Abstract

A temporal wavelet analysis method is proposed for velocity, displacement, and three-dimensional surface-profile measurement of a continuously deforming object by use of the shadow moiré technique. A grating is placed close to a deforming object, and its shadow is observed through the grating. The moiré fringe patterns, generated by the interference of the grating lines and their shadows, are captured by a high-speed CCD camera with a telecentric gauging lens. Instantaneous frequency of gray-value variation is evaluated point by point with the continuous wavelet transform. From the instantaneous frequency of each point on the object, the velocity, displacement, and high-quality surface profile at different instants can be retrieved. In this application, two specimens are tested to demonstrate the validity of the proposed method: One is a small coin with a rigid body motion, and the other is a simply supported beam subjected to a central point load. The results are compared with those obtained from temporal Fourier-transform and mechanical stylus methods.

© 2004 Optical Society of America

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  9. T. Yoshizawa, T. Tomisawa, “Shadow moiré topography by means of the phase-shift method,” Opt. Eng. 32, 1668–1674 (1993).
    [CrossRef]
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    [CrossRef]
  11. J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surface,” Opt. Lasers Eng. 36, 29–40 (2001).
    [CrossRef]
  12. R. Henan, A. Tagliaferri, R. Torroba, “A contouring approach using single grating digital shadow moiré with a phase stepping technique,” Optik 110, 199–201 (1999).
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    [CrossRef] [PubMed]
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  20. C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
    [CrossRef]
  21. C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
    [CrossRef]
  22. C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
    [CrossRef]
  23. H. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
    [CrossRef]
  24. M. Takeda, H. Yamamoto, “Fourier-transform speckle profilometry: three-dimensional shape measurements of diffuse objects with large height steps and/or spatially isolated surfaces,” Appl. Opt. 33, 7829–7837 (1994).
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  25. L. H. Jin, Y. Otani, T. Yoshizawa, “Shadow moiré profilometry by frequency sweeping,” Opt. Eng. 40, 1383–1386 (2001).
    [CrossRef]
  26. C. Quan, Y. Fu, C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230, 23–33 (2004).
    [CrossRef]
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    [CrossRef]
  29. L. R. Watkins, S. M. Tan, T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
    [CrossRef]
  30. A. Federico, G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
    [CrossRef]
  31. J. Fang, C. Y. Xiong, Z. L. Yang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
    [CrossRef]
  32. Y. Morimoto, M. Fujigaki, S. Yoneyama, “Shape, stress, and strain measurement using phase analysis of grating or fringe patterns,” in Third International Conference on Experimental Mechanics, X. Wu, Y. Qin, J. Fang, J. Ke, eds., Proc. SPIE4537, 47–52 (2002).
    [CrossRef]
  33. K. Qian, H. S. Seah, A. Asundi, “Instantaneous frequency and its application to strain extraction in moiré interferometry,” Appl. Opt. 42, 6504–6513 (2003).
    [CrossRef]
  34. K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
    [CrossRef]
  35. X. Colonna de Lega, “Continuous deformation measurement using dynamic phase-shifting and wavelet transform,” in Applied Optics and Optoelectronics 1996, K. T. V. Grattan, ed. (Institute of Physics, Bristol, UK, 1996), pp. 261–267.
  36. M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
  37. S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

2004 (1)

C. Quan, Y. Fu, C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230, 23–33 (2004).
[CrossRef]

2003 (2)

K. Qian, H. S. Seah, A. Asundi, “Instantaneous frequency and its application to strain extraction in moiré interferometry,” Appl. Opt. 42, 6504–6513 (2003).
[CrossRef]

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

2002 (1)

A. Federico, G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
[CrossRef]

2001 (3)

J. Fang, C. Y. Xiong, Z. L. Yang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

L. H. Jin, Y. Otani, T. Yoshizawa, “Shadow moiré profilometry by frequency sweeping,” Opt. Eng. 40, 1383–1386 (2001).
[CrossRef]

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surface,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

2000 (1)

L. Jin, Y. Kodera, T. Yoshizawa, Y. Otani, “Shadow moiré profilometry using the phase-shifting method,” Opt. Eng. 39, 2119–2123 (2000).
[CrossRef]

1999 (2)

L. R. Watkins, S. M. Tan, T. H. Barnes, “Determination of interferometer phase distributions by use of wavelets,” Opt. Lett. 24, 905–907 (1999).
[CrossRef]

R. Henan, A. Tagliaferri, R. Torroba, “A contouring approach using single grating digital shadow moiré with a phase stepping technique,” Optik 110, 199–201 (1999).

1998 (3)

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

1997 (2)

H. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

X. Xie, M. J. Lalor, D. R. Burton, M. M. Shaw, “Four-map absolute distance contouring,” Opt. Eng. 36, 2517–1520 (1997).
[CrossRef]

1995 (1)

1994 (2)

1993 (2)

T. Yoshizawa, T. Tomisawa, “Shadow moiré topography by means of the phase-shift method,” Opt. Eng. 32, 1668–1674 (1993).
[CrossRef]

J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

1982 (2)

1977 (1)

1973 (1)

B. Dessus, M. Leblanc, “The ‘fringe method’ and its application to the measurement of deformations, vibrations, contour lines and differences of objects,” Opto-electronics 5, 369–391 (1973).
[CrossRef]

1971 (2)

1970 (3)

1969 (1)

T. Tsuruta, Y. Itoh, “Interferometric generation of counter lines on opaque objects,” Opt. Commun. 1, 34–36 (1969).
[CrossRef]

Allen, J. B.

Arai, Y.

Asundi, A.

Barnes, T. H.

Boone, P.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surface,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Bremand, F.

Burton, D. R.

X. Xie, M. J. Lalor, D. R. Burton, M. M. Shaw, “Four-map absolute distance contouring,” Opt. Eng. 36, 2517–1520 (1997).
[CrossRef]

Cherbuliez, M.

M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).

Colonna de Lega, X.

M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).

X. Colonna de Lega, “Continuous deformation measurement using dynamic phase-shifting and wavelet transform,” in Applied Optics and Optoelectronics 1996, K. T. V. Grattan, ed. (Institute of Physics, Bristol, UK, 1996), pp. 261–267.

Daubechies, I.

I. Daubechies, Ten Lectures on Wavelets (Society for Industrial and Applied Mathematics, Philadelphia, Pa., 1992).
[CrossRef]

Degrieck, J.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surface,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Dessus, B.

B. Dessus, M. Leblanc, “The ‘fringe method’ and its application to the measurement of deformations, vibrations, contour lines and differences of objects,” Opto-electronics 5, 369–391 (1973).
[CrossRef]

Durelli, A. J.

Fang, J.

J. Fang, C. Y. Xiong, Z. L. Yang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

Federico, A.

A. Federico, G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
[CrossRef]

Franze, B.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

H. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Fu, Y.

C. Quan, Y. Fu, C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230, 23–33 (2004).
[CrossRef]

Fujigaki, M.

Y. Morimoto, M. Fujigaki, S. Yoneyama, “Shape, stress, and strain measurement using phase analysis of grating or fringe patterns,” in Third International Conference on Experimental Mechanics, X. Wu, Y. Qin, J. Fang, J. Ke, eds., Proc. SPIE4537, 47–52 (2002).
[CrossRef]

Fujimoto, J.

Haible, P.

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

H. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Henan, R.

R. Henan, A. Tagliaferri, R. Torroba, “A contouring approach using single grating digital shadow moiré with a phase stepping technique,” Optik 110, 199–201 (1999).

Hovanesian, J. D.

Hung, Y. Y.

Huntley, J. M.

J. M. Huntley, H. Saldner, “Temporal phase-unwrapping algorithm for automated interferogram analysis,” Appl. Opt. 32, 3047–3052 (1993).
[CrossRef] [PubMed]

J. M. Huntley, “Challenges in phase unwrapping,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier Science, Oxford, UK, 2000), pp. 37–44.

Ina, H.

Itoh, Y.

T. Tsuruta, Y. Itoh, “Interferometric generation of counter lines on opaque objects,” Opt. Commun. 1, 34–36 (1969).
[CrossRef]

Jacquot, P.

M. Cherbuliez, P. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).

Jin, L.

L. Jin, Y. Kodera, T. Yoshizawa, Y. Otani, “Shadow moiré profilometry using the phase-shifting method,” Opt. Eng. 39, 2119–2123 (2000).
[CrossRef]

Jin, L. H.

L. H. Jin, Y. Otani, T. Yoshizawa, “Shadow moiré profilometry by frequency sweeping,” Opt. Eng. 40, 1383–1386 (2001).
[CrossRef]

Joenathan, C.

Johnson, W. O.

Jonathan, C.

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

Kadooka, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Kaufmann, G. H.

A. Federico, G. H. Kaufmann, “Evaluation of the continuous wavelet transform method for the phase measurement of electronic speckle pattern interferometry fringes,” Opt. Eng. 41, 3209–3216 (2002).
[CrossRef]

Kobayashi, S.

Kodera, Y.

L. Jin, Y. Kodera, T. Yoshizawa, Y. Otani, “Shadow moiré profilometry using the phase-shifting method,” Opt. Eng. 39, 2119–2123 (2000).
[CrossRef]

Kunoo, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Lagarde, A.

Lalor, M. J.

X. Xie, M. J. Lalor, D. R. Burton, M. M. Shaw, “Four-map absolute distance contouring,” Opt. Eng. 36, 2517–1520 (1997).
[CrossRef]

Leblanc, M.

B. Dessus, M. Leblanc, “The ‘fringe method’ and its application to the measurement of deformations, vibrations, contour lines and differences of objects,” Opto-electronics 5, 369–391 (1973).
[CrossRef]

Liang, C. Y.

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing (Academic, San Diego, Calif., 1998).

Mauvoisin, G.

Meadows, D. M.

Morimoto, Y.

Y. Morimoto, M. Fujigaki, S. Yoneyama, “Shape, stress, and strain measurement using phase analysis of grating or fringe patterns,” in Third International Conference on Experimental Mechanics, X. Wu, Y. Qin, J. Fang, J. Ke, eds., Proc. SPIE4537, 47–52 (2002).
[CrossRef]

Nagayasu, T.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Ono, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Otani, Y.

L. H. Jin, Y. Otani, T. Yoshizawa, “Shadow moiré profilometry by frequency sweeping,” Opt. Eng. 40, 1383–1386 (2001).
[CrossRef]

L. Jin, Y. Kodera, T. Yoshizawa, Y. Otani, “Shadow moiré profilometry using the phase-shifting method,” Opt. Eng. 39, 2119–2123 (2000).
[CrossRef]

Qian, K.

Quan, C.

C. Quan, Y. Fu, C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230, 23–33 (2004).
[CrossRef]

Saldner, H.

Seah, H. S.

Shaw, M. M.

X. Xie, M. J. Lalor, D. R. Burton, M. M. Shaw, “Four-map absolute distance contouring,” Opt. Eng. 36, 2517–1520 (1997).
[CrossRef]

Tagliaferri, A.

R. Henan, A. Tagliaferri, R. Torroba, “A contouring approach using single grating digital shadow moiré with a phase stepping technique,” Optik 110, 199–201 (1999).

Takasaki, H.

Takeda, M.

Tan, S. M.

Tay, C. J.

C. Quan, Y. Fu, C. J. Tay, “Determination of surface contour by temporal analysis of shadow moiré fringes,” Opt. Commun. 230, 23–33 (2004).
[CrossRef]

Tiziani, H.

H. Tiziani, B. Franze, P. Haible, “Wavelength-shift speckle interferometry for absolute profilometry using a mode-hop free external cavity diode laser,” J. Mod. Opt. 44, 1485–1496 (1997).
[CrossRef]

Tiziani, H. J.

C. Joenathan, B. Franze, P. Haible, H. J. Tiziani, “Speckle interferometry with temporal phase evaluation for measuring large-object deformation,” Appl. Opt. 37, 2608–2614 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Large in-plane displacement measurement in dual-beam speckle interferometry using temporal phase measurement,” J. Mod. Opt. 45, 1975–1984 (1998).
[CrossRef]

C. Jonathan, B. Franze, P. Haible, H. J. Tiziani, “Novel temporal Fourier transform speckle pattern shearing interferometer,” Opt. Eng. 37, 1790–1795 (1998).
[CrossRef]

H. J. Tiziani, “Spectral and temporal phase evaluation for interferometry and speckle applications,” in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi, D. Inaudi, eds. (Elsevier Science, Oxford, UK, 2000), pp. 323–343.

Tomisawa, T.

T. Yoshizawa, T. Tomisawa, “Shadow moiré topography by means of the phase-shift method,” Opt. Eng. 32, 1668–1674 (1993).
[CrossRef]

Torroba, R.

R. Henan, A. Tagliaferri, R. Torroba, “A contouring approach using single grating digital shadow moiré with a phase stepping technique,” Optik 110, 199–201 (1999).

Tsuruta, T.

T. Tsuruta, Y. Itoh, “Interferometric generation of counter lines on opaque objects,” Opt. Commun. 1, 34–36 (1969).
[CrossRef]

Uda, N.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45–51 (2003).
[CrossRef]

Van Paepegem, W.

J. Degrieck, W. Van Paepegem, P. Boone, “Application of digital phase-shift shadow moiré to micro deformation measurements of curved surface,” Opt. Lasers Eng. 36, 29–40 (2001).
[CrossRef]

Wasowski, J.

J. Wasowski, “Moiré topographic maps,” Opt. Commun. 2, 321–323 (1970).
[CrossRef]

Watkins, L. R.

Xie, X.

X. Xie, M. J. Lalor, D. R. Burton, M. M. Shaw, “Four-map absolute distance contouring,” Opt. Eng. 36, 2517–1520 (1997).
[CrossRef]

Xiong, C. Y.

J. Fang, C. Y. Xiong, Z. L. Yang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

Yamada, T.

Yamamoto, H.

Yang, Z. L.

J. Fang, C. Y. Xiong, Z. L. Yang, “Digital transform processing of carrier fringe patterns from speckle-shearing interferometry,” J. Mod. Opt. 48, 507–520 (2001).
[CrossRef]

Yokozeki, S.

Yoneyama, S.

Y. Morimoto, M. Fujigaki, S. Yoneyama, “Shape, stress, and strain measurement using phase analysis of grating or fringe patterns,” in Third International Conference on Experimental Mechanics, X. Wu, Y. Qin, J. Fang, J. Ke, eds., Proc. SPIE4537, 47–52 (2002).
[CrossRef]

Yoshizawa, T.

L. H. Jin, Y. Otani, T. Yoshizawa, “Shadow moiré profilometry by frequency sweeping,” Opt. Eng. 40, 1383–1386 (2001).
[CrossRef]

L. Jin, Y. Kodera, T. Yoshizawa, Y. Otani, “Shadow moiré profilometry using the phase-shifting method,” Opt. Eng. 39, 2119–2123 (2000).
[CrossRef]

T. Yoshizawa, T. Tomisawa, “Shadow moiré topography by means of the phase-shift method,” Opt. Eng. 32, 1668–1674 (1993).
[CrossRef]

Appl. Opt. (12)

J. B. Allen, D. M. Meadows, “Removal of unwanted patterns from moiré contour maps by grid translation techniques,” Appl. Opt. 10, 210–212 (1971).
[CrossRef] [PubMed]

Y. Arai, S. Yokozeki, T. Yamada, “Fringe-scanning method using a general function for shadow moiré,” Appl. Opt. 34, 4877–4882 (1995).
[CrossRef] [PubMed]

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Figures (12)

Fig. 1
Fig. 1

Schematic layout of the shadow moiré system.

Fig. 2
Fig. 2

Experimental setup.

Fig. 3
Fig. 3

Typical moiré fringe patterns of a simply supported beam at different instants: (a) 0.4 s, (b) 0.64 s, and (c) 0.88 s.

Fig. 4
Fig. 4

(a) Gray values of points A and B. (b) Modulus of the Morlet wavelet transform at point A. (c) Modulus of the Morlet wavelet transform at point B.

Fig. 5
Fig. 5

(a) Instantaneous velocity and (b) displacement at points A and B.

Fig. 6
Fig. 6

Displacement of the beam between two instants T 1 = 0.4 s and T 2 = 0.8 s by use of (a) temporal wavelet analysis and (b) temporal Fourier analysis.

Fig. 7
Fig. 7

Comparison of displacement at cross section C-C between wavelet and Fourier analyses. FT, Fourier transform.

Fig. 8
Fig. 8

(a) Wrapped phase map, (b) phase map after unwrapping, and (c) 3-D plot of instantaneous surface profile at t 2 = 0.8 s.

Fig. 9
Fig. 9

(a) Area of interest on a coin (specimen 2) and typical moiré fringe patterns at (b) 0.4 s and (c) 0.8 s.

Fig. 10
Fig. 10

(a) Gray values of points D and E. (b) Plot of moduli of Morlet wavelet transform at points D and E.

Fig. 11
Fig. 11

(a) Wrapped phase map, (b) phase map after unwrapping, and (c) reconstructed 3-D plot of instantaneous surface profile of specimen 2 at t 2 = 0.8 s.

Fig. 12
Fig. 12

Comparison of the surface profile of specimen 2 at cross section F-F between the wavelet and the mechanical stylus methods.

Equations (14)

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

Ix, y=ax, y+bx, ycos2πhx, ydpl+hx, y,
Ix, y=ax, y+bx, ycoskhx, y,
Ix, y; t=ax, y; t+bx, y; tcosϕxyt=ax, y; t+bx, y; tcoskhx, y; t+h0x, y,
WSa, b=-+ stΨa,b*tdt,
Ψa,bt=1a ψt-ba, bR, a>0,
st=1CΨ-+0+ WSa, bψt-badaa2db,
CΨ=-+|Ψˆω|2ωdω<+,
Ψt=gtexpiω0t,
Wxya, b=a2 Axyb(gˆaζ-ϕxyb+b, ζ)expiϕxyb,
ω02|ϕxyb|2|Axyb||Axyb|  1,
ω02|ϕxyb||ϕxyb|2  1.
ϕxyb=ζrb=ω0/arb,
Wxyarb, barb2 Axybgˆ0expiϕxyb.
ϕxyb=tan-1ImWxyarb, bReWxyarb, b,

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