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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  1. J. Kühn, T. Colomb, F. Montfort, F. Charrière, Y. Emery, E. Cuche, P. Marquet, and C. Depeursinge, “Real-time dual-wavelength digital holographic microscopy with a single hologram acquisition,” Opt. Express 15, 7231-7242 (2007).
    [CrossRef] [PubMed]
  2. T. Shimobaba, Y. Sato, J. Miura, M. Takenouchi, and T. Ito, “Real-time digital holographic microscopy using the graphic processing unit,” Opt. Express 16, 11776-11781 (2008).
    [CrossRef] [PubMed]
  3. B. Kemper and G. von Bally, “Digital holographic microscopy for live cell applications and technical inspection,” Appl. Opt. 47, A52-A61 (2008).
    [CrossRef] [PubMed]
  4. P. Langehanenberg, B. Kemper, D. Dirksen, and G. von Bally, “Autofocusing in digital holographic phase contrast microscopy on pure phase objects for live cell imaging,” Appl. Opt. 47, D176-D182 (2008).
    [CrossRef] [PubMed]
  5. E. Cuche, P. Marquet, and C. Depeursinge, “Simultaneous amplitude-contrast and quantitative phase-contrast microscopy by numerical reconstruction of Fresnel off-axis holograms,” Appl. Opt. 38, 6994-7001 (1999).
    [CrossRef]
  6. L. Yu and M. K. Kim, “Wavelength-scanning digital interference holography for tomographic three-dimensional imaging by use of the angular spectrum method,” Opt. Lett. 30, 2092-2094 (2005).
    [CrossRef] [PubMed]
  7. I. Yamaguchi and T. Zhang, “Phase-shifting digital holography,” Opt. Lett. 22, 1268-1270 (1997).
    [CrossRef] [PubMed]
  8. T. Colomb, J. Kühn, F. Charrière, C. Depeursinge, P. Marquet, and N. Aspert, “Total aberrations compensation in digital holographic microscopy with a reference conjugated hologram,” Opt. Express 14, 4300-4306 (2006).
    [CrossRef] [PubMed]
  9. D. Lebrun, S. Belaïd, and C. Özkul, “Hologram reconstruction by use of optical wavelet transform,” Appl. Opt. 38, 3730-3734 (1999).
    [CrossRef]
  10. J. Zhong and J. Weng, “Phase retrieval of optical fringe patterns from the ridge of a wavelet transform,” Opt. Lett. 30, 2560-2562 (2005).
    [CrossRef] [PubMed]
  11. M. Liebling, T. Blu, and M. Unser, “Fresnelets: new multiresolution wavelet bases for digital holography,” IEEE Trans. Image Process. 11, 1-14 (2002).
    [CrossRef]
  12. J. Weng, J. Zhong, and C. Hu, “Digital reconstruction based on angular spectrum diffraction with the ridge of wavelet transform in holographic phase-contrast microscopy,” Opt. Express 16, 21971-21981 (2008).
    [CrossRef] [PubMed]
  13. W. L. Anderson and H. Diao, “Two-dimensional wavelet transform and application to holographic particle velocimetry,” Appl. Opt. 34, 249-255 (1995).
    [CrossRef] [PubMed]
  14. 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]
  15. 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]
  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]
  18. A. Z. Abid, M. A. Gdeisat, D. R. Burton, M. J. Lalor, and F. Lilley, “Spatial fringe pattern analysis using the two-dimensional continuous wavelet transform employing a cost function,” Appl. Opt. 46, 6120-6126 (2007).
    [CrossRef] [PubMed]
  19. Zh. Li, B. Gu, and G. Yang,“Slowly varying amplitude approximation appraised by transfer-matrix approach,” Phys. Rev. B 60, 10644-10647 (1999).
    [CrossRef]

2008

2007

2006

2005

2003

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

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

1997

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

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.

Exp. Mech.

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.

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.

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

Opt. Lett.

Phys. Rev. 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]

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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)

Tables Icon

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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