Abstract

Measurement of curvature and twist is an important aspect in the study of object deformation. In recent years, several methods have been proposed to determine curvature and twist of a deformed object using digital shearography. Here we propose a novel method to determine the curvature and twist of a deformed object using digital holography and a complex phasor. A sine/cosine transformation method and two-dimensional short time Fourier transform are proposed subsequently to process the wrapped phase maps. It is shown that high-quality phase maps corresponding to curvature and twist can be obtained. An experiment is conducted to demonstrate the validity of the proposed method.

© 2008 Optical Society of America

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  1. F. P. Chiang and T. Y. Kao, “An optical method of generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251-260 (1979).
    [CrossRef]
  2. F. S. Chau and J. Zhou, “Direct measurement of curvature and twist of plates using digital shearography,” Opt. Lasers Eng. 39, 431-440 (2003).
    [CrossRef]
  3. Y. Y. Hung and C. Y. Liang, “Image-shearing camera for direct measurement of surface strains,” Appl. Opt. 18, 1046-1051(1979).
    [CrossRef] [PubMed]
  4. D. K. Sharma, R. S. Sirohi, and M. P. Kothiyal, “Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer,” Appl. Opt. 23, 1542-1546(1984).
    [CrossRef] [PubMed]
  5. D. K. Sharma, N. K. Mohan, and R. S. Sirohi, “A holographic speckle shearing technique for the measurement of out-of-plane displacement, slope and curvature,” Opt. Commun. 57, 230-235 (1986).
    [CrossRef]
  6. K. V. Sriram, M. P. Kothiyal, and R. S. Sirohi, “Talbot interferometry in noncollimated illumination for curvature and focal length measurements,” Appl. Opt. 31, 75-79 (1992).
    [CrossRef] [PubMed]
  7. P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251-1263 (1991).
    [CrossRef]
  8. C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91-98 (1994).
    [CrossRef]
  9. C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Multiple-image shearography: a direct method to determine curvatures,” Appl. Opt. 34, 2202-2206 (1995).
    [CrossRef] [PubMed]
  10. P. K. Rastogi, “Measurement of curvature and twist of a deformed object by electronic speckle-shearing pattern interferometry,” Opt. Lett. 21, 905-907 (1996).
    [CrossRef] [PubMed]
  11. K. F. Wang, A. K. Tieu, and E. B. Li, “Simultaneous measurement of pure curvature and twist distribution fields by a five-aperture shearing and two-Fourier filtering technique,” Appl. Opt. 39, 2577-2583 (2000).
    [CrossRef]
  12. S. Grilli, P. Ferraro, S. D. Nicola, A. Finizio, G. Pierattini, and R. Meucci, “Whole optical wavefields reconstruction by digital holography,” Opt. Express 9, 294-302 (2001).
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  13. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  14. E. Cuche, F. Bevilacqua, and C. Depeursinge, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291-293 (1999).
    [CrossRef]
  15. M. Takeda, H. Ina, and S. Kobayashi, “Fourier-transform method of fringe-pattern analysis for computer-based topography and interferometry,” J. Opt. Soc. Am. 72, 156-160 (1982).
    [CrossRef]
  16. S. Mallat, A Wavelet Tour of Signal Processing, 2nd ed.(Academic, 1999).
  17. M. M. Hossain, D. S. Mehta, and C. Shakher, “Information reduction using lensless Fourier transform digital composite holography,” Opt. Laser Technol. 40, 120-128 (2008).
    [CrossRef]
  18. D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).
  19. Q. Kemao, “Two-dimensional windowed Fourier transform for fringe pattern analysis: principles, applications and implementations,” Opt. Lasers Eng. 45, 304-317 (2007).
    [CrossRef]
  20. B. Ströbel, “Processing of interferometric phase maps as complex-valued phasor images,” Appl. Opt. 35, 2192-2198(1996).
    [CrossRef] [PubMed]
  21. C. Liu, “Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method,” Opt. Eng. 42, 3443-3446 (2003).
    [CrossRef]

2008 (1)

M. M. Hossain, D. S. Mehta, and C. Shakher, “Information reduction using lensless Fourier transform digital composite holography,” Opt. Laser Technol. 40, 120-128 (2008).
[CrossRef]

2007 (1)

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

2003 (2)

C. Liu, “Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method,” Opt. Eng. 42, 3443-3446 (2003).
[CrossRef]

F. S. Chau and J. Zhou, “Direct measurement of curvature and twist of plates using digital shearography,” Opt. Lasers Eng. 39, 431-440 (2003).
[CrossRef]

2001 (1)

2000 (1)

1999 (1)

1997 (1)

1996 (2)

1995 (1)

1994 (1)

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91-98 (1994).
[CrossRef]

1992 (1)

1991 (1)

P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251-1263 (1991).
[CrossRef]

1986 (1)

D. K. Sharma, N. K. Mohan, and R. S. Sirohi, “A holographic speckle shearing technique for the measurement of out-of-plane displacement, slope and curvature,” Opt. Commun. 57, 230-235 (1986).
[CrossRef]

1984 (1)

1982 (1)

1979 (2)

F. P. Chiang and T. Y. Kao, “An optical method of generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251-260 (1979).
[CrossRef]

Y. Y. Hung and C. Y. Liang, “Image-shearing camera for direct measurement of surface strains,” Appl. Opt. 18, 1046-1051(1979).
[CrossRef] [PubMed]

Bevilacqua, F.

Chau, F. S.

F. S. Chau and J. Zhou, “Direct measurement of curvature and twist of plates using digital shearography,” Opt. Lasers Eng. 39, 431-440 (2003).
[CrossRef]

Chiang, F. P.

F. P. Chiang and T. Y. Kao, “An optical method of generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251-260 (1979).
[CrossRef]

Cuche, E.

Depeursinge, C.

Ferraro, P.

Finizio, A.

Ghiglia, D. C.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).

Grilli, S.

Hossain, M. M.

M. M. Hossain, D. S. Mehta, and C. Shakher, “Information reduction using lensless Fourier transform digital composite holography,” Opt. Laser Technol. 40, 120-128 (2008).
[CrossRef]

Hung, Y. Y.

Ina, H.

Kao, T. Y.

F. P. Chiang and T. Y. Kao, “An optical method of generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251-260 (1979).
[CrossRef]

Kemao, Q.

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

Kobayashi, S.

Kothiyal, M. P.

Li, E. B.

Liang, C. Y.

Lin, Q. Y.

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Multiple-image shearography: a direct method to determine curvatures,” Appl. Opt. 34, 2202-2206 (1995).
[CrossRef] [PubMed]

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91-98 (1994).
[CrossRef]

Liu, C.

C. Liu, “Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method,” Opt. Eng. 42, 3443-3446 (2003).
[CrossRef]

Mallat, S.

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

Mehta, D. S.

M. M. Hossain, D. S. Mehta, and C. Shakher, “Information reduction using lensless Fourier transform digital composite holography,” Opt. Laser Technol. 40, 120-128 (2008).
[CrossRef]

Meucci, R.

Mohan, N. K.

D. K. Sharma, N. K. Mohan, and R. S. Sirohi, “A holographic speckle shearing technique for the measurement of out-of-plane displacement, slope and curvature,” Opt. Commun. 57, 230-235 (1986).
[CrossRef]

Nicola, S. D.

Pierattini, G.

Pritt, M. D.

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).

Rastogi, P. K.

P. K. Rastogi, “Measurement of curvature and twist of a deformed object by electronic speckle-shearing pattern interferometry,” Opt. Lett. 21, 905-907 (1996).
[CrossRef] [PubMed]

P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251-1263 (1991).
[CrossRef]

Shakher, C.

M. M. Hossain, D. S. Mehta, and C. Shakher, “Information reduction using lensless Fourier transform digital composite holography,” Opt. Laser Technol. 40, 120-128 (2008).
[CrossRef]

Shang, H. M.

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Multiple-image shearography: a direct method to determine curvatures,” Appl. Opt. 34, 2202-2206 (1995).
[CrossRef] [PubMed]

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91-98 (1994).
[CrossRef]

Sharma, D. K.

D. K. Sharma, N. K. Mohan, and R. S. Sirohi, “A holographic speckle shearing technique for the measurement of out-of-plane displacement, slope and curvature,” Opt. Commun. 57, 230-235 (1986).
[CrossRef]

D. K. Sharma, R. S. Sirohi, and M. P. Kothiyal, “Simultaneous measurement of slope and curvature with a three-aperture speckle shearing interferometer,” Appl. Opt. 23, 1542-1546(1984).
[CrossRef] [PubMed]

Sirohi, R. S.

Sriram, K. V.

Ströbel, B.

Takeda, M.

Tay, C. J.

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Multiple-image shearography: a direct method to determine curvatures,” Appl. Opt. 34, 2202-2206 (1995).
[CrossRef] [PubMed]

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91-98 (1994).
[CrossRef]

Tieu, A. K.

Toh, S. L.

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Multiple-image shearography: a direct method to determine curvatures,” Appl. Opt. 34, 2202-2206 (1995).
[CrossRef] [PubMed]

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91-98 (1994).
[CrossRef]

Wang, K. F.

Yamaguchi, I.

Zhang, T.

Zhou, J.

F. S. Chau and J. Zhou, “Direct measurement of curvature and twist of plates using digital shearography,” Opt. Lasers Eng. 39, 431-440 (2003).
[CrossRef]

Appl. Opt. (6)

Int. J. Solids Struct. (1)

F. P. Chiang and T. Y. Kao, “An optical method of generating slope and curvature contours of bent plates,” Int. J. Solids Struct. 15, 251-260 (1979).
[CrossRef]

J. Mod. Opt. (1)

P. K. Rastogi, “Visualization and measurement of slope and curvature fields using holographic interferometry: an application to flaw detection,” J. Mod. Opt. 38, 1251-1263 (1991).
[CrossRef]

J. Opt. Soc. Am. (1)

Opt. Commun. (1)

D. K. Sharma, N. K. Mohan, and R. S. Sirohi, “A holographic speckle shearing technique for the measurement of out-of-plane displacement, slope and curvature,” Opt. Commun. 57, 230-235 (1986).
[CrossRef]

Opt. Eng. (1)

C. Liu, “Simultaneous measurement of displacement and its spatial derivatives with a digital holographic method,” Opt. Eng. 42, 3443-3446 (2003).
[CrossRef]

Opt. Express (1)

Opt. Laser Technol. (2)

M. M. Hossain, D. S. Mehta, and C. Shakher, “Information reduction using lensless Fourier transform digital composite holography,” Opt. Laser Technol. 40, 120-128 (2008).
[CrossRef]

C. J. Tay, S. L. Toh, H. M. Shang, and Q. Y. Lin, “Direct determination of second-order derivatives in plate bending using multiple-exposure shearography,” Opt. Laser Technol. 26, 91-98 (1994).
[CrossRef]

Opt. Lasers Eng. (2)

F. S. Chau and J. Zhou, “Direct measurement of curvature and twist of plates using digital shearography,” Opt. Lasers Eng. 39, 431-440 (2003).
[CrossRef]

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

Opt. Lett. (3)

Other (2)

D. C. Ghiglia and M. D. Pritt, Two-Dimensional Phase Unwrapping: Theory, Algorithm, and Software (Wiley, 1998).

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

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Figures (8)

Fig. 1
Fig. 1

(a) Schematic experimental setup: VBS, variable beam splitter; SF, spatial filter; (b) schematic of a plate loaded at center; (c) shift direction for numerical determination of curvature 2 d / x 2 ; (d) shift direction for numerical determination of twist 2 d / x y .

Fig. 2
Fig. 2

Coordinate system for the numerical reconstruction of a recorded hologram.

Fig. 3
Fig. 3

Flow chart of measurement of curvature and twist using the proposed method.

Fig. 4
Fig. 4

(a) Wrapped phase map obtained by the CP method; (b) 3D plot of a continuous phase map for deformation of a centrally loaded plate.

Fig. 5
Fig. 5

(a) Phase map corresponding to curvature 2 d / x 2 without filtering; (b) phase map obtained by use of CP; (c) filtered phase map by use of sine/cosine transformation for (b); (d) 3D plot of an unwrapped phase map.

Fig. 6
Fig. 6

(a) Phase map corresponding to curvature 2 d / y 2 without filtering; (b) phase map obtained by CP; (c) filtered phase map for (b); (d) 3D plot of an unwrapped phase map.

Fig. 7
Fig. 7

(a) Phase map corresponding to twist 2 d / x y without filtering; (b) filtered phase map obtained by use of sine/cosine transformation; (c) phase map obtained by use of CP; (d) filtered phase map obtained by sine/cosine transformation filtering on the phase map in (c); (e) unwrapped phase map; (f) 3D plot of the continuous phase map.

Fig. 8
Fig. 8

(a) Phase map corresponding to curvature 2 d / x 2 filtered by STFT; (b) phase map corresponding to curvature 2 d / y 2 filtered by STFT; (c) phase map corresponding to twist 2 d / x y filtered by STFT.

Equations (20)

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E ( x , y ) = C exp [ i π λ L ( x 2 + y 2 ) ] I 1 [ H ( ξ , η ) ] ,
E ( m , n ) = C exp [ i π λ L ( m 2 M 2 Δ ξ 2 + n 2 N 2 Δ η 2 ) ] k = 0 M 1 l = 0 N 1 H ( k Δ ξ , l Δ η ) exp [ i ( 2 π m k M + 2 π n l N ) ] ,
φ ( m , n ) = arctan Im [ E ( m , n ) ] Re [ E ( m , n ) ] ,
I ( m , n ) = | E ( m , n ) | 2 .
f 2 λ sin ( θ max 2 ) ,
A ( m , n ) = E ( m , n , 2 ) E * ( m , n , 1 ) = a ( m , n , 2 ) a ( m , n , 1 ) exp { i [ φ ( m , n , 2 ) φ ( m , n , 1 ) ] } = a ( m , n ) exp { i [ Δ φ ( m , n ) ] } ,
Δ φ ( m , n ) = arctan Im [ E ( m , n , 2 ) E * ( m , n , 1 ) ] / 9 Re [ E ( m , n , 2 ) E * ( m , n , 1 ) ] / 9 ,
Γ ( m , n ) = A ( m , n ) A * ( m , n ) = a ( m + δ x , n ) a ( m , n ) exp { i [ Δ φ ( m + δ x , n ) Δ φ ( m , n ) ] } = a ( m , n ) exp { i [ Δ Δ φ ( m , n ) ] } ,
Δ Δ φ ( m , n ) = arctan Im [ A ( m , n ) A * ( m , n ) ] / 9 Re [ A ( m , n ) A * ( m , n ) ] / 9 ,
Ω ( m , n ) = Γ ( m , n ) Γ * ( m , n ) = a ( m + δ x , n ) a ( m , n ) exp { i [ Δ Δ φ ( m + δ x , n ) Δ Δ φ ( m , n ) ] } = a ( m , n ) exp ( i { Δ [ Δ Δ φ ( m , n ) ] } ) ,
Δ [ Δ Δ φ ( m , n ) ] = arctan Im [ Γ ( m , n ) Γ * ( m , n ) ] / 9 Re [ Γ ( m , n ) Γ * ( m , n ) ] / 9 ,
Δ [ Δ Δ φ ( m , n ) ] 4 π λ [ d x ( m + δ x , n ) d x ( m , n ) ] δ x 4 π λ 2 d x 2 δ x 2 ,
Δ [ Δ Δ φ ( m , n ) ] ¯ = arctan S ( m , n ) / 9 C ( m , n ) / 9 ,
S ( x , y ) = sin { Δ [ Δ Δ φ ( x , y ) ] } ,
C ( x , y ) = cos { Δ [ Δ Δ φ ( x , y ) ] } .
f ( x , y ) = C ( x , y ) + i S ( x , y ) = exp ( i { Δ [ Δ Δ φ ( x , y ) ] } ) .
S f ( u , v , ξ , η ) = + + f ( x , y ) g ( x u , y v ) exp ( i ξ x i η y ) d x d y ,
S f ( u , v , ξ , η ) ¯ = { S f ( u , v , ξ , η ) if | S f ( u , v , ξ , η ) | thrd 0 if | S f ( u , v , ξ , η ) | < thrd ,
f ( x , y ) ¯ = 1 4 π 2 + + η l η u ξ l ξ u S f ( u , v , ξ , η ) ¯ g ( x u , y v ) exp ( i ξ x + i η y ) d ξ d η d u d v ,
Δ [ Δ Δ φ ( x , y ) ] ¯ = arctan Im f ( x , y ) ¯ Re f ( x , y ) ¯ ,

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