Abstract

In-line holograms of particle fields are rich in their data content, and therefore analysis of such holograms is complex. Conventionally, such analysis is carried out by focusing on the particles in the reconstructed three-dimensional image of the original volume. This task is tedious, even if it is carried out by automated means. Analysis methods that eliminate the need for focusing are far more efficient. In the presented method a signal-processing approach, Wigner analysis, is applied directly to the holograms without any reconstruction. The Wigner distribution, which is a space-frequency representation, gives the three-dimensional locations of small particles quite accurately. The proposed method is implemented digitally, and the results are shown.

© 1992 Optical Society of America

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References

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  1. G. Haussmann, W. Lauterborn, “Determination of size and position of fast moving gas bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
    [Crossref] [PubMed]
  2. R. Bexon, J. Gibbs, G. D. Bishop, “Automatic assessment of aerosol holograms,”J. Aerosol Sci.7, 397–407 (1976).
    [Crossref]
  3. L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
    [Crossref]
  4. G. Liu, P. D. Scott, “Phase retrieval and twin-image elimination for in-line Fresnel holograms,” J. Opt. Soc. Am. A 4, 159–165 (1987).
    [Crossref]
  5. H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
    [Crossref]
  6. C. S. Vikram, M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149–153 (1984).
    [Crossref]
  7. C. S. Vikram, M. L. Billet, “On the problem of automated analysis of particle holograms: proposal for direct diffraction measurements without holography,” Optik 73, 160–162 (1986).
  8. T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).
  9. T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).
  10. W. Mecklenbrauker, “A tutorial on non-parametric bilinear time-frequency signal representations,” in Signal Processing, Les Houches Session 45, J. L. Lacoume, T. S. Durrani, R. Stora, eds. (North-Holland, Amsterdam, 1987), pp. 277–327.
  11. T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The aliasing problem in discrete-time Wigner distributions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1067–1072 (1983).
    [Crossref]
  12. B. Boashash, B. Escudie, “Wigner-Ville analysis of asymptotic signals,” Signal Process. 8, 315–327 (1985).
    [Crossref]
  13. B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,”IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
    [Crossref]
  14. M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).
  15. G. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
    [Crossref]
  16. H. H. Szu, “Two-dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).
    [Crossref]

1988 (1)

B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,”IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
[Crossref]

1987 (2)

1986 (1)

C. S. Vikram, M. L. Billet, “On the problem of automated analysis of particle holograms: proposal for direct diffraction measurements without holography,” Optik 73, 160–162 (1986).

1985 (2)

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
[Crossref]

B. Boashash, B. Escudie, “Wigner-Ville analysis of asymptotic signals,” Signal Process. 8, 315–327 (1985).
[Crossref]

1984 (1)

C. S. Vikram, M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149–153 (1984).
[Crossref]

1983 (1)

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The aliasing problem in discrete-time Wigner distributions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1067–1072 (1983).
[Crossref]

1982 (1)

H. H. Szu, “Two-dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).
[Crossref]

1980 (3)

G. Haussmann, W. Lauterborn, “Determination of size and position of fast moving gas bubbles in liquids by digital 3-D image processing of hologram reconstructions,” Appl. Opt. 19, 3529–3535 (1980).
[Crossref] [PubMed]

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

1976 (1)

G. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[Crossref]

Bexon, R.

R. Bexon, J. Gibbs, G. D. Bishop, “Automatic assessment of aerosol holograms,”J. Aerosol Sci.7, 397–407 (1976).
[Crossref]

Billet, M. L.

C. S. Vikram, M. L. Billet, “On the problem of automated analysis of particle holograms: proposal for direct diffraction measurements without holography,” Optik 73, 160–162 (1986).

C. S. Vikram, M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149–153 (1984).
[Crossref]

Bishop, G. D.

R. Bexon, J. Gibbs, G. D. Bishop, “Automatic assessment of aerosol holograms,”J. Aerosol Sci.7, 397–407 (1976).
[Crossref]

Boashash, B.

B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,”IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
[Crossref]

B. Boashash, B. Escudie, “Wigner-Ville analysis of asymptotic signals,” Signal Process. 8, 315–327 (1985).
[Crossref]

Born, M.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Caulfield, H. J.

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
[Crossref]

Claasen, T. A. C. M.

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The aliasing problem in discrete-time Wigner distributions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1067–1072 (1983).
[Crossref]

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

Escudie, B.

B. Boashash, B. Escudie, “Wigner-Ville analysis of asymptotic signals,” Signal Process. 8, 315–327 (1985).
[Crossref]

Gibbs, J.

R. Bexon, J. Gibbs, G. D. Bishop, “Automatic assessment of aerosol holograms,”J. Aerosol Sci.7, 397–407 (1976).
[Crossref]

Haussmann, G.

Lauterborn, W.

Liu, G.

Mecklenbrauker, W.

W. Mecklenbrauker, “A tutorial on non-parametric bilinear time-frequency signal representations,” in Signal Processing, Les Houches Session 45, J. L. Lacoume, T. S. Durrani, R. Stora, eds. (North-Holland, Amsterdam, 1987), pp. 277–327.

Mecklenbrauker, W. F. G.

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The aliasing problem in discrete-time Wigner distributions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1067–1072 (1983).
[Crossref]

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

Onural, L.

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[Crossref]

Scott, P. D.

Szu, H. H.

H. H. Szu, “Two-dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).
[Crossref]

Thompson, B. J.

G. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[Crossref]

Tyler, G.

G. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[Crossref]

Vikram, C. S.

C. S. Vikram, M. L. Billet, “On the problem of automated analysis of particle holograms: proposal for direct diffraction measurements without holography,” Optik 73, 160–162 (1986).

C. S. Vikram, M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149–153 (1984).
[Crossref]

Wolf, E.

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

Appl. Opt. (1)

Appl. Phys. B (1)

C. S. Vikram, M. L. Billet, “Far-field holography at non-image planes for size analysis of small particles,” Appl. Phys. B 33, 149–153 (1984).
[Crossref]

IEEE Trans. Acoust. Speech Signal Process. (2)

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The aliasing problem in discrete-time Wigner distributions,”IEEE Trans. Acoust. Speech Signal Process. ASSP-31, 1067–1072 (1983).
[Crossref]

B. Boashash, “Note on the use of the Wigner distribution for time-frequency analysis,”IEEE Trans. Acoust. Speech Signal Process. 36, 1518–1521 (1988).
[Crossref]

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

Opt. Acta (1)

G. Tyler, B. J. Thompson, “Fraunhofer holography applied to particle size analysis: a reassessment,” Opt. Acta 23, 685–700 (1976).
[Crossref]

Opt. Eng. (3)

H. H. Szu, “Two-dimensional optical processing of one-dimensional acoustic data,” Opt. Eng. 21, 804–813 (1982).
[Crossref]

L. Onural, P. D. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
[Crossref]

H. J. Caulfield, “Automated analysis of particle holograms,” Opt. Eng. 24, 462–463 (1985).
[Crossref]

Optik (1)

C. S. Vikram, M. L. Billet, “On the problem of automated analysis of particle holograms: proposal for direct diffraction measurements without holography,” Optik 73, 160–162 (1986).

Philips J. Res. (2)

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 1: continuous-time signals,” Philips J. Res. 35, 217–250 (1980).

T. A. C. M. Claasen, W. F. G. Mecklenbrauker, “The Wigner distribution—a tool for time-frequency signal analysis. Part 2: discrete-time signals,” Philips J. Res. 35, 276–300 (1980).

Signal Process. (1)

B. Boashash, B. Escudie, “Wigner-Ville analysis of asymptotic signals,” Signal Process. 8, 315–327 (1985).
[Crossref]

Other (3)

M. Born, E. Wolf, Principles of Optics (Pergamon, New York, 1975).

W. Mecklenbrauker, “A tutorial on non-parametric bilinear time-frequency signal representations,” in Signal Processing, Les Houches Session 45, J. L. Lacoume, T. S. Durrani, R. Stora, eds. (North-Holland, Amsterdam, 1987), pp. 277–327.

R. Bexon, J. Gibbs, G. D. Bishop, “Automatic assessment of aerosol holograms,”J. Aerosol Sci.7, 397–407 (1976).
[Crossref]

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Figures (8)

Fig. 1
Fig. 1

In-line hologram recording model: The input object distribution is 2-D for the sake of simplicity. The 3-D model is obtained by superposing the fields corresponding to many 2-D object distributions located at different depths.

Fig. 2
Fig. 2

Space-frequency pattern of the kernel g(x) exhibited by Wg(x, ω). The undesired cross term is ignored.

Fig. 3
Fig. 3

Discrete computation of the hologram, IzD(n, m), of the 2-D input object distribution, azD(n, m), located at a distance z from the hologram plane (DFT, discrete Fourier transform; IDFT, inverse discrete Fourier transform.

Fig. 4
Fig. 4

(a) Simulated 1-D hologram: The object is 5 pixels wide, and its center is located at 128. The hologram size (the horizontal axis) is 512; 0 is at the left, and 511 is at the right. There is no variation along the vertical direction. The value of the distance parameter a is 1. (b) Result of Wigner analysis of the hologram in (a): The horizontal axis is n and the vertical axis is (π/N)m. n ∈ [0, 511], m ∈ [−255, 256], and N = 512. Since everything is discrete and normalized, there is no need for physical units.

Fig. 5
Fig. 5

Result of Wigner analysis of a 1-D hologram: There are three particles in this hologram, located at the intersection points of the three bright cross shapes. The depth is the same for all three particles.

Fig. 6
Fig. 6

Wigner analysis of a 1-D hologram with two particles located at different depths: The one at the left is closer to the hologram plane. The exact depths can be easily computed from the slopes of the crosses.

Fig. 7
Fig. 7

Simulations for 2-D holograms: (a) Three 2-D objects located at the same depth but at different (x, y) coordinates. (b) Simulated hologram of the object distribution given in (a); the bright line shows the location of a 1-D horizontal profile of this hologram. (c) Result of the Wigner analysis of the 1-D profile of (b). Note that the intersections of the crosses coincide with the particle locations. Since the depth of the particles are the same, the slopes of the crosses are also the same. (d) Vertical profile location. (e) Wigner analysis of the profile of (d); the left-hand side of this figure corresponds to the top of the vertical profile.

Fig. 8
Fig. 8

Application of the Wigner analysis to an optical hologram: (a) Portion of an optical hologram of a hairlike object. This portion is digitized. (b) Result of the Wigner analysis applied to a horizontal profile of the hologram of (a). The normalized depth variable a is found to be 0.851 from the slopes. (c) Digital reconstruction of the hologram focusing on to the computed depth. As a result of the accurate depth information from the analysis of (b) there was no need to search for the correct depth.

Equations (39)

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ψ z ( x , y ) = B j λ z exp ( j 2 π λ z ) - - [ 1 - a ( ξ , η ) ] × exp { j π λ z [ ( x - ξ ) 2 + ( y - η ) 2 ] } d ξ d η ,
ψ z ( x , y ) = B exp ( j 2 π λ z ) { [ 1 - a ( x , y ) ] * * h z ( x , y ) } ,
h z ( x , y ) = 1 j λ z exp [ j π λ z ( x 2 + y 2 ) ] .
ψ z ( x , y ) = [ 1 - a ( x , y ) ] * * ( h z ( x , y ) .
ψ z ( x ) = B exp ( j 2 π λ z ) - [ 1 - a ( ξ ) ] 1 ( λ z ) 1 / 2 × exp { j [ π λ z ( x - ξ ) 2 - π 4 ] } d ξ .
ψ z ( x ) = [ 1 - a ( x ) ] * h 2 ( x ) ,
h z ( x ) = 1 ( λ z ) 1 / 2 exp [ j ( π λ z x 2 - π 4 ) ] .
W f , g ( t , ω ) = - f ( t + τ 2 ) g * ( t - τ 2 ) exp ( - j ω τ ) d τ .
W f , g ( n , θ ) = 2 k = - exp ( - j 2 k θ ) f ( n + k ) g * ( n - k ) ,
W f ( n , θ ) W f ( n , θ + π ) ,             n , θ .
W f + g ( m , θ ) = W f ( n , θ ) + W g ( n , θ ) + 2 Re { W f , g ( n , θ ) } .
f ( n ) = 0 ,             n < n a or n > n b ,
W f ( n , θ ) = 0 ,             n < n a or n > n b .
I z ( x ) = 1 - a ( x ) * h z ( x ) - a * ( x ) * h z * ( x ) .
I z ( x ) = 1 - a ( x ) * 2 Re { h z ( x ) } = 1 - a ( x ) * g ( x ) ,
g ( x ) = 2 Re { h z ( x ) } = 2 ( λ z ) 1 / 2 cos ( π λ z x 2 - π 4 ) .
W f ( t , ω ) = 2 π δ ( ω - α t ) .
W g ( x , ω ) = 2 π λ z δ ( ω - 2 π λ z x ) + 2 π λ z δ ( ω + 2 π λ z x ) + 2 2 ( λ z ) 1 / 2 cos ( 2 π λ z x 2 - λ z 2 π ω 2 - π 4 ) .
W J ( x , ω ) = - W a ( α , ω ) W g ( x - α , ω ) d α .
W J ( x , ω ) = 2 W g ( x - x 0 , ω ) .
a ( x ) = δ ( x - x 1 ) + δ ( x - x 2 ) .
W a ( x , ω ) = 2 δ ( x - x 1 ) + 2 δ ( x - x 2 ) + 4 × cos [ ω ( x 2 - x 1 ) ] δ ( x - x 1 + x 2 2 ) .
W J ( x , ω ) = 2 W g ( x - x 1 , ω ) + 2 W g ( x - x 2 , ω ) + 4 × cos [ ω ( x 2 - x 1 ) ] W g ( x - x 1 + x 2 2 , ω ) ,
a ( x ) = i = 1 n δ ( x - x i ) ,
W a ( x , ω ) = 2 i = 1 n δ ( x - x i ) + 4 i = 1 n j = i + 1 n cos [ ω ( x - x i ) ] × δ ( x - x i + x j 2 ) .
W f ( x , y , μ , ν ) = - - f ( x + α / 2 , y + β / 2 ) × f * ( x - α / 2 , y - β / 2 ) × exp [ - j ( α μ + β ν ) ] d α d β .
W J ( n , π N m ) = 2 k = - exp ( - j 2 π N k m ) J ( n + k ) J ( n - k ) ,
W J ( n , π N m ) = 2 k = - N / 2 + 1 N / 2 - 1 y n ( k ) exp ( - j 2 π N k m ) ,
y n ( k ) = J ( n + k ) J ( n - k )
W J ( n , π N m ) = 2 k = 0 N / 2 - 1 y n ( k ) exp ( - j 2 π N k m ) + 2 × k = N / 2 + 1 N - 1 y n ( k - N ) exp ( - j 2 π N k m ) .
z n ( k ) = { 2 y n ( k ) 0 k N / 2 - 1 0 k = N / 2 2 y n ( k - N ) N / 2 + 1 k N - 1 ,
W J ( n , π N m ) = 2 k = 0 N - 1 z n ( k ) exp ( - j 2 π N k m ) ,
W ( m ) = k = 0 N - 1 z n ( k ) exp ( - j 2 π N k m ) , m { 0 , 1 , , N - 1 } ,
W J ( n , π N m ) = { W ( m ) 0 m N / 2 W ( m + N ) - N / 2 + 1 m - 1 .
h z D ( n ) = K exp [ j ( π λ z X 2 n 2 - π 4 ) ] ,             n = 0 , , 511 ,
ψ z D ( n ) = [ 1 - a D ( n ) ] * h z D ( n ) ,
α 2 π N = π λ z X 2 .
h z D ( n ) = K exp [ j ( α 2 π N n 2 - π 4 ) ] .
H z D ( k ) = { exp [ - j ( 1 / α 2 ) ( π / N ) k 2 0 k N / 2 exp [ - j ( 1 / α 2 ) ( π / N ) ( N - k ) 2 ] N / 2 + 1 k N - 1 0 else .

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