Abstract

Moire interferometry is an effective experimental technique for measurement of in-plane deformation. However, it is information on the derivatives of the deformation, i.e., strains, that is usually desired in experimental mechanics. It is shown that the desired strains are the instantaneous frequencies of the fringe pattern and that either an energy operator or wavelet ridges can be used to extract the instantaneous frequencies from a single fringe pattern. The energy operator is a pixelwise processor; thus the strain extraction can be done on the fly, but it is sensitive to noise. The wavelet ridges extract the local features in the fringe pattern. The strain extraction is thus insensitive to noise, and good results are obtainable at the cost of longer computation time. The two methods can thus be chosen for different needs in strain analysis. The properties of the two methods as well as their applications to a real fringe pattern are given. The effectiveness of the proposed methods is illustrated by their comparison with traditional methods.

© 2003 Optical Society of America

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References

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  1. D. Post, B. Han, P. Ifju, High Sensitivity Moiré (Springer-Verlag, Berlin, 1994).
    [CrossRef]
  2. D. W. Robinson, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).
  3. A. Asundi, M. T. Cheung, “Moiré of moiré interferometry,” Exp. Tech. 11(8), 28–30 (1987).
    [CrossRef]
  4. A. K. Asundi, “Moiré methods using computer-generated gratings,” Opt. Eng. 32, 107–116 (1993).
    [CrossRef]
  5. A. Asundi, W. Jun, “Strain contouring using Gabor filters: principle and algorithm,” Opt. Eng. 41, 1400–1405 (2002).
    [CrossRef]
  6. S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Norwood, Mass., 1996).
  7. P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
    [CrossRef]
  8. P. Maragos, A. C. Bovik, “Demodulation of images modeled by amplitude-frequency modulations using multidimensional energy seperation,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineering, Piscataway, N.J., 1994), Vol. 3, pp. 421–425.
  9. P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
    [CrossRef]
  10. P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulation with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
    [CrossRef]
  11. M. Servin, M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson eds. (Marcel Dekker, New York, 2001), pp. 373–426.
  12. J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineering, Piscataway, N.J., 1990), Vol. 1, pp. 381–384.
    [CrossRef]
  13. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, San Diego, Fla., 1999).
  14. N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
    [CrossRef]
  15. P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
    [CrossRef]
  16. W. Jun, A. Asundi, “Strain contouring with Gabor filters: filter bank design,” Appl. Opt. 41, 7229–7236 (2002).
    [CrossRef] [PubMed]

2002 (2)

A. Asundi, W. Jun, “Strain contouring using Gabor filters: principle and algorithm,” Opt. Eng. 41, 1400–1405 (2002).
[CrossRef]

W. Jun, A. Asundi, “Strain contouring with Gabor filters: filter bank design,” Appl. Opt. 41, 7229–7236 (2002).
[CrossRef] [PubMed]

1996 (1)

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

1993 (3)

A. K. Asundi, “Moiré methods using computer-generated gratings,” Opt. Eng. 32, 107–116 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulation with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

1992 (1)

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

1987 (1)

A. Asundi, M. T. Cheung, “Moiré of moiré interferometry,” Exp. Tech. 11(8), 28–30 (1987).
[CrossRef]

Asundi, A.

A. Asundi, W. Jun, “Strain contouring using Gabor filters: principle and algorithm,” Opt. Eng. 41, 1400–1405 (2002).
[CrossRef]

W. Jun, A. Asundi, “Strain contouring with Gabor filters: filter bank design,” Appl. Opt. 41, 7229–7236 (2002).
[CrossRef] [PubMed]

A. Asundi, M. T. Cheung, “Moiré of moiré interferometry,” Exp. Tech. 11(8), 28–30 (1987).
[CrossRef]

Asundi, A. K.

A. K. Asundi, “Moiré methods using computer-generated gratings,” Opt. Eng. 32, 107–116 (1993).
[CrossRef]

Bovik, A. C.

P. Maragos, A. C. Bovik, “Demodulation of images modeled by amplitude-frequency modulations using multidimensional energy seperation,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineering, Piscataway, N.J., 1994), Vol. 3, pp. 421–425.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

Cheung, M. T.

A. Asundi, M. T. Cheung, “Moiré of moiré interferometry,” Exp. Tech. 11(8), 28–30 (1987).
[CrossRef]

Delprat, N.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Escudié, B.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Guillemain, P.

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Hahn, S. L.

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Norwood, Mass., 1996).

Han, B.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré (Springer-Verlag, Berlin, 1994).
[CrossRef]

Ifju, P.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré (Springer-Verlag, Berlin, 1994).
[CrossRef]

Jun, W.

W. Jun, A. Asundi, “Strain contouring with Gabor filters: filter bank design,” Appl. Opt. 41, 7229–7236 (2002).
[CrossRef] [PubMed]

A. Asundi, W. Jun, “Strain contouring using Gabor filters: principle and algorithm,” Opt. Eng. 41, 1400–1405 (2002).
[CrossRef]

Kaiser, J. F.

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulation with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineering, Piscataway, N.J., 1990), Vol. 1, pp. 381–384.
[CrossRef]

Kronland-Martinet, R.

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Kujawinska, M.

M. Servin, M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson eds. (Marcel Dekker, New York, 2001), pp. 373–426.

Mallat, S.

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, San Diego, Fla., 1999).

Maragos, P.

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulation with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, A. C. Bovik, “Demodulation of images modeled by amplitude-frequency modulations using multidimensional energy seperation,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineering, Piscataway, N.J., 1994), Vol. 3, pp. 421–425.

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

Post, D.

D. Post, B. Han, P. Ifju, High Sensitivity Moiré (Springer-Verlag, Berlin, 1994).
[CrossRef]

Quatieri, T. F.

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulation with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

Servin, M.

M. Servin, M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson eds. (Marcel Dekker, New York, 2001), pp. 373–426.

Tchamitchian, P.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Torrésani, B.

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

Appl. Opt. (1)

Exp. Tech. (1)

A. Asundi, M. T. Cheung, “Moiré of moiré interferometry,” Exp. Tech. 11(8), 28–30 (1987).
[CrossRef]

IEEE Trans. Inf. Theory (1)

N. Delprat, B. Escudié, P. Guillemain, R. Kronland-Martinet, P. Tchamitchian, B. Torrésani, “Asymptotic wavelet and Gabor analysis: extraction of instantaneous frequencies,” IEEE Trans. Inf. Theory 38, 644–664 (1992).
[CrossRef]

IEEE Trans. Signal Process. (2)

P. Maragos, J. F. Kaiser, T. F. Quatieri, “On amplitude and frequency demodulation using energy operators,” IEEE Trans. Signal Process. 41, 1532–1550 (1993).
[CrossRef]

P. Maragos, J. F. Kaiser, T. F. Quatieri, “Energy separation in signal modulation with application to speech analysis,” IEEE Trans. Signal Process. 41, 3024–3051 (1993).
[CrossRef]

Opt. Eng. (2)

A. K. Asundi, “Moiré methods using computer-generated gratings,” Opt. Eng. 32, 107–116 (1993).
[CrossRef]

A. Asundi, W. Jun, “Strain contouring using Gabor filters: principle and algorithm,” Opt. Eng. 41, 1400–1405 (2002).
[CrossRef]

Proc. IEEE (1)

P. Guillemain, R. Kronland-Martinet, “Characterization of acoustic signals through continuous linear time-frequency representations,” Proc. IEEE 84, 561–585 (1996).
[CrossRef]

Other (8)

D. Post, B. Han, P. Ifju, High Sensitivity Moiré (Springer-Verlag, Berlin, 1994).
[CrossRef]

D. W. Robinson, G. T. Reid, eds., Interferogram Analysis: Digital Fringe Pattern Measurement Techniques (Institute of Physics, Bristol, UK, 1993).

S. L. Hahn, Hilbert Transforms in Signal Processing (Artech House, Norwood, Mass., 1996).

P. Maragos, A. C. Bovik, T. F. Quatieri, “A multidimensional energy operator for image processing,” in Visual Communications and Image Processing ’92, P. Maragos, ed., Proc. SPIE1818, 177–186 (1992).
[CrossRef]

P. Maragos, A. C. Bovik, “Demodulation of images modeled by amplitude-frequency modulations using multidimensional energy seperation,” in Proceedings of the IEEE International Conference on Image Processing (Institute of Electrical and Electronics Engineering, Piscataway, N.J., 1994), Vol. 3, pp. 421–425.

M. Servin, M. Kujawinska, “Modern fringe pattern analysis in interferometry,” in Handbook of Optical Engineering, D. Malacara, B. J. Thompson eds. (Marcel Dekker, New York, 2001), pp. 373–426.

J. F. Kaiser, “On a simple algorithm to calculate the ‘energy’ of a signal,” in Proceedings of the IEEE International Conference on Acoustics, Speech and Signal Processing (Institute of Electrical and Electronics Engineering, Piscataway, N.J., 1990), Vol. 1, pp. 381–384.
[CrossRef]

S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed. (Academic, San Diego, Fla., 1999).

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Figures (7)

Fig. 1
Fig. 1

Real moire fringe pattern and its amplitude spectrum: (a) fringe pattern, (b) amplitude spectrum.

Fig. 2
Fig. 2

Strain extraction by the moire of moire method: strain contours in (a) the x direction (1-pixel shift), (b) the y direction (1-pixel shift), (c) the x direction (40-pixel shift), and (d) the x direction (40-pixel shift).

Fig. 3
Fig. 3

Strain in the x direction extracted by the energy operator: (a) raw strain data obtained by the energy operator, (b) smoothed strain field, (c) 3D view of the strain, and (d) contour of the strain.

Fig. 4
Fig. 4

Strain in the y direction extracted by the energy operator: (a) raw strain data obtained by the energy operator, (b) smoothed strain field, (c) 3D view of the strain, and (d) contour of the strain.

Fig. 5
Fig. 5

Heisenberg boxes of two wavelet atoms. Smaller scales decrease the time spread but increase the frequency support, which is shifted toward higher frequencies.

Fig. 6
Fig. 6

Strain in the x direction extracted by wavelet ridges: (a) strain obtained by wavelet ridges, (b) 3D view of the strain, and (c) Contour of the strain.

Fig. 7
Fig. 7

Strain in the y direction extracted by wavelet ridges: (a) strain obtained by wavelet ridges, (b) 3D view of the strain, and (c) Contour of the strain.

Tables (1)

Tables Icon

Table 1 Comparison of Methods of Strain Extraction

Equations (40)

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

gx, y=ax, y+bx, ycosφx, y,
ux, y=p2π φx, y,
εxx=ux, yx=p2πφx, yx,
εxy1=ux, yy=p2πφx, yy.
ft=btcosφt,
ωt=φtt.
fx, y=bx, ycosφx, y,
ωxx, y=φx, yx,
ωyx, y=φx, yy.
Ψft=dfdt2-f d2fdt2.
ft=B cosωct,
Ψf=B2ωc2,
Ψdfdt=B2ωc4.
ωc=ΨdfdtΨf½.
ωtΨdfdtΨf½.
Φfm, n=2f2m, n-fm-1, nfm+1, n-fm, n-1fm, n+1;
fm, n=bm, ncosφm, n,
ωxm, narcsin×Φfm+1, n-fm-1, n4Φfm, n1/2,
ωym, narcsin×Φfm, n+1-fm, n-14Φfm, n1/2.
-+ ψtdt=0.
ψu,st=1s ψt-us.
Wfu, s=f, ψu,s=-+ ftψu,s*tdt.
Pwfu, s=|Wfu, s|2.
ψt=gtexpjηt,
gt=1σ2π1/4exp-t22σ2.
-+-+ ψx, ydxdy=0.
ψu,v,s1,s2x, y=1s1s2 ψx-us1, y-vs2.
Wfu, v, s1, s2=f, ψu,v,s1,s2=-+-+ fx, yψu,v,s1,s2*x, ydxdy.
Pwfu, v, s1, s2=|Wfu, v, s1, s2|2.
ψx, y=ψxψy=gxgyexpjη1x+jη2y,
Wfu, v, s1, s2 =fx, y, ψu,v,s1,s2x, y=ax, y, ψu,v,s1,s2x, y+½ bx, yexpjφx, y, ψu,v,s1,s2x, y +½ bx, yexp-jφx, y, ψu,v,s1,s2x, y=Ia+Iφ+I-φ.
Ia=-+-+ ax, ygs1x-ugs2y-v×exp-jξ1x-uexp-jξ2y-vdxdy=-+-+ ax+u, y+vgs1xgs2y×exp-jξ1xexp-jξ2ydxdy,
Ia=s1s2 au, vĝs1ξ1ĝs2ξ20,
Iφ=½ -+-+ bx, yexpjφx, ygs1x-ugs2y-vexp-jξ1x-u-jξ2y-vdxdy=½ -+-+ bx+u, y+vexpjφx+u, y+vgs1xgs2yexp-jξ1xexp-jξ2ydxdy.
φx+u, y+vφu, v+φu, vu x+φu, vv y.
Iφs1s22 bu, vexpjφu, vĝs1ξ1-φu, vuĝs2ξ2-φu, vv.
I-φs1s22 bu, vexp-jφu, vĝs1ξ1+φu, vuĝs2ξ2+φu, vv.
Pwfu, v, s1, s2=|Ia+Iφ+I-φ|2 |Iφ|2=s1s24 b2u, vĝs1ξ1-φu, vu2ĝs2ξ2-φu, vv2.
ξ1=η1/s1,
ξ2=η2/s2.

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