Abstract

We describe a numerical reconstruction technique for digital holography by means of the two- dimensional Gabor wavelet transform (2D-GWT). Applying the 2D-GWT to digital holography, the object wave can be reconstructed by calculating the wavelet coefficients of the hologram at the peak of the 2D-GWT automatically. At the same time the effect of the zero-order diffraction image and the twin image are eliminated without spatial filtering. Comparing the numerical reconstruction of a holographic image by the analysis of the one-dimensional Gabor wavelet transform (1D-GWT) with the 2D-GWT, we show that the 2D-GWT method is superior to the 1D-GWT method, especially when the fringes of the hologram are not just along the y direction. The theory and the results of a simulation and experiments are shown.

© 2009 Optical Society of America

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    [CrossRef]
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    [CrossRef]
  16. Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram and processing,” Opt. Eng. 45, 1-5 (2006).
    [CrossRef]
  17. M. A. Gdeisat, D. R. Burton, and M. J. Lalor, “Spatial carrier fringe pattern demodulation by use of a two-dimensional continuous wavelet transform,” Appl. Opt. 45, 8722-8732(2006).
    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
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    [CrossRef]

2008 (4)

2007 (2)

2006 (3)

2005 (2)

2003 (1)

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

2002 (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 11, 1-14 (2002).
[CrossRef]

1999 (3)

1997 (2)

I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268-1270 (1997).
[CrossRef] [PubMed]

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).
[CrossRef]

1995 (1)

Abid, A. Z.

Anderson, W. L.

Aspert, N.

Belaïd, S.

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730-3734 (1999).
[CrossRef]

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).
[CrossRef]

Blu, T.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 11, 1-14 (2002).
[CrossRef]

Burton, D. R.

Charrière, F.

Colomb, T.

Cuche, E.

Depeursinge, C.

Diao, H.

Dirksen, D.

Emery, Y.

Gdeisat, M. A.

Gu, B.

Zh. Li, B. Gu, and G. Yang,“Slowly varying amplitude approximation appraised by transfer-matrix approach,” Phys. Rev. B 60, 10644-10647 (1999).
[CrossRef]

Hu, C.

Ito, T.

Kadooka, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Kemper, B.

Kim, M. K.

Kühn, J.

Kunoo, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Lalor, M. J.

Langehanenberg, P.

Lebrun, D.

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730-3734 (1999).
[CrossRef]

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).
[CrossRef]

Li, Zh.

Zh. Li, B. Gu, and G. Yang,“Slowly varying amplitude approximation appraised by transfer-matrix approach,” Phys. Rev. B 60, 10644-10647 (1999).
[CrossRef]

Liebling, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 11, 1-14 (2002).
[CrossRef]

Lilley, F.

Ma, H.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram and processing,” Opt. Eng. 45, 1-5 (2006).
[CrossRef]

Marquet, P.

Miura, J.

Montfort, F.

Nagayasu, T.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Ono, K.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Özkul, C.

D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730-3734 (1999).
[CrossRef]

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).
[CrossRef]

Sato, Y.

Shimobaba, T.

Takenouchi, M.

Uda, N.

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

Unser, M.

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 11, 1-14 (2002).
[CrossRef]

von Bally, G.

Wang, Z.

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram and processing,” Opt. Eng. 45, 1-5 (2006).
[CrossRef]

Weng, J.

Yamaguchi, I.

Yang, G.

Zh. Li, B. Gu, and G. Yang,“Slowly varying amplitude approximation appraised by transfer-matrix approach,” Phys. Rev. B 60, 10644-10647 (1999).
[CrossRef]

Yu, L.

Zhang, T.

Zhong, J.

Appl. Opt. (7)

Exp. Mech. (1)

K. Kadooka, K. Kunoo, N. Uda, K. Ono, and T. Nagayasu, “Strain analysis for moiré interferometry using the two-dimensional continuous wavelet transform,” Exp. Mech. 43, 45-51 (2003).
[CrossRef]

IEEE Trans. Image Process. (1)

M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 11, 1-14 (2002).
[CrossRef]

Opt. Eng. (2)

S. Belaïd, D. Lebrun, and C. Özkul, “Application of two-dimensional wavelet transform to hologram analysis: visualization of glass fibers in a turbulent flame,” Opt. Eng. 36, 1947-1951 (1997).
[CrossRef]

Z. Wang and H. Ma, “Advanced continuous wavelet transform algorithm for digital interferogram and processing,” Opt. Eng. 45, 1-5 (2006).
[CrossRef]

Opt. Express (4)

Opt. Lett. (3)

Phys. Rev. B (1)

Zh. Li, B. Gu, and G. Yang,“Slowly varying amplitude approximation appraised by transfer-matrix approach,” Phys. Rev. B 60, 10644-10647 (1999).
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Holography of a simulation phase object of 256 × 256 pixels: (a) the amplitude and (b) the phase of the object, (c) the digital hologram generated by the computer, and (d) the spectrum on the logarithmic coordinates.

Fig. 2
Fig. 2

Analysis by the 2D-GWT: (a) the spectrum on the logarithmic coordinates, (b) the amplitude, and (c) the unwrapped phase of the reconstructed wave.

Fig. 3
Fig. 3

Reconstructed phase at the 128th row: (a) the simulated phase and the reconstructed phase at the 128th row by the analysis of the 2D-GWT, (b) the error at the 128th row by the analysis of the 2D-GWT.

Fig. 4
Fig. 4

Apparatus for digital holography experiment.

Fig. 5
Fig. 5

USAF 1951.

Fig. 6
Fig. 6

(a) Hologram and (b) its spectrum on the logarithmic coordinates.

Fig. 7
Fig. 7

Holography analyzed by the 1D-GWT: (a) the spectrum on the logarithmic coordinates, (b) the amplitude, and (c) the unwrapped phase of the reconstructed wave.

Fig. 8
Fig. 8

Holography analyzed by the 2D-GWT: (a) the spectrum on the logarithmic coordinates, (b) the amplitude and (c) the unwrapped phase of the reconstructed wave, and (d) the unwrapped phase in the three-dimensional view.

Fig. 9
Fig. 9

Holography of an onion specimen: (a) the hologram with an onion specimen; the wrapped phase of the reconstructed wave by the analysis of (b) the 1D-GWT and (c) the 2D-GWT corresponding to the local interference pattern inside the black box.

Fig. 10
Fig. 10

Holography of an onion specimen: (a) the hologram with the onion specimen [12], (b) the amplitude and (c) the unwrapped phase of the reconstructed wave by the analysis of the 1D-GWT [12], and (d) the amplitude and (e) the unwrapped phase of the reconstructed wave by the analysis of the 2D-GWT.

Tables (1)

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Table 1 Modulus of the Wavelet Coefficients at the Position (128,128)

Equations (23)

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W f ( s , θ , a , b ) = f ( x , y ) ψ s , θ * ( x , y , a , b ) d x d y ,
ψ s , θ ( x , y , a , b ) = 1 s ψ ( x a s , x b s , θ ) ,
ψ ( x , y ) = 1 π 4 2 π γ exp [ ( 2 π / γ ) 2 ( x 2 + y 2 ) 2 + j 2 π ( x + y ) ] ,
ψ s , θ ( x , y , a , b ) = 1 π 4 2 π γ exp { ( 2 π / γ ) 2 [ ( x a ) 2 + ( y b ) 2 ] 2 s 2 } exp { j 2 π ( x a ) cos θ + ( y b ) sin θ s } .
| W f ( s , θ , a , b ) | = [ imag ( W f ( s , θ , a , b ) ) ] 2 + [ real ( W f ( s , θ , a , b ) ) ] 2 ,
ψ ( s , θ , a , b ) = arctan { imag [ W f ( s , θ , a , b ) ] real [ W f ( s , θ , a , b ) ] } ,
O ( x , y ) = o ( x , y ) exp [ j ϕ ( x , y ) ] ,
R ( x , y ) = R 0 · exp [ j 2 π λ ( x cos α + y cos β ) ] ,
I ( x , y ) = R R * + O O * + O R * + O * R = | R 0 | 2 + | o ( x , y ) | 2 + o ( x , y ) R 0 exp { j [ 2 π λ ( x cos α + y cos β ) + ϕ ( x , y ) ] } + o ( x , y ) R 0 exp { j [ 2 π λ ( x cos α + y cos β ) + ϕ ( x , y ) ] } .
φ ( x , y ) = 2 π λ ( x cos α + y cos β ) + ϕ ( x , y ) .
I ( x , y ) = A ( x , y ) + o ( x , y ) R 0 exp { j φ ( x , y ) } + o ( x , y ) R 0 exp { j φ ( x , y ) } .
φ ( x , y ) = φ ( a , b ) + [ ( x a ) x + ( y b ) y ] φ ( a , b ) + 1 2 ! [ ( x a ) x + ( y b ) y ] 2 φ ( a , b ) + ,
φ ( x , y ) = φ ( a , b ) + [ ( x a ) x + ( y b ) y ] φ ( a , b ) .
{ f x = 1 2 π [ φ ( a , b ) ] x f y = 1 2 π [ φ ( a , b ) ] y .
{ f x = cos α T f y = cos β T = sin α T ,
φ ( x , y ) = φ ( a , b ) + 2 π ( x a ) cos α + ( y b ) sin α T .
I ( x , y ) = A ( x , y ) + o ( x , y ) R 0 exp { j [ 2 π ( x a ) cos α + ( y b ) sin α T + φ ( a , b ) ] } + o ( x , y ) R 0 exp { j [ 2 π ( x a ) cos α + ( y b ) sin α T + φ ( a , b ) ] } .
W ( s , θ , a , b ) = I ( x , y ) ψ s , θ * ( x , y , a , b ) d x d y = W 1 ( s , θ , a , b ) + W 2 ( s , θ , a , b ) + W 3 ( s , θ , a , b ) .
W 1 ( s , θ , a , b ) = π γ 2 4 4 A exp ( γ 2 ) , W 2 ( s , θ , a , b ) = π γ 2 64 4 o R 0 exp { γ 2 [ ( s T 1 ) 2 + 2 s T ( 1 cos ( α θ ) ) ] } exp { j φ ( a , b ) } , W 3 ( s , θ , a , b ) = π γ 2 64 4 o R 0 exp { γ 2 [ ( s T + 1 ) 2 2 s T ( 1 cos ( α θ ) ) ] } exp { j φ ( a , b ) } .
W peak ( a , b ) = π γ 2 4 4 A exp ( γ 2 ) + π γ 2 64 4 o R 0 exp { j φ ( a , b ) } + π γ 2 64 4 o R 0 exp ( 4 γ 2 ) exp { j φ ( a , b ) } .
W peak ( x , y ) = π γ 2 64 4 o R 0 exp { j [ 2 π λ ( x cos α + y cos β ) + ϕ ( x , y ) ] } .
U peak ( x , y ) = W peak ( x , y ) exp [ j 2 π λ ( x cos α + y cos β ) ] = π γ 2 64 4 R 0 2 o exp [ ϕ ( x , y ) ] .
ϕ ( x , y ) = { ( x 128 ) 2 + ( y 128 ) 2 12 where     ( x 128 ) 2 + ( y 128 ) 2 96 2 ( x , y Z + ) 0 others .

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