Abstract

In recent years, optical interferometry has been applied to the whole-field, noncontact measurement of vibrating or continuously deforming objects. In many cases, a high resolution measurement of kinematic (displacement, velocity, and acceleration, etc.) and deformation parameters (strain, curvature, and twist, etc.) can give useful information on the dynamic response of the objects concerned. Different signal processing algorithms are applied to two types of interferogram sequences, which were captured by a high-speed camera using different interferometric setups: (1) a speckle or fringe pattern sequence with a temporal carrier and (2) a wrapped phase map sequence. These algorithms include Fourier transform, windowed Fourier transform, wavelet transform, and even a combination of two of these techniques. We will compare these algorithms using the example of a 1D temporal evaluation of interferogram sequences and extend these algorithms to 2D and 3D processing, so that accurate kinematic and deformation parameters of moving objects can be evaluated with different types of optical interferometry.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
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    [CrossRef]
  31. Y. Fu, G. Pedrini, and W. Osten, "Vibration measurement by temporal Fourier analyses of digital hologram sequence," Appl. Opt. 46, 5719-5727 (2007).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]

2007 (3)

2006 (2)

2005 (4)

2004 (2)

Y. Fu, C. J. Tay, C. Quan, and L. J. Chen, "Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry," Opt. Eng. 43, 2780-2787 (2004).
[CrossRef]

K. Qian, "Windowed Fourier transform for fringe pattern analysis," Appl. Opt. 43, 2695-2702 (2004).
[CrossRef]

2003 (2)

K. Qian, S. H. Seah, and A. Asundi, "Phase-shifting windowed Fourier ridges for determination of phase derivatives," Opt. Lett. 28, 1657-1659 (2003).
[CrossRef] [PubMed]

G. H. Kaufmann, "Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier," Opt. Commun. 217, 141-149 (2003).
[CrossRef]

2002 (1)

A. Federico and 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]

1999 (4)

1998 (2)

1997 (1)

G. Pedrini, H. Tiziani, and Y. Zou, "Digital double pulse-TV-holography," Opt. Lasers Eng. 26, 199-219 (1997).
[CrossRef]

1996 (2)

X. Colonna de Lega and P. Jacquot, "Interferometric deformation measurement using object induced dynamic phase shifting," in Optical Inspection and Micromeasurements, C. Goreuki, ed., Proc. SPIE 2782, 169-179 (1996).
[CrossRef]

X. Colonna de Lege and P. Jacquot, "Deformation measurement with object-induced dynamic phase shifting," Appl. Opt. 35, 5115-5121 (1996).
[CrossRef]

1993 (1)

1986 (1)

1982 (1)

1967 (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

1965 (1)

Appl. Opt. (12)

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

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

X. Colonna de Lege and P. Jacquot, "Deformation measurement with object-induced dynamic phase shifting," Appl. Opt. 35, 5115-5121 (1996).
[CrossRef]

D. I. Farrant, G. H. Kaufmann, J. N. Petzing, J. R. Tyrer, B. F. Oreb, and D. Kerr, "Measurement of transient deformations with dual-pulse addition electronic speckle-pattern interferometry," Appl. Opt. 37, 7259-7267 (1998).
[CrossRef]

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

S. Schedin, G. Pedrini, H. J. Tiziani, and F. M. Santoyo, "Simultaneous three-dimensional dynamic deformation measurements with pulsed digital holography," Appl. Opt. 38, 7056-7062 (1999).
[CrossRef]

K. Qian, "Windowed Fourier transform for fringe pattern analysis," Appl. Opt. 43, 2695-2702 (2004).
[CrossRef]

Y. Fu, C. J. Tay, C. Quan, and H. Miao, "Wavelet analysis of speckle patterns with a temporal carrier," Appl. Opt. 44, 959-965 (2005).
[CrossRef] [PubMed]

S. Nakadate, "Vibration measurement using phase-shifting time-average holographic interferometry," Appl. Opt. 25, 4155-4161 (1986).
[CrossRef] [PubMed]

G. Pedrini, W. Osten, and M. E. Gusev, "High-speed digital holographic interferometry for vibration measurement," Appl. Opt. 45, 3456-3462 (2006).
[CrossRef] [PubMed]

Y. Fu, G. Pedrini, and W. Osten, "Vibration measurement by temporal Fourier analyses of digital hologram sequence," Appl. Opt. 46, 5719-5727 (2007).
[CrossRef] [PubMed]

K. Qian, T. H. N. Le, F. Lin, and H. S. Seah, "Comparative analysis on some filters for wrapped phase maps," Appl. Opt. 46, 7412-7418 (2007).
[CrossRef]

Appl. Phys. Lett. (1)

J. W. Goodman and R. W. Lawrence, "Digital image formation from electronically detected holograms," Appl. Phys. Lett. 11, 77-79 (1967).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (2)

H. A. Aebischer and S. Waldner, "A simple and effective method for filtering speckle-interferometric phase fringe patterns," Opt. Commun. 162, 205-210 (1999).
[CrossRef]

G. H. Kaufmann, "Phase measurement in temporal speckle pattern interferometry using the Fourier transform method with and without a temporal carrier," Opt. Commun. 217, 141-149 (2003).
[CrossRef]

Opt. Eng. (3)

A. Federico and 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]

K. Qian and H. S. Seah, "Two-dimensional windowed Fourier frames for noise reduction in fringe pattern analysis," Opt. Eng. 44, 075601 (2005).
[CrossRef]

Y. Fu, C. J. Tay, C. Quan, and L. J. Chen, "Temporal wavelet analysis for deformation and velocity measurement in speckle interferometry," Opt. Eng. 43, 2780-2787 (2004).
[CrossRef]

Opt. Lasers Eng. (2)

K. Qian, "Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations," Opt. Lasers Eng. 45, 304-317 (2007).
[CrossRef]

G. Pedrini, H. Tiziani, and Y. Zou, "Digital double pulse-TV-holography," Opt. Lasers Eng. 26, 199-219 (1997).
[CrossRef]

Opt. Lett. (5)

Proc. SPIE (1)

X. Colonna de Lega and P. Jacquot, "Interferometric deformation measurement using object induced dynamic phase shifting," in Optical Inspection and Micromeasurements, C. Goreuki, ed., Proc. SPIE 2782, 169-179 (1996).
[CrossRef]

Other (5)

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

http://www.photron.com/content.cfm?n=products&id=SlowMotionVideo.

H. J. Tiziani, "Spectral and temporal phase evaluation for interferometry and speckle applications," in Trends in Optical Nondestructive Testing and Inspection, P. K. Rastogi and D. Inaudi, eds. (Elsevier Science B. V., 2000), pp. 323-343.
[CrossRef]

X. Colonna 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 1666 (Swiss Federal Institute of Technology, 1997).

M. Cherbuliez and P. Jacquot, "Phase computation through wavelet analysis: yesterday and nowadays," in Fringe 2001, W. Osten and W. Juptner, eds. (Elsevier, 2001), pp. 154-162.

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Figures (7)

Fig. 1
Fig. 1

(a) Real part of a windowed Fourier kernel at different frequencies: 0.1π, 0.2π, and 0.3π. The window size is set as σ = 20 . (b) Spectrum of windowed Fourier kernels in (a). (c) Real part of a complex Morlet wavelet at different frequencies: 0.1π, 0.2π, and 0.3π ( a = 20 , 10, and 6.67). (d) Spectrum of complex Morlet wavelets in (c).

Fig. 2
Fig. 2

(Color online) (a) Simulated intensity variation on a vibrating object with a temporal carrier; (b) theoretical phase value of signal in (a); (c) first-order derivative of the phase value; (d) Fourier spectrum of the signal; (e) first-order derivative of the phase obtained by Fourier bandpass filtering and a numerical differentiation; (f) modulus of the WFT when σ = 20 and the corresponding ridge; (g) modulus of the WT and the corresponding ridge; and (h) ridge obtained by a combination of FFT and WT processing.

Fig. 3
Fig. 3

(a) Some typical wrapped phase maps at different instants obtained by image-plane digital holography; (b) phase variation at point A; (c) spectrum of the exponential phase signal; (d) first-order derivatives of the phase after FT filtering; (e) modulus of the WFT and corresponding ridge; and (f) WFT ridge proportional to the acceleration.

Fig. 4
Fig. 4

Three-dimensional plot of the instantaneous (a) displacement, (b) velocity, and (c) acceleration at frame number 100.

Fig. 5
Fig. 5

(a) Simulated wrapped phase map with noise on a fully clamped circular plate, loaded by uniform pressure; (b) theoretical value of d w / d x and simulated noise signal; (c) absolute errors of d w / d x measurement by WFT with different window sizes; and (d) comparison of theoretical curvature value with WFT results using different window sizes.

Fig. 6
Fig. 6

(a) Typical ESPI fringes at frame 300; (b) wrapped phase map obtained by temporal Fourier analysis at frame 300; (c) grayscale map indicating the instantaneous spatial distribution of d w / d x ; (d) grayscale map indicating the instantaneous spatial distribution of d w / d y ; (e) grayscale map indicating the instantaneous spatial distribution of curvature d 2 w / d x 2 ; and (f) grayscale map indicating the instantaneous spatial distribution of twist d 2 w / d x d y .

Fig. 7
Fig. 7

(a) Wrapped phase obtained by 3D Fourier filtering and (b) wrapped phase obtained by 3D windowed Fourier filtering.

Equations (19)

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f I ( x , y ; t ) = I b ( x , y ; t ) + A ( x , y ; t ) cos ( φ ( x , y ; t ) ) ,
f II ( x , y ; t ) = exp ( j ϕ ( x , y ; t ) ) ,
f ^ ( ξ ) = + f ( t ) exp ( j ξ t ) d t ,
f ( t ) = 1 2 π + f ^ ( ξ ) exp ( j ξ t ) d ξ ,
f ^ ( ξ ) = D C ( ξ ) + C ^ ( ξ ) + C ^ * ( ξ ) ,
φ ( t ) = arctan Im ( C ( t ) ) Re ( C ( t ) ) ,
S f ( u , ξ ) = + f ( t ) g u , ξ * ( t ) d t ,
f ( t ) = 1 2 π + + S f ( u , ξ ) g u , ξ ( t ) d ξ d u ,
g u , ξ ( t ) = g ( t u ) exp ( j ξ t ) .
g ( t ) = exp ( t 2 / 2 σ 2 ) ,
S f I I ( u , ξ ) = s 2 A ( u ) exp ( j [ φ ( u ) ξ u ] ) ( g ^ ( s [ ξ φ ( u ) ] ) + ε ( u , ξ ) ) ,
ξ ( u ) = φ ( u ) ,
W f ( a , b ) = 1 a + f ( t ) Ψ * ( t b a ) d t = 1 a + f ( t ) Ψ a b * ( t ) d t ,
Ψ ( t ) = g ( t ) exp ( i ω 0 t ) ,
Ψ a b ( t ) = Ψ ( t b a ) = exp ( ( t b ) 2 2 a 2 ) exp ( i ω 0 a ( t b ) ) .
ξ = ω 0 a = 2 π a .
W f ( a , b ) = a 2 A ( b ) { g ^ [ a ( ξ φ ( b ) ) ] + ε ( b , ξ ) } exp [ i φ ( b ) ] ,
φ ( b ) = ξ r b = ω 0 a r b ,
f I I ( t ) = U ( x , y ; t n ) U * ( x , y ; t n 1 ) = exp ( j φ ( x , y ; t n ) ) exp ( j φ ( x , y ; t n 1 ) ) ,

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