Abstract

When a reconstruction is performed on a digital holographic image that contains small objects at different depths, diffraction that is due to out-of-focus objects disrupts the visibility of the nearby focused objects. We propose a method to substitute for focused object amplitudes other amplitudes that will reduce propagation diffraction effects when other depths are investigated. The replacement amplitudes are computed by use of an algorithm that reduces the highest spatial frequencies of the resultant image. The theoretical aspects of the method are presented, and results for simulated and experimental examples are shown.

© 2005 Optical Society of America

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References

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  1. K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Chap. 5.
    [CrossRef]
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    [CrossRef] [PubMed]
  3. U. Schnars, W. Jüptner, “Direct recording of holograms by a CCD target and numerical reconstruction,” Appl. Opt. 33, 179–181 (1994).
    [CrossRef] [PubMed]
  4. E. Cuche, F. Belivacqua, C. Despeuringe, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
    [CrossRef]
  5. F. Dubois, L. Joannes, J.-C. Legros, “Improved three dimensional imaging with digital holography microscope using a partial spatial coherent source,” Appl. Opt. 38, 7085–7094 (1999).
    [CrossRef]
  6. F. Dubois, O. Monnom, C. Yourassowsky, J.-C. Legros, “Border processing in digital holography by extension of the digital hologram and reduction of higher spatial frequencies,” Appl. Opt. 41, 2621–2626 (2002).
    [CrossRef] [PubMed]
  7. E. Cuche, P. Marquet, C. Despeuringe, “Aperture apodization using cubic spline interpolation: application in digital holography microscopy,” Opt. Commun. 182, 59–69 (2000).
    [CrossRef]
  8. L. Yu, L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. 18, 1033–1045 (2001).
    [CrossRef]
  9. M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
    [CrossRef]
  10. F. Dubois, C. Minetti, O. Monnom, C. Yourassowsky, J.-C. Legros, P. Kischel, “Pattern recognition with a digital holographic microscope working in partially coherent illumination,” Appl. Opt. 41, 4108–4119 (2002).
    [CrossRef] [PubMed]

2002 (2)

2001 (1)

L. Yu, L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. 18, 1033–1045 (2001).
[CrossRef]

2000 (1)

E. Cuche, P. Marquet, C. Despeuringe, “Aperture apodization using cubic spline interpolation: application in digital holography microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

1999 (2)

1997 (1)

1994 (1)

1980 (1)

Belivacqua, F.

Cai, L.

L. Yu, L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. 18, 1033–1045 (2001).
[CrossRef]

Creath, K.

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Chap. 5.
[CrossRef]

Cuche, E.

E. Cuche, P. Marquet, C. Despeuringe, “Aperture apodization using cubic spline interpolation: application in digital holography microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

E. Cuche, F. Belivacqua, C. Despeuringe, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

Despeuringe, C.

E. Cuche, P. Marquet, C. Despeuringe, “Aperture apodization using cubic spline interpolation: application in digital holography microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

E. Cuche, F. Belivacqua, C. Despeuringe, “Digital holography for quantitative phase-contrast imaging,” Opt. Lett. 24, 291–293 (1999).
[CrossRef]

Dubois, F.

Joannes, L.

Jüptner, W.

Kischel, P.

Legros, J.-C.

Marquet, P.

E. Cuche, P. Marquet, C. Despeuringe, “Aperture apodization using cubic spline interpolation: application in digital holography microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

Minetti, C.

Monnom, O.

Nazarathy, M.

Schnars, U.

Shamir, J.

Yamaguchi, I.

Yourassowsky, C.

Yu, L.

L. Yu, L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. 18, 1033–1045 (2001).
[CrossRef]

Zhang, T.

Appl. Opt. (4)

J. Opt. Soc. Am. (2)

M. Nazarathy, J. Shamir, “Fourier optics described by operator algebra,” J. Opt. Soc. Am. 70, 150–159 (1980).
[CrossRef]

L. Yu, L. Cai, “Iterative algorithm with a constraint condition for numerical reconstruction of a three-dimensional object from its hologram,” J. Opt. Soc. Am. 18, 1033–1045 (2001).
[CrossRef]

Opt. Commun. (1)

E. Cuche, P. Marquet, C. Despeuringe, “Aperture apodization using cubic spline interpolation: application in digital holography microscopy,” Opt. Commun. 182, 59–69 (2000).
[CrossRef]

Opt. Lett. (2)

Other (1)

K. Creath, “Phase measurement interferometry techniques,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1988), Chap. 5.
[CrossRef]

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Figures (17)

Fig. 1
Fig. 1

Intensity of the simulated image.

Fig. 2
Fig. 2

Intensity of the propagation performed at d = 400 µm.

Fig. 3
Fig. 3

Selecting values that form a closed loop about a focused object.

Fig. 4
Fig. 4

Values copied in a 64 × 64 blank image.

Fig. 5
Fig. 5

Calculating the remaining values of the image.

Fig. 6
Fig. 6

Calculated values copied in the initial image.

Fig. 7
Fig. 7

Propagation performed identically as for the initial image.

Fig. 8
Fig. 8

Image of beads flowing in water.

Fig. 9
Fig. 9

Image of beads at d = 0 µm.

Fig. 10
Fig. 10

Image of beads at d = 120 µm (no processing).

Fig. 11
Fig. 11

Image of beads at d = 0 µm; the focused beads have been replaced by processed values.

Fig. 12
Fig. 12

Image of the remaining beads at d = 120 µm (with processing).

Fig. 13
Fig. 13

Image of fibrosarcoma cells embedded in three-dimensional collagen.

Fig. 14
Fig. 14

Out-of-focus cell A, to be processed (arrow), above focused cell B.

Fig. 15
Fig. 15

Reconstruction of out-of-focus cell A.

Fig. 16
Fig. 16

Focused cell A is replaced by the processed values.

Fig. 17
Fig. 17

Inverse reconstruction to refocus cell B.

Equations (16)

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x o ( s Δ , t Δ ) = exp ( j k d ) F s , t U , V 1 Q U , V U , V d × F U , V s , t + 1 x i ( s Δ , t Δ ) ,
F ν , η α , β ± 1 = 1 N α = 0 N 1 β = 0 N 1 exp [ j 2 π N ( α ν + β η ) ]
Q U , V U , V d = exp [ j k λ 2 d 2 N 2 Δ 2 ( U 2 + V 2 ) ]
{ ( C + D ) + M ( C + D ) minimum d i = 0 c i 0 , i = 0 N 2 1 ,
F α , β = 1 N exp [ j 2 π N ( s U + t N ) ] .
C = F c , D = F d .
M α , α = { U 2 + V 2 U < N / 2 , V < N / 2 U 2 + ( N V ) 2 U < N / 2 , V N / 2 ( N U ) 2 + V 2 U N / 2 , V < N / 2 ( N U ) 2 + ( N V ) 2 U N / 2 , V N / 2 .
d i = j = 0 N 2 1 F i , j + D j = 0 , c i 0 , i = 0 N 2 1 ,
{ ( C + D ) + M ( C + D ) minimum E + D = 0 .
E r T D r + E i T D i = 0 , E r T D i E i T D r = 0 ,
T = ( C + D ) + M ( C + D ) Γ r T ( E r T D r + E i T D i ) Γ i T ( E r T D i E i T D r ) ,
T ( D r α ) = 0 , T ( D i α ) = 0.
2 M ( C r + D r ) ( E r Γ r E i Γ i ) = 0 , 2 M ( C i + D i ) ( E i Γ i + E r Γ r ) = 0 .
2 M ( C + D ) E Γ = 0 ,
D = ½ M 1 E Γ C .
D = M 1 E ( E + M 1 E ) 1 E + C C .

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