Abstract

A simple derivation is given for the signal-to-noise ratio (SNR) in images reconstructed from incoherent holograms. Dependence is shown to be on the hologram SNR, object complexity, and the number of pixels in the detector. Reconstruction of involved objects becomes possible with high dynamic range detectors such as charge-coupled devices. We have produced such white light holograms by means of a rotational shear interferometer combined with a chromatic corrector. A digital inverse transform recreated the object.

© 1988 Optical Society of America

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References

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    [CrossRef]
  11. F. Roddier, “Rotation-Shearing Interferometry,” in High Angular Resolution Stellar Interferometry, IAU Coll. 50, J. Davis, W. J. Tango, Eds. (U. Sydney, 1978).
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    [CrossRef]
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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  27. W. T. Rhodes, A. A. Sawchuk, “Incoherent Image Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, Berlin, 1981).
  28. J. Hogbom, “Aperture Synthesis with a Non-regular Distribution of Interferometer Baselines,” Astrophys. J. Suppl. 15, 417 (1974).

1985 (5)

1984 (2)

N. George, S. Wang, “Cosinusoidal Transforms in White Light,” Appl. Opt. 23, 787 (1984).
[CrossRef] [PubMed]

N. George, S. Wang, “Cosinusoidal Transforms in White Light,” Appl Opt. 23, 787 (1984).
[CrossRef] [PubMed]

1983 (1)

K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
[CrossRef]

1980 (1)

C. Roddier, F. Roddier, F. Martin, A. Baranne, R. Brun, “Twin-Image Holography with Spectrally Broad Light,” J. Opt. 11, 49 (1980).
[CrossRef]

1978 (2)

J. B. Breckinridge, “A White-Light Amplitude Interferometer with 180-degrees Shear,” Opt. Eng. 17, 156 (1978).
[CrossRef]

F. Roddier, C. Roddier, J. Demarcq, “A Rotation Shearing Interferometer with Phase-Compensated Roof Prisms,” J. Opt. Paris 9, 145 (1978).
[CrossRef]

1974 (2)

J. Hogbom, “Aperture Synthesis with a Non-regular Distribution of Interferometer Baselines,” Astrophys. J. Suppl. 15, 417 (1974).

J. B. Breckinridge, “Two-Dimensional White Light Coherence Interferometer,” Appl. Opt. 13, 2760 (1974).
[CrossRef] [PubMed]

1972 (1)

1970 (2)

P. Connes, “Astronomical Fourier Spectroscopy,” Ann. Rev. Astron. Astrophys. 8, 209 (1970).
[CrossRef]

S. Lowenthal, J. Serres, H. Arsenault, “Resolution and Film-Grain Noise in Fourier Transform Holograms Recorded with Coherent or Spatially Incoherent Light,” Opt. Commun. 1, 438 (1970).
[CrossRef]

1969 (1)

1966 (2)

1965 (2)

A. Lohmann, “Wavefront Reconstruction for Incoherent Objects,” J. Opt. Soc. Am. 55, 1556 (1965).
[CrossRef]

G. W. Stroke, R. C. Restrick, “Holography with Spatially Incoherent Light,” Appl. Phys. Lett. 7, 229 (1965).
[CrossRef]

Angell, D. K.

Arsenault, H.

S. Lowenthal, J. Serres, H. Arsenault, “Resolution and Film-Grain Noise in Fourier Transform Holograms Recorded with Coherent or Spatially Incoherent Light,” Opt. Commun. 1, 438 (1970).
[CrossRef]

Baranne, A.

C. Roddier, F. Roddier, F. Martin, A. Baranne, R. Brun, “Twin-Image Holography with Spectrally Broad Light,” J. Opt. 11, 49 (1980).
[CrossRef]

Breckinridge, J. B.

Brun, R.

C. Roddier, F. Roddier, F. Martin, A. Baranne, R. Brun, “Twin-Image Holography with Spectrally Broad Light,” J. Opt. 11, 49 (1980).
[CrossRef]

Cochran, G.

Connes, P.

P. Connes, “Astronomical Fourier Spectroscopy,” Ann. Rev. Astron. Astrophys. 8, 209 (1970).
[CrossRef]

Demarcq, J.

F. Roddier, C. Roddier, J. Demarcq, “A Rotation Shearing Interferometer with Phase-Compensated Roof Prisms,” J. Opt. Paris 9, 145 (1978).
[CrossRef]

George, N.

Hagerott, E. C.

Hogbom, J.

J. Hogbom, “Aperture Synthesis with a Non-regular Distribution of Interferometer Baselines,” Astrophys. J. Suppl. 15, 417 (1974).

Itoh, K.

K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
[CrossRef]

Kozma, A.

Leibowitz, E.

Leith, E. N.

Lohmann, A.

Lowenthal, S.

S. Lowenthal, J. Serres, H. Arsenault, “Resolution and Film-Grain Noise in Fourier Transform Holograms Recorded with Coherent or Spatially Incoherent Light,” Opt. Commun. 1, 438 (1970).
[CrossRef]

Martin, F.

C. Roddier, F. Roddier, F. Martin, A. Baranne, R. Brun, “Twin-Image Holography with Spectrally Broad Light,” J. Opt. 11, 49 (1980).
[CrossRef]

Massey, N.

Mertz, L.

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

Murty, M. V. R. K.

Ohtsuka, Y.

K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
[CrossRef]

Papoulis, A.

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Restrick, R. C.

G. W. Stroke, R. C. Restrick, “Holography with Spatially Incoherent Light,” Appl. Phys. Lett. 7, 229 (1965).
[CrossRef]

Rhodes, W. T.

W. T. Rhodes, A. A. Sawchuk, “Incoherent Image Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, Berlin, 1981).

Ribak, E.

Roddier, C.

C. Roddier, F. Roddier, F. Martin, A. Baranne, R. Brun, “Twin-Image Holography with Spectrally Broad Light,” J. Opt. 11, 49 (1980).
[CrossRef]

F. Roddier, C. Roddier, J. Demarcq, “A Rotation Shearing Interferometer with Phase-Compensated Roof Prisms,” J. Opt. Paris 9, 145 (1978).
[CrossRef]

F. Roddier, C. Roddier, “Imaging with a Muiti-Mirror Telescope,” in Optical Telescopes of the Future, Geneva 1977, F. Pacini, W. Richter, R. N. Wilson, Eds. (ESO-CERN, Location, 1978).

C. Roddier, F. Roddier, “Imaging with a Coherence Interferometer in Optical Astronomy,” in Formation of Images from Spatial Coherence Functions in Astronomy, IAU Coll. 49, Van Schooneveld, Ed. (Reidel, Norwell, MA, 1979).
[CrossRef]

Roddier, F.

C. Roddier, F. Roddier, F. Martin, A. Baranne, R. Brun, “Twin-Image Holography with Spectrally Broad Light,” J. Opt. 11, 49 (1980).
[CrossRef]

F. Roddier, C. Roddier, J. Demarcq, “A Rotation Shearing Interferometer with Phase-Compensated Roof Prisms,” J. Opt. Paris 9, 145 (1978).
[CrossRef]

C. Roddier, F. Roddier, “Imaging with a Coherence Interferometer in Optical Astronomy,” in Formation of Images from Spatial Coherence Functions in Astronomy, IAU Coll. 49, Van Schooneveld, Ed. (Reidel, Norwell, MA, 1979).
[CrossRef]

F. Roddier, “Rotation-Shearing Interferometry,” in High Angular Resolution Stellar Interferometry, IAU Coll. 50, J. Davis, W. J. Tango, Eds. (U. Sydney, 1978).

F. Roddier, C. Roddier, “Imaging with a Muiti-Mirror Telescope,” in Optical Telescopes of the Future, Geneva 1977, F. Pacini, W. Richter, R. N. Wilson, Eds. (ESO-CERN, Location, 1978).

Sawchuk, A. A.

W. T. Rhodes, A. A. Sawchuk, “Incoherent Image Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, Berlin, 1981).

Serres, J.

S. Lowenthal, J. Serres, H. Arsenault, “Resolution and Film-Grain Noise in Fourier Transform Holograms Recorded with Coherent or Spatially Incoherent Light,” Opt. Commun. 1, 438 (1970).
[CrossRef]

Stroke, G. W.

G. W. Stroke, R. C. Restrick, “Holography with Spatially Incoherent Light,” Appl. Phys. Lett. 7, 229 (1965).
[CrossRef]

Wang, S.

Ann. Rev. Astron. Astrophys. (1)

P. Connes, “Astronomical Fourier Spectroscopy,” Ann. Rev. Astron. Astrophys. 8, 209 (1970).
[CrossRef]

Appl Opt. (1)

N. George, S. Wang, “Cosinusoidal Transforms in White Light,” Appl Opt. 23, 787 (1984).
[CrossRef] [PubMed]

Appl. Opt. (10)

Appl. Phys. Lett. (1)

G. W. Stroke, R. C. Restrick, “Holography with Spatially Incoherent Light,” Appl. Phys. Lett. 7, 229 (1965).
[CrossRef]

Astrophys. J. Suppl. (1)

J. Hogbom, “Aperture Synthesis with a Non-regular Distribution of Interferometer Baselines,” Astrophys. J. Suppl. 15, 417 (1974).

J. Opt. (1)

C. Roddier, F. Roddier, F. Martin, A. Baranne, R. Brun, “Twin-Image Holography with Spectrally Broad Light,” J. Opt. 11, 49 (1980).
[CrossRef]

J. Opt. Paris (1)

F. Roddier, C. Roddier, J. Demarcq, “A Rotation Shearing Interferometer with Phase-Compensated Roof Prisms,” J. Opt. Paris 9, 145 (1978).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (2)

S. Lowenthal, J. Serres, H. Arsenault, “Resolution and Film-Grain Noise in Fourier Transform Holograms Recorded with Coherent or Spatially Incoherent Light,” Opt. Commun. 1, 438 (1970).
[CrossRef]

K. Itoh, Y. Ohtsuka, “Interferometric Imaging of a Thermally Luminous Two-Dimensional Object,” Opt. Commun. 48, 75 (1983).
[CrossRef]

Opt. Eng. (1)

J. B. Breckinridge, “A White-Light Amplitude Interferometer with 180-degrees Shear,” Opt. Eng. 17, 156 (1978).
[CrossRef]

Other (7)

F. Roddier, C. Roddier, “Imaging with a Muiti-Mirror Telescope,” in Optical Telescopes of the Future, Geneva 1977, F. Pacini, W. Richter, R. N. Wilson, Eds. (ESO-CERN, Location, 1978).

C. Roddier, F. Roddier, “Imaging with a Coherence Interferometer in Optical Astronomy,” in Formation of Images from Spatial Coherence Functions in Astronomy, IAU Coll. 49, Van Schooneveld, Ed. (Reidel, Norwell, MA, 1979).
[CrossRef]

F. Roddier, “Rotation-Shearing Interferometry,” in High Angular Resolution Stellar Interferometry, IAU Coll. 50, J. Davis, W. J. Tango, Eds. (U. Sydney, 1978).

L. Mertz, Transformations in Optics (Wiley, New York, 1965).

A. Papoulis, Probability, Random Variables and Stochastic Processes (McGraw-Hill, New York, 1965).

Photometrics, Ltd., 2010 N. Forbes Blvd., Suite 103, Tucson, AZ 85745.

W. T. Rhodes, A. A. Sawchuk, “Incoherent Image Processing,” in Optical Information Processing, S. H. Lee, Ed. (Springer-Verlag, Berlin, 1981).

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Figures (3)

Fig. 1
Fig. 1

Incoherent hologram recorded with broadband white light. The dc bias has been removed (background grey level is zero, dark is negative, bright is positive). The object consisted of 40 bright dots.

Fig. 2
Fig. 2

Reconstructed image obtained by computing the 2-D Fourier transform of the hologram displayed in Fig. 1. The picture is a negative with a reduced dynamic range to better show the noise distribution.

Fig. 3
Fig. 3

Ratio of the SNR in the reconstructed image to the SNR in the hologram as a function of the number of bright pixels in the image taken as a measure of its complexity. A minus one slope is shown for comparison with theory.

Equations (16)

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I ( u ) = O ( 0 ) + Re O ( k R u / λ ) ,
( k / λ ) 2 i ( k R 1 x / λ ) = O ( 0 ) δ ( x ) + 1 2 [ o ( x ) + 0 ( x ) ] ,
| N I ( u ) | 2 d u = ( λ / k ) 2 | n O ( x ) | 2 d x ,
N ¯ I 2 A = ( λ / k ) 2 n ¯ O 2 a ,
o ¯ = 1 a o ( x ) d x = O ( 0 ) a = I ¯ a ,
SNR rec = o ¯ n ¯ O .
SNR rec = a a A λ k I ¯ N ¯ I .
SNR rec = a A 1 M I ¯ N ¯ I = a a 1 M SNR hol ,
SNR rec = P / M M SNR hol = P M SNR hol
M = a A = A o ¯ o ( x ) d x .
SNR hol = N / P ,
SNR rec = N M ,
( SNR rec ) max = P M ( SNR hol ) max
SNR rec = 2048 × 400 10 , 000 = 82 .
I ( u ) = O ( 0 ) + Re O ( m R u ) .
D ( u ) = Re O ( m R u ) ,

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