Abstract

The equations to reconstruct an image plane from a hologram are developed. This development is carried out for planes parallel to the hologram, which allows fast computation through the use of fast Fourier transforms. Algorithms for a digital computer are developed so images can be reconstructed, both with and without the Fresnel approximation, from a digitized hologram without the need for three-dimensional optical reconstruction equipment. Examples of holographically recorded images of marine micro-organisms are shown. A computational method for counting the number of micro-organisms in the holographically recorded volume is developed, and an example is provided.

© 2002 Optical Society of America

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References

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  1. J. R. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).
  2. P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge U. Press, Cambridge, England, 1996).
    [CrossRef]
  3. E. Malkiel, O. Alquaddoomi, J. Katz, “Measurements of plankton distribution in the ocean using submersible holography,” Meas. Sci. Technol. 10, 1142–1152 (1999).
    [CrossRef]
  4. J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–3819 (1997).
    [CrossRef]
  5. Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
    [CrossRef]
  6. E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
    [CrossRef]
  7. U. Schnars, T. Kreis, W. Jueptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
    [CrossRef]
  8. T. Zhang, I. Yamaguchi, “Three-dimensional microscopy with phase-shifting digital holography,” Opt. Lett. 23, 1221–1223 (1998).
    [CrossRef]
  9. R. Owen, A. Zozulya, “In-line holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
    [CrossRef]
  10. S. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing (California Technical, San Diego, Calif., 1997).
  11. C. Vikram, Particle Field Holography (Cambridge U. Press, Cambridge, England, 1992).
    [CrossRef]
  12. W. Li, “Computational reconstruction of images from optical holograms,” M. S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 2001).
  13. J. Katz, P. Donaghay, S. King, K. Russell, “Submersible holocamera for detection of particle characteristics and motions in the ocean,” Deep-Sea Res. 46, 1455–1481 (1999).
    [CrossRef]
  14. L. Onural, P. Scott, “Digital decoding of in-line holograms,” Opt. Eng. 26, 1124–1132 (1987).
    [CrossRef]
  15. Image Processing Toolbox For Use with Matlab (Math Works, Natick, Mass., 1997).

2000 (3)

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

E. Cuche, P. Marquet, C. Depeursinge, “Spatial filtering for zero-order and twin-image elimination in digital off-axis holography,” Appl. Opt. 39, 4070–4075 (2000).
[CrossRef]

R. Owen, A. Zozulya, “In-line holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

1999 (2)

J. Katz, P. Donaghay, S. King, K. Russell, “Submersible holocamera for detection of particle characteristics and motions in the ocean,” Deep-Sea Res. 46, 1455–1481 (1999).
[CrossRef]

E. Malkiel, O. Alquaddoomi, J. Katz, “Measurements of plankton distribution in the ocean using submersible holography,” Meas. Sci. Technol. 10, 1142–1152 (1999).
[CrossRef]

1998 (1)

1997 (1)

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–3819 (1997).
[CrossRef]

1996 (1)

U. Schnars, T. Kreis, W. Jueptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

1987 (1)

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

Alquaddoomi, O.

E. Malkiel, O. Alquaddoomi, J. Katz, “Measurements of plankton distribution in the ocean using submersible holography,” Meas. Sci. Technol. 10, 1142–1152 (1999).
[CrossRef]

Cuche, E.

Depeursinge, C.

Donaghay, P.

J. Katz, P. Donaghay, S. King, K. Russell, “Submersible holocamera for detection of particle characteristics and motions in the ocean,” Deep-Sea Res. 46, 1455–1481 (1999).
[CrossRef]

Goodman, J. R.

J. R. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

Hariharan, P.

P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge U. Press, Cambridge, England, 1996).
[CrossRef]

Jueptner, W.

U. Schnars, T. Kreis, W. Jueptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

Katz, J.

J. Katz, P. Donaghay, S. King, K. Russell, “Submersible holocamera for detection of particle characteristics and motions in the ocean,” Deep-Sea Res. 46, 1455–1481 (1999).
[CrossRef]

E. Malkiel, O. Alquaddoomi, J. Katz, “Measurements of plankton distribution in the ocean using submersible holography,” Meas. Sci. Technol. 10, 1142–1152 (1999).
[CrossRef]

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–3819 (1997).
[CrossRef]

King, S.

J. Katz, P. Donaghay, S. King, K. Russell, “Submersible holocamera for detection of particle characteristics and motions in the ocean,” Deep-Sea Res. 46, 1455–1481 (1999).
[CrossRef]

Kreis, T.

U. Schnars, T. Kreis, W. Jueptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

Li, W.

W. Li, “Computational reconstruction of images from optical holograms,” M. S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 2001).

Malkiel, E.

E. Malkiel, O. Alquaddoomi, J. Katz, “Measurements of plankton distribution in the ocean using submersible holography,” Meas. Sci. Technol. 10, 1142–1152 (1999).
[CrossRef]

Marquet, P.

Meng, H.

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

Onural, L.

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

Owen, R.

R. Owen, A. Zozulya, “In-line holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

Pu, Y.

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

Russell, K.

J. Katz, P. Donaghay, S. King, K. Russell, “Submersible holocamera for detection of particle characteristics and motions in the ocean,” Deep-Sea Res. 46, 1455–1481 (1999).
[CrossRef]

Schnars, U.

U. Schnars, T. Kreis, W. Jueptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

Scott, P.

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

Smith, S.

S. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing (California Technical, San Diego, Calif., 1997).

Tao, B.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–3819 (1997).
[CrossRef]

Vikram, C.

C. Vikram, Particle Field Holography (Cambridge U. Press, Cambridge, England, 1992).
[CrossRef]

Yamaguchi, I.

Zhang, J.

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–3819 (1997).
[CrossRef]

Zhang, T.

Zozulya, A.

R. Owen, A. Zozulya, “In-line holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

Appl. Opt. (1)

Deep-Sea Res. (1)

J. Katz, P. Donaghay, S. King, K. Russell, “Submersible holocamera for detection of particle characteristics and motions in the ocean,” Deep-Sea Res. 46, 1455–1481 (1999).
[CrossRef]

Exp. Fluids (2)

J. Zhang, B. Tao, J. Katz, “Turbulent flow measurements in a square duct with hybrid holographic PIV,” Exp. Fluids 23, 373–3819 (1997).
[CrossRef]

Y. Pu, H. Meng, “An advanced off-axis holographic particle image velocimetry (HPIV) system,” Exp. Fluids 29, 184–197 (2000).
[CrossRef]

Meas. Sci. Technol. (1)

E. Malkiel, O. Alquaddoomi, J. Katz, “Measurements of plankton distribution in the ocean using submersible holography,” Meas. Sci. Technol. 10, 1142–1152 (1999).
[CrossRef]

Opt. Eng. (3)

U. Schnars, T. Kreis, W. Jueptner, “Digital recording and numerical reconstruction of holograms: reduction of the spatial frequency spectrum,” Opt. Eng. 35, 977–982 (1996).
[CrossRef]

R. Owen, A. Zozulya, “In-line holographic sensor for monitoring and characterizing marine particulates,” Opt. Eng. 39, 2187–2197 (2000).
[CrossRef]

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

Opt. Lett. (1)

Other (6)

Image Processing Toolbox For Use with Matlab (Math Works, Natick, Mass., 1997).

J. R. Goodman, Introduction to Fourier Optics, 2nd ed. (McGraw-Hill, New York, 1996).

P. Hariharan, Optical Holography: Principles, Techniques, and Applications (Cambridge U. Press, Cambridge, England, 1996).
[CrossRef]

S. Smith, The Scientist and Engineer’s Guide to Digital Signal Processing (California Technical, San Diego, Calif., 1997).

C. Vikram, Particle Field Holography (Cambridge U. Press, Cambridge, England, 1992).
[CrossRef]

W. Li, “Computational reconstruction of images from optical holograms,” M. S. thesis (Massachusetts Institute of Technology, Cambridge, Mass., 2001).

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Figures (12)

Fig. 1
Fig. 1

Digitizing an optically reconstructed image from an in-line hologram (courtesy of E. Malkiel, The Johns Hopkins University).

Fig. 2
Fig. 2

Spectra in one dimension for an image and its in-line and off-axis holograms. (a) Image, (b) in-line hologram, and (c) off-axis hologram.

Fig. 3
Fig. 3

Image reconstructions from a hologram of a large copepod. (a) Optical reconstruction (courtesy of E. Malkiel). (b) Computational reconstruction that uses the exact kernel. (c) Computational reconstruction that uses the Fresnel approximation. (d) Computational reconstruction with twin-image reduction.

Fig. 4
Fig. 4

Image reconstructions from a hologram of a linear diatom. (a) Optical reconstruction (courtesy of J. Zhang). (b) Computational reconstruction that uses the Fresnel approximation.

Fig. 5
Fig. 5

Image reconstructions from a hologram of a helical diatom. (a) Optical reconstruction (courtesy of Dr. E. Malkiel). (b) Computational reconstruction with the hologram digitized at 126 pixels/mm. (c) Computational reconstruction with the hologram digitized at 315 pixels/mm.

Fig. 6
Fig. 6

Image reconstructions of a copepod whose center plane is at an angle to the reconstruction plane. (a) Optical reconstruction. (b) Computational reconstruction with the hologram digitized at 315 pixels/mm.

Fig. 7
Fig. 7

Image reconstructions of a copepod with and without spatial bandpass filtering from a hologram digitized at 126 pixels/mm. (a) Computational reconstruction without bandpass filtering. (b) Computational reconstruction with the bandpass filtering described in the text.

Fig. 8
Fig. 8

Summation of parallel planes through a holographic image, spaced 2 mm apart. (a) Each image has intensity scaled to full range. (b) Analytic sum for N ps used.

Fig. 9
Fig. 9

Image of Fig. 8(b) after low-frequency removal.

Fig. 10
Fig. 10

Result of applying an 0.38 threshold level to the image of Fig. 9.

Fig. 11
Fig. 11

Results of an 0.1 mm × 0.1 mm erosion process applied to the image of Fig. 10.

Fig. 12
Fig. 12

Image of Fig. 11 after dilation of each point to a square of dimensions 0.16 mm × 0.16 mm.

Equations (36)

Equations on this page are rendered with MathJax. Learn more.

Ixh, yh=|rxh, yh+oxh, yh|2=|rxh, yh|2+|oxh, yh|2+rxh, yho*xh, yh+r*xh, yhoxh, yh.
rxh, yh=A expikxh sin θ,
oxh, yh=---ρxo, yo, zozo2+xo-xh2+yo-yh21/2×exp-ikzo2+xo-xh2+yo-yh21/2×dxodyodzo.
phxh, yh=A expikxh sin θ×---ρxo, yo, zozo2+xo-xh2+yo-yh21/2×expikzo2+xo-xh2+yo-yh21/2dxodyodzo+A exp-ikxh sin θ×---ρxo, yo, zozo2+xo-xh2+yo-yh21/2×exp-ikzo2+xo-xh2+yo-yh21/2dxodyodzo.
phpxh, yh=expikxh sin θ×---ρxo, yo, zozoexpikzo×expik2zoxo-xh2+yo-yh2dxodyodzo+exp-ikxh sin θ×---ρxo, yo, zozoexp-ikzo×exp-ik2zoxo-xh2+yo-yh2dxodyodzo.
no=exp-ikxh sin θ×exp-ik zo2+xo-xh2+yo-yh21/2zo2+xo-xh2+yo-yh21/2.
nmxr, yr; zo=expikxh sin θ×expikzo2+xr-xh2+yr-yh21/2zo2+xr-xh2+yr-yh21/2.
pzxr, yr=expikxh sin θ-- phxh, yh×expikz2+xr-xh2+yr-yh21/2z2+xr-xh2+yr-yh21/2dxhdyh.
pzpxr, yr=expikxh sin θexpikz1z-- phxh, yh×expik2zxr-xh2+yr-yh2dxhdyh.
phzxh, yh; zo=1zoexp-ikzo-- ρxo, yo, zo×exp-ik2zoxo-xh2+yo-yh2dxodyo.
pRxr, yr; zr, zo=1zr-- phzxh, yh; zo×expik2zrxh-xr2+yh-yr2dxhdyh.
pRxr, yr; zr, zo=1zozr---- ρxo, yo; zo×expik2zrxh-xr2+yh-yr2exp-ik2zo×xo-xh2+yo-yh2dxhdyhdxodyo.
pRxr, yr; zr, zo=1zozr-- ρxo, yo; zo×αxr, xo; zr, zo×αyr, yo; zr, zodxodyo,
αxr, xo; zr, zo=-expik2zrxh-xr2-ik2zoxo-xh2dxh,
αyr, yo; zr, zo=-expik2zryh-yr2-ik2zoyo-yh2dyh.
αxr, xo; zo, zo=-expik2zoxh-xr2-xo-xh2dxh=expik2zoxr2-xo2×-exp-ik2zoxr-xoxhdxh=4πzrkexpik2zoxr2-xo2×δxr-xo,
αyr, yo; zo, zo=4πzokexpik2zoyr2-yo2×δyr-yo.
pRxr, yr; zr=zo=4πk2-- ρxo, yo; zo×expik2zrxr2+yr2-xo2-yo2×δxr-xoδyr-yodxodyo=4πk2 ρxr, yr; zo.
αxr, xo; zr, zo=-expik2zrxh2×exp-ik2zoxo-xr-xh2dxh=-1expik2zrx2×exp-ik2zox2x=xo-xr=2πizrzokzr-zo1/2×expik2zr-zoxo-xr2.
expix22β=2πiβ1/2 exp-i2π2βfx2.
αyr, yo; zr, zo=2πizrzokzr-zo1/2×expik2zr-zoyo-yr2.
pRxr, yo; zr, zo=2πikzr-zo-- ρxo, yo; zo×expik2zr-zoxr-xo2+yr-yo2dxodyo.
pzx, y=phx, ynx, y, z,
nx, y, z=expikz2+x2+y21/2z2+x2+y21/2.
pzpx, y=phx, ynpx, y, z,
npx, y, z=1zexpik2zx2+y2.
Pzfx, fy=Phfx, fyNfx, fy, z,
Pzpfx, fy=Phfx, fyNpfx, fy, z.
pzx, y=-1Pzfx, fy,
pzhx, y=-1Pzhfx, fy.
Npfx, fy, z=1z--expik2zx2+y2×exp-i2πxfx+yfydxdy=i2πkexp-i2π2zkfx2+fy2.
wfx, fy=exp-fx2+fy212-fo22σ2,
nsx, y, z=m=1M nx, y, mδz,
Nsx, y, z=m=1M Nfx, fy, mδz.
Nps=i2πkm=1Mexp-i 2π2mδzkfx2+fy2= i2πkQ-QM+11-QQ1i2πk MQ=1,
Q=exp-i 2π2δzkfx2+fy2.

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