Abstract

Most standard temporal-phase-shifting (TPS) algorithms evaluate the phase by computing a windowed Fourier transform (WFT) of the intensity signal at the carrier frequency of the system. However, displacement of the specimen during image acquisition may cause the peak of the transform to shift away from the carrier frequency, leading to phase errors and even unwrapping failure. We present a novel TPS method that searches for the peak of the WFT and evaluates the phase at that frequency instead of at the carrier frequency. The performance of this method is compared with that of standard algorithms by using numerical simulations. Experimental results from high-speed speckle interferometry studies of carbon fiber panels are also presented.

© 2003 Optical Society of America

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References

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    [CrossRef]
  4. X. C. de Lega, “Processing of non-stationary interference patterns: adapted phase-shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1997).
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    [CrossRef] [PubMed]
  12. J. M. Huntley, “Suppression of phase errors from vibration in phase-shifting interferometry,” J. Opt. Soc. Am. A 15, 2233–2241 (1998).
    [CrossRef]
  13. P. D. Ruiz, J. M. Huntley, J. Shen, C. R. Coggrave, G. H. Kaufmann, “Vibration-induced phase errors in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 40, 2117–2125 (2001).
    [CrossRef]
  14. P. D. Ruiz, J. M. Huntley, J. Shen, C. R. Coggrave, G. H. Kaufmann, “Effects of random vibration in high-speed phase-shifting speckle pattern interferometry,” Appl. Opt. 41, 3941–3949 (2002).
    [CrossRef] [PubMed]
  15. P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
    [CrossRef]
  16. M. Cherbuliez, P. M. Jacquot, X. Colonna de Lega, “Wavelet processing of interferometric signals and fringe patterns,” in Wavelet Applications in Signal and Image Processing VII, M. A. Unser, A. Aldroubi, A. F. Laine, eds., Proc. SPIE3813, 692–702 (1999).
    [CrossRef]
  17. B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
    [CrossRef]
  18. E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

2002 (1)

2001 (1)

2000 (2)

1999 (2)

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

J. M. Huntley, G. H. Kaufmann, D. Kerr, “Phase-shifted dynamic speckle pattern interferometry at 1 kHz,” Appl. Opt. 38, 6556–6563 (1999).
[CrossRef]

1998 (1)

1996 (2)

Y. Surrel, “Design of algorithms for phase measurements by the use of phase stepping,” Appl. Opt. 35, 51–60 (1996).
[CrossRef] [PubMed]

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

1993 (1)

1990 (1)

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

1987 (1)

1983 (1)

1966 (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Amick, C. H.

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Barton, J. S.

Bessason, B.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Buckberry, C.

Burow, R.

Carré, P.

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Cherbuliez, M.

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

Coggrave, C. R.

Colonna de Lega, X.

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

de Lega, X. C.

X. C. de Lega, “Processing of non-stationary interference patterns: adapted phase-shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1997).

Eiju, T.

Elssner, K. E.

Frøystein, H. A.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Grzanna, J.

Haible, P.

Hariharan, P.

Huntley, J. M.

Jacquot, P. M.

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

Jones, J. D. C.

Kato, J.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Kaufmann, G. H.

Kerr, D.

Kilpatrick, J. M.

Kolbjørnsen, H.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Kothiyal, M. P.

Liu, J. Y.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Madshus, C.

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Merkel, K.

Moore, A. J.

Oreb, B. F.

Reeves, M.

Ruiz, P. D.

Saldner, H.

Schwider, J.

Shen, J.

Spolaczyk, R.

Sturz, D. H.

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Surrel, Y.

Tiziani, H. J.

Ungar, E. E.

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Yamaguchi, I.

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Appl. Opt. (8)

J. Opt. Soc. Am. A (1)

Meas. Sci. Technol. (1)

B. Bessason, C. Madshus, H. A. Frøystein, H. Kolbjørnsen, “Vibration criteria for metrology laboratories,” Meas. Sci. Technol. 10, 1009–1014 (1999).
[CrossRef]

Metrologia (1)

P. Carré, “Installation et utilisation du comparateur photoélectrique et interférentiel du Bureau International des Poids et Mesures,” Metrologia 2, 13–23 (1966).
[CrossRef]

Opt. Eng. (1)

I. Yamaguchi, J. Y. Liu, J. Kato, “Active phase-shifting interferometers for shape and deformation measurements,” Opt. Eng. 35, 2930–2937 (1996).
[CrossRef]

Opt. Lett. (1)

Sound Vib. (1)

E. E. Ungar, D. H. Sturz, C. H. Amick, “Vibration control design of high technology facilities,” Sound Vib. 7, 20–27 (1990).

Other (4)

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

X. C. de Lega, “Processing of non-stationary interference patterns: adapted phase-shifting algorithms and wavelet analysis. Application to dynamic deformation measurements by holographic and speckle interferometry,” Ph.D. dissertation (Ecole Polytechnique Fédérale de Lausanne, Lausanne, Switzerland, 1997).

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

D. W. Robinson, G. T. Reid, eds., Interferogram Analysis (Institute of Physics, Bristol, 1993).

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Figures (5)

Fig. 1
Fig. 1

Schematic view of the windowed Fourier transform of the intensity signal at an arbitrary time t for which the ridge appears at kr(t). The transform of the 16-frame Hanning window is shown centered on each delta function. This case corresponds to an object moving with velocity z˙=λ/8(kr-1) μm s-1; the effect of the velocity is to shift the peak from kt=1 to kt=kr(t).

Fig. 2
Fig. 2

Velocity spectrum of the pseudorandom vibrations used in the numerical simulations.

Fig. 3
Fig. 3

Results of the analysis of a pseudo-random vibration signal having a spectrum similar to that shown in Fig. 2. (a) Intensity signal. (b) Gray-scale representation of the WFT of I(t) in the kt versus t domain measured with the ridge-searching algorithm. (c) Ridge trajectory kr(t). (d) Intensity signal modulation Im. (e) Original input phase compared to the unwrapped phase change evaluated through standard (S-TPS) and ridge-searching (RS-TPS) algorithms. In both cases a 32-frame Hanning window was used.

Fig. 4
Fig. 4

Dynamic high-speed, phase-shifting speckle interferometer showing a Pockels cell (P) in the reference beam, high-voltage driver (D), function generator (G), frame store (F), Sun computer (Sun), carbon composite laminate (O), vacuum chamber (VC), 90:10 beam splitters (BS), mirrors (M), and lenses (L). Details of the synchronization of the system’s electronics are given in Ref. 5. Internal delaminations in the composite material are revealed when it is vacuum loaded.

Fig. 5
Fig. 5

Phase profile measured from the surface of a carbon fiber composite panel containing a subsurface delamination in the presence of mechanical vibration without vibration isolation by use of 64-frame Hanning window algorithms. Phase evaluated over kt=1 (S-TPS) and over the ridge kr(t) (RS-TPS).

Tables (2)

Tables Icon

Table 1 Numerical Results of Unwrapping Success Rate a

Tables Icon

Table 2 Numerical results of the rms phase-change error σ¯ a

Equations (30)

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I(t)=I0+Im cos(Φ+ϕt),t=0, 1,, Nt-1,
ΔΦˆw(t, 0)=tan-1N(t)D(0)-D(t)N(0)D(t)D(0)+N(t)N(0),t=0, 1,, Nt-M,
N(t)=Im[Z(t)],D(t)=Re[Z(t)],
Z(t)=t=0M-1a(t)I(t+t)+it=0M-1b(t)I(t+t)exp(-iϕt),
a(t)+ib(t)=w(t)exp(-iϕt),t=0, 1, 2,, M-1,
I˜(kt, t)=t=0M-1w(t)I(t+t)exp(-i2πktt/N),
w(t)=12+12cos2πMt-M-12,t=0, 1,, M-2.
I˜(kt, t)=-w(τ)I(t+τ)n=-δ(τ-n)×exp(-i2πktτ/N)dτ,
I˜(kt, t)=w˜(kt)*[A(kt)+B+(kt, t)+B-(kt, t)],
A(kt)=I0j=-δ(kt-jN),
B+(kt, t)=Im2exp(iϕt)- exp(iΦ)×exp[-i2πτ(kt-1)/N]×n=-δ(τ-n)dτ,
B-(kt, t)=Im2exp(-iϕt)- exp(-iΦ)×exp[-i2πτ(kt+1)/N]×n=-δ(τ-n)dτ,
A(kt)=I0δ(kt),
B+(kt, t)=Im2exp[i(Φ+ϕt)]δ(kt-1),
B-(kt, t)=Im2exp[-i(Φ+ϕt)]δ(kt+1).
Φd(t+δt)Φd(t)+Φ˙d(t)δt,
Φ˙d(t)=4πλz˙(t).
B+(kt, t)=Im2exp{i[Φs+Φd(t)+ϕt]}×δkt-1+Φ˙d(t)N2π,
B-(kt, t)=Im2exp{-i[Φs+Φd(t)+ϕt]}×δkt+1+Φ˙d(t)N2π.
kr(t)=1+Φ˙d(t)N2π.
I˜[kr(t),t]=w˜[kr(t)]I0+Im2w˜(0)×exp{i[Φs+Φd(t)+ϕt]}+Im2w˜[2kr(t)]×exp{i[Φs+Φd(t)+ϕt]}.
I˜[kr(t), t]=Im2w˜(0)exp{i[Φs+Φd(t)+ϕt]}.
Z(t)=I˜[kr(t), t]exp(-iϕt).
Φ˙d(t)=π2[kr(t)-1],
ΔΦˆdwt+M2, 0ΔΦˆdw(t, 0)+πM4[kr(t)-1].
S(f )=S0,0<ff0S0f0/f,f0<f,
σ(t, tj)=1Ll=0L-1[ΔΦˆu(t, tj, l)-ΔΦˆu(t, tj)]21/2,
ΔΦˆu(t, tj)=1Ll=0L-1ΔΦˆu(t, tj, l).
σ¯=1sjst=1stj=1sjσ(t, tj),
|z˙|<1-4Mλϕ4π.

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