Abstract

A numerical reconstruction technique of digital holography based on angular spectrum diffraction by means of the ridge of Gabor wavelet transform (GWT) is presented. Appling the GWT, the object wave can be reconstructed by calculating the wavelet coefficients of the hologram at the ridge of the GWT automatically even if the spectrum of the virtual image is disturbed by the other spectrum. It provides a way to eliminate the effect of the zero-order and the twin-image terms without the spatial filtering. In particular, based on the angular spectrum theory, GWT is applied to the digital holographic phase-contrast microscopy on biological specimens. The theory, the results of a simulation and an experiment of an onion specimen are shown.

© 2008 Optical Society of America

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References

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2008 (2)

B. Kemper and G. von Bally, "Digital holographic microscopy for live cell applications and technical inspection," Appl. Opt. 47, A52-A61 (2008).
[CrossRef] [PubMed]

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, 176-182 (2008).
[CrossRef]

2006 (3)

2005 (4)

2004 (2)

2003 (1)

A. Cesar and K. Taeeeui, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

2002 (1)

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Proc. 11, 1-14 (2002).

2001 (1)

H. Jeong, "Analysis of plate wave propagation in anisotropic laminates using a wavelet transform," NDT & E Int. 34, 185-190(2001).
[CrossRef]

1997 (1)

Aspert, N.

Blu, T.

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Proc. 11, 1-14 (2002).

Carl, D.

Cesar, A.

A. Cesar and K. Taeeeui, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Charrière, F.

Colomb, T.

Cuche, E.

Depeursinge, C.

Dirksen, D.

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, 176-182 (2008).
[CrossRef]

Emery, Y.

Jeong, H.

H. Jeong, "Analysis of plate wave propagation in anisotropic laminates using a wavelet transform," NDT & E Int. 34, 185-190(2001).
[CrossRef]

Kemper, B.

Kim, M. K.

Kühn, J.

Langehanenberg, P.

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, 176-182 (2008).
[CrossRef]

Liebling, M.

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Proc. 11, 1-14 (2002).

Lo, C.

Magistretti, P. J.

Mann, C. J.

Marquet, P.

Montfort, F.

Rappaz, B.

Taeeeui, K.

A. Cesar and K. Taeeeui, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Unser, M.

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Proc. 11, 1-14 (2002).

von Bally, G.

Weng, J.

Wernicke, G.

Yamaguchi, I.

Yu, L.

Zhang, T.

Zhong, J.

Appl. Opt. (5)

IEEE Trans. Image Proc. (1)

M. Liebling, T. Blu, and M. Unser, "Fresnelets: new multiresolution wavelet bases for digital holography," IEEE Trans. Image Proc. 11, 1-14 (2002).

J. Opt. Soc. Am. A (1)

NDT & E Int. (1)

H. Jeong, "Analysis of plate wave propagation in anisotropic laminates using a wavelet transform," NDT & E Int. 34, 185-190(2001).
[CrossRef]

Opt. Eng. (1)

A. Cesar and K. Taeeeui, "Determination of strains from fringe patterns using space-frequency representations," Opt. Eng. 42, 3182-3193 (2003).
[CrossRef]

Opt. Express (2)

Opt. Lett. (4)

Other (1)

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

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Figures (6)

Fig. 1.
Fig. 1.

(a) the Gabor wavelet and (b) its Fourier transform.

Fig. 2.
Fig. 2.

Holography of a simulation phase object of 256×256 pixels. (a) the amplitude and (b) the phase of the object; (c) the hologram; (d) the amplitude and (e) the unwrapped phase of the reconstructed wave by the SFTF; (f) the modulus of the wavelet coefficients at the 120th row; (g) the amplitude and (h) the unwrapped phase of the reconstructed wave by the GWT; (i) the simulated phase and the phase at the 120th row by the SFTF and the GWT; (j) the error at the 120th row by the SFTF and the GWT; (k) the first and second derivative of the simulated phase at the 120th row.

Fig. 3.
Fig. 3.

Apparatus for digital holography experiment.

Fig. 4.
Fig. 4.

Hologram. (a) the image of the onion specimen; (b), (c) are the respectively hologram with and without the onion specimen; (d) the spectrum of the hologram with the onion specimen on the logarithmic coordinates, where the small black box represents the square filter of size 15 pixels, the big black box represents the square filter of size 30 pixels and the white line window represents the manual spatial filter.

Fig. 5.
Fig. 5.

The reconstructed wave by the analysis of the SFTF. (a) the amplitude and (b) the phase of that with the square filter of size 15 pixels; (c) the amplitude and (d) the phase of that with the square filter of size 30 pixels; (e) the amplitude and (f) the phase of that with the manual spatial filter.

Fig. 6.
Fig. 6.

Holography of an onion specimen by the analysis of the GWT. (a) the modulus of the GWT at the 256th row of the hologram; (b) the spectrum, (c) the amplitude, (d) the wrapped phase and (e) the unwrapped phase of the reconstructed wave.

Equations (21)

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W ( a , b ) = f ( x ) ψ a , b * ( x ) d x = < f ( x ) , ψ a , b ( x ) >
ψ a , b ( x ) = 1 a ψ ( x b a )
{ ψ ( x ) = 1 π 4 2 π γ exp [ ( 2 π γ ) 2 x 2 2 + j 2 π x ] ψ ( ω ) = 2 π π 4 γ 2 π exp [ ( γ 2 π ) 2 2 ( ω 2 π ) 2 ]
W ( a , b ) = [ imag ( W ( a , b ) ) ] 2 + [ real ( W ( a , b ) ) ] 2
O ( x , y ) = o ( x , y ) · exp [ j ϕ ( x , y ) ]
R ( x , y ) = R 0 · exp [ j 2 π sin θ λ x ]
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 sin θ λ x + ϕ ( x , y ) ] }
+ o ( x , y ) R 0 exp { j [ 2 π sin θ λ x + ϕ ( x , y ) ] }
I ( x ) = A ( x ) + R 0 o ( x ) exp { j φ ( x ) } + R 0 o ( x ) exp { j φ ( x ) }
φ ( x ) = [ 2 π sin θ λ b + ϕ ( b ) ] + [ 2 π sin θ λ + ϕ ( b ) ] ( x b ) + ϕ ( b ) 2 ! ( x b ) 2 +
φ ( x ) [ 2 π sin θ λ b + ϕ ( b ) ] + [ 2 π sin θ λ + ϕ ( b ) ] ( x b )
W ( a , b ) = I ( x ) ψ a , b * ( x ) d x = W 1 ( a , b ) + W 2 ( a , b ) + W 3 ( a , b )
{ W 1 ( a , b ) = γ 2 π 4 A exp ( γ 2 ) W 2 ( a , b ) = γ 4 π 4 R 0 o exp { γ 2 [ ( sin θ λ + ϕ ( b ) 2 π ) a 1 ] 2 } exp { j [ 2 π sin θ λ b + ϕ ( b ) ] } W 3 ( a , b ) = γ 4 π 4 R 0 o exp { γ 2 [ ( sin θ λ + ϕ ( b ) 2 π ) a + 1 ] 2 } exp { j [ 2 π sin θ λ b + ϕ ( b ) ] }
a = 1 sin θ λ + ϕ ( b ) 2 π
W ridge ( b ) = γ 2 π 4 A exp ( γ 2 ) + γ 4 π 4 R 0 o exp { j [ 2 π sin θ λ b + ϕ ( b ) ] }
+ γ 4 π 4 R 0 o exp ( 4 γ 2 ) exp { j [ 2 π sin θ λ b + ϕ ( b ) ] }
W ridge ( x ) = γ 4 π 4 R 0 o exp { j [ 2 π sin θ λ x + ϕ ( x ) ] }
U ridge ( x ) = W ridge ( x ) R 0 exp [ j 2 π sin θ λ x ] = γ 4 π 4 R 0 2 o exp [ j ϕ ( x ) ]
A ( ξ , η ; z ) = A ( ξ , η , 0 ) · exp [ j 2 π z λ 1 ( λ ξ ) 2 ( λ η ) 2 ]
ϕ ( x , y ) = { 1 5 ( 1 x ) 2 exp [ x 2 ( y + 1 ) 2 ] 2 3 ( x 5 x 3 y 5 ) exp [ x 2 y 2 ] 1 45 exp [ ( x + 1 ) 1 y 2 ] where x = m 40 , y = n 40 , ( m 120 ) 2 + ( n 120 ) 2 40 2 ( m , n Z ) 0 others

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