 0 (I)arccos|Δν|-|Δν|1-|Δν|21/2-arcsin|ν|if Q  0 II-|ν|1-|ν|21/2+2|νxΔνy-νyΔνx|
iı˜  J1πDν|r|/πDν|r|2,
Δνcc=Dν1-rFWHM0/rFWHM.
Δνcc/|Δν|=1+0.0441-|Δν|+0.0221-|Δν|2.
bβ=a0+a1 cos2β-βmax+a2 cos4β-βmax.
|Δν|=Dν1-bmax-11+0.044bmax-1+0.022bmax-2.
pI, Ĩ=1π2|I|221-|ccĨI|2×exp-|I|2+|Ĩ|2-2 ReccĨI* I*Ĩ|I|21-|ccĨI |2|.
L=j,k pIj, k, Ĩj, k,
 ln LΔνx= ln LΔνy=0
σΔz=k+1-ccμ10ccμ2gpol1+cos θ1-ccγ3/4ccγ+0.4 1-ccμ3/4ccμ,
Z=DTD-1DTΔz.

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