Abstract

A novel approach for the determination of large-scale out-of-plane displacements from digital Fourier holograms is presented. The proposed method is invariant to lateral object shifts. It is based on the determination of the scaling of the reconstructed image that occurs when the recording distance is changed. For a precise determination of the scaling factor, we utilize the Mellin transform. After the discussion of mathematical and computational issues, experimental results are presented to verify the theoretical considerations. The results show that displacements of at least up to 8.4% from the initial recording distance can be detected with this approach. The displacements could be determined with a deviation of typically less than 1.0% from the set values.

© 2007 Optical Society of America

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References

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    [CrossRef]
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    [CrossRef]
  6. C. Pérez-López, M. del Socorro Hernández-Montes, and F. Mendoza-Santoyo, "Detection of micromechanical deformation under rigid body displacement using twin-pulsed 3D digital holography," in Eighth International Symposium on Laser Metrology, R. Rodriguez-Vera and F. Mendoza-Santoyo, eds., Proc. SPIE 5776, pp. 757-762 (2005).
    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef]
  11. T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, Weinheim, 2004).
    [CrossRef]
  12. C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).
    [CrossRef]
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    [CrossRef]
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2006 (3)

2005 (2)

C. Pérez-López, M. del Socorro Hernández-Montes, and F. Mendoza-Santoyo, "Detection of micromechanical deformation under rigid body displacement using twin-pulsed 3D digital holography," in Eighth International Symposium on Laser Metrology, R. Rodriguez-Vera and F. Mendoza-Santoyo, eds., Proc. SPIE 5776, pp. 757-762 (2005).
[CrossRef]

B. Javidi and D. Kim, "Three-dimensional-object recognition by use of single-exposure on-axis digital holography," Opt. Lett. 30, 236-238 (2005).
[CrossRef] [PubMed]

2004 (1)

2003 (1)

2002 (1)

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

2001 (2)

Y. Frauel, E. Tajahuerce, M.-A. Castro, and B. Javidi, "Distortion-tolerant three-dimensional object recognition with digital holography," Appl. Opt. 40, 3887-3893 (2001).
[CrossRef]

C. Pérez-López and F. M. Santoyo, "Rigid body motion removal by derotating digital holograms," in Fourth Iberoamerican Meeting on Optics and Seventh Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, and M. C. Marconi, eds., Proc. SPIE 4419, pp. 226-230 (2001).
[CrossRef]

2000 (2)

B. Javidi and E. Tajahuerce, "Three-dimensional object recognition by use of digital holography," Opt. Lett. 25, 610-612 (2000).
[CrossRef]

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).
[CrossRef]

1999 (2)

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, "Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology," Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

T. M. Lehmann, C. Gönner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

1997 (1)

1996 (1)

G. Cristóbal and L. Cohen, "Scale in images," in Advanced Signal Processing Algorithms, Architectures, and Implementations VI, F. T. Luk, ed., Proc. SPIE 2846, pp. 251-261 (1996).
[CrossRef]

Baumbach, T.

Castro, M.-A.

Cohen, L.

G. Cristóbal and L. Cohen, "Scale in images," in Advanced Signal Processing Algorithms, Architectures, and Implementations VI, F. T. Luk, ed., Proc. SPIE 2846, pp. 251-261 (1996).
[CrossRef]

Cristóbal, G.

G. Cristóbal and L. Cohen, "Scale in images," in Advanced Signal Processing Algorithms, Architectures, and Implementations VI, F. T. Luk, ed., Proc. SPIE 2846, pp. 251-261 (1996).
[CrossRef]

del Socorro Hernández-Montes, M.

C. Pérez-López, M. del Socorro Hernández-Montes, and F. Mendoza-Santoyo, "Detection of micromechanical deformation under rigid body displacement using twin-pulsed 3D digital holography," in Eighth International Symposium on Laser Metrology, R. Rodriguez-Vera and F. Mendoza-Santoyo, eds., Proc. SPIE 5776, pp. 757-762 (2005).
[CrossRef]

Frauel, Y.

Füzessy, Z.

Gombkötö, B.

Gönner, C.

T. M. Lehmann, C. Gönner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

Javidi, B.

Jüptner, W.

Jüptner, W. P. O.

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Kebbel, V.

Kim, D.

Kiss, M.

Kolenovic, E.

Kornis, J.

Kovács, P.

Kreis, T.

T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, Weinheim, 2004).
[CrossRef]

Lehmann, T. M.

T. M. Lehmann, C. Gönner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

Mendoza-Santoyo, F.

C. Pérez-López, M. del Socorro Hernández-Montes, and F. Mendoza-Santoyo, "Detection of micromechanical deformation under rigid body displacement using twin-pulsed 3D digital holography," in Eighth International Symposium on Laser Metrology, R. Rodriguez-Vera and F. Mendoza-Santoyo, eds., Proc. SPIE 5776, pp. 757-762 (2005).
[CrossRef]

Osten, W.

Pérez-López, C.

C. Pérez-López, M. del Socorro Hernández-Montes, and F. Mendoza-Santoyo, "Detection of micromechanical deformation under rigid body displacement using twin-pulsed 3D digital holography," in Eighth International Symposium on Laser Metrology, R. Rodriguez-Vera and F. Mendoza-Santoyo, eds., Proc. SPIE 5776, pp. 757-762 (2005).
[CrossRef]

C. Pérez-López and F. M. Santoyo, "Rigid body motion removal by derotating digital holograms," in Fourth Iberoamerican Meeting on Optics and Seventh Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, and M. C. Marconi, eds., Proc. SPIE 4419, pp. 226-230 (2001).
[CrossRef]

Santoyo, F. M.

C. Pérez-López and F. M. Santoyo, "Rigid body motion removal by derotating digital holograms," in Fourth Iberoamerican Meeting on Optics and Seventh Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, and M. C. Marconi, eds., Proc. SPIE 4419, pp. 226-230 (2001).
[CrossRef]

Schnars, U.

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Seebacher, S.

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, "Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology," Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

Sjödahl, M.

Spitzer, K.

T. M. Lehmann, C. Gönner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

Svanbro, A.

Tajahuerce, E.

von Kopylow, C.

Wagner, C.

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).
[CrossRef]

C. Wagner, S. Seebacher, W. Osten, and W. Jüptner, "Digital recording and numerical reconstruction of lensless Fourier holograms in optical metrology," Appl. Opt. 38, 4812-4820 (1999).
[CrossRef]

Yaroslavsky, L.

Yaroslavsky, L. P.

Appl. Opt. (8)

IEEE Trans. Med. Imaging (1)

T. M. Lehmann, C. Gönner, and K. Spitzer, "Survey: interpolation methods in medical image processing," IEEE Trans. Med. Imaging 18, 1049-1075 (1999).
[CrossRef]

Meas. Sci. Technol. (1)

U. Schnars and W. P. O. Jüptner, "Digital recording and numerical reconstruction of holograms," Meas. Sci. Technol. 13, R85-R101 (2002).
[CrossRef]

Opt. Eng. (1)

C. Wagner, W. Osten, and S. Seebacher, "Direct shape measurement by digital wavefront reconstruction and multiwavelength contouring," Opt. Eng. 39, 79-85 (2000).
[CrossRef]

Opt. Lett. (2)

Proc. SPIE (3)

C. Pérez-López and F. M. Santoyo, "Rigid body motion removal by derotating digital holograms," in Fourth Iberoamerican Meeting on Optics and Seventh Latin American Meeting on Optics, Lasers, and Their Applications, V. L. Brudny, S. A. Ledesma, and M. C. Marconi, eds., Proc. SPIE 4419, pp. 226-230 (2001).
[CrossRef]

C. Pérez-López, M. del Socorro Hernández-Montes, and F. Mendoza-Santoyo, "Detection of micromechanical deformation under rigid body displacement using twin-pulsed 3D digital holography," in Eighth International Symposium on Laser Metrology, R. Rodriguez-Vera and F. Mendoza-Santoyo, eds., Proc. SPIE 5776, pp. 757-762 (2005).
[CrossRef]

G. Cristóbal and L. Cohen, "Scale in images," in Advanced Signal Processing Algorithms, Architectures, and Implementations VI, F. T. Luk, ed., Proc. SPIE 2846, pp. 251-261 (1996).
[CrossRef]

Other (2)

J. W. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, 1996).

T. Kreis, Handbook of Holographic Interferometry, 1st ed. (Wiley-VCH, Weinheim, 2004).
[CrossRef]

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Figures (3)

Fig. 1
Fig. 1

Recording geometry of the digital Fourier-holography setup.

Fig. 2
Fig. 2

Schematic view of the processing steps on the reconstructed intensity images.

Fig. 3
Fig. 3

Phase differences Δ ϕ M from the Mellin transforms for various displacements Δ d .

Tables (1)

Tables Icon

Table 1 Scaling Factors and Displacements Evaluated from the Phase Differences of the Mellin Transforms

Equations (29)

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b ˜ ( x , y ) exp ( i 2 π d λ ) × exp [ i π λ d ( x 2 + y 2 ) ] 1 [ h ( ξ , η ) ] ( x λ d , y λ d ) ,
x λ d = x and x λ d = y ,
1 [ h ( ξ , η ) ] ( x , y ) exp ( i 2 π d λ ) exp [ i π λ d ( x 2 + y 2 ) ] × b ˜ ( λ d x , λ d y ) .
I ˜ ( x , y ) = | b ˜ ( λ d x , λ d y ) | 2
Δ ϕ = 2 π Δ d λ + π λ Δ d ( x 2 + y 2 ) + arg [ b ˜ ( λ d 1 x , λ d 1 y ) ] arg [ b ˜ ( λ d 2 x , λ d 2 y ) ] ,
Δ d = d 1 d 2 .
b ˜ ( λ d 2 x , λ d 2 y ) = b ˜ ( a λ d 1 x , a λ d 1 y ) ,
a = d 2 d 1 .
M ( s ) = 0 g ( ξ ) ξ s 1 d ξ .
M a ( s ) = 0 g ( a ξ ) ξ s 1 d ξ = 0 g ( ξ ) ( ξ a ) s 1 d ξ a = a s 0 g ( ξ ) ξ s 1 d ξ = exp ( s   ln   a ) M ( s ) .
M a ( i 2 π f ) = exp ( i 2 π f   ln   a ) M ( i 2 π f ) ,
ξ = exp ( u )
M ( i 2 π f ) = g [ exp ( u ) ] exp ( i 2 π f u ) d u .
ξ = b   exp ( c u ) ,
M ( i 2 π f / c ) = c   exp ( i 2 π f / c   ln   b ) g [ b   exp ( c u ) ] × exp ( i 2 π f u ) d u .
| [ I ( x , y ) ] | 2 ( ξ , η ) = | [ I ( x + x s , y + y s ) ] | 2 ( ξ , η ) ,
| [ I ( x , y ) ] | 2 ( ξ a , η a ) | [ I ( a x , a y ) ] | 2 ( ξ , η ) .
a = 1 a = d 1 d 2 ,
Δ ϕ M = arg [ M ( i 2 π f m / c , i 2 π f n / c ) ] arg [ M a ( i 2 π f m / c , i 2 π f n / c ) ] = 2 π / c   ln   a ( f m + f n ) .
b   exp ( c · 0 ) = N 1 ,
b   exp [ c ( N 1 ) ] = ε ,
b = N 1 and c = 1 N 1   ln   ε N 1 .
Δ f m , Δ f n = 1 N ,
f m = m N and Δ f n = n N .
Δ ϕ M = 2 π N 1 N ( ln   ε N 1 ) 1   ln   α ( m + n ) ,
k ( ξ ) = w B ( ξ ) sin ( π ξ ) π ξ ,
w B ( ξ ) = 0.42323 + 0.49755   cos ( π ξ / r ) + 0.07922   cos ( 2 π ξ / r )
k ( ξ , η ) = w B ( ξ ) w B ( η ) sin ( π ξ ) π ξ sin ( π η ) π η
w F F T ( m , n ) = [ 1 cos ( 2 π N 1 m ) ] [ 1 cos ( 2 π N 1 n ) ]

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