Abstract

The performance of an imaging technique relying on the spatial correlation of laser-speckle intensity measurements is evaluated on the basis of theoretical analysis, computer simulation, and laboratory results. A theoretical expression for the signal-to-noise ratio of the recovered imaging target’s power spectrum is used to estimate the imaging performance expected in the computer simulation and laboratory experiment. Power-spectrum estimates for an imaging target, obtained both in the laboratory and through simulation, are compared with the theoretical results and with the true spectrum of the target. Images recovered from the simulation data and the laboratory data are also compared. Our results suggest that the signal-to-noise ratio expression provides an accurate means for estimating the recoverable frequency content of a simple target.

© 1991 Optical Society of America

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References

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  1. P. S. Idell, J. R. Fienup, R. S. Goodman, “Image Synthesis from Nonimaged Laser-Speckle Patterns,” Opt. Lett. 12, 858–860 (1987).
    [CrossRef] [PubMed]
  2. J. R. Fienup, P. S. Idell, “Imaging Correlography with Sparse Arrays of Detectors,” Opt. Eng., 27, 778–784 (1988).
  3. P. S. Idell, J. D. Gonglewski, D. G. Voelz, J. Knopp, “Image Synthesis from Nonimaged Laser-Speckle Patterns: Experimental Verification,” Opt. Lett. 14, 154–156 (1989).
    [CrossRef] [PubMed]
  4. P. D. Henshaw, D. E. B. Lees, “Electronically Agile Multiple Aperture Imager Receiver,” Opt. Eng. 27, 793–800 (1988).
  5. M. Nieto-Vesperinas, M. J. Perez-Ilzarbe, R. Navarro, “Object Reconstruction from Experimental Far-Field Data Using Phase Retrieval Algorithms,” in Signal Recovery and Synthesis III, Vol. 15 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 146–149.
  6. J. R. Fienup, “Phase Retrieval Algorithms: A Comparison,” Appl. Opt. 21, 2758–2769 (1982).
    [CrossRef] [PubMed]
  7. J. R. Fienup, C. C. Wackerman, “Phase Retrieval Stagnation Problems and Solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
    [CrossRef]
  8. R. Hanbury Brown, R. Q. Twiss, “A Test of a New Type of Stellar Interferometer on Sirius,” Nature (London) 178, 1046–1048 (1956).
    [CrossRef]
  9. R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).
  10. M. Elbaum, M. King, M. Greenebaum, Laser Correlography: Transmission of High-Resolution Object Signatures Through the Turbulent Atmosphere, Tech. Rep. T-1/306-3-11 (Riverside Research Institute, New York, 1974).
  11. R. Q. Twiss, “Applications of Intensity Interferometry in Physics and Astronomy,” Opt. Acta 16, 423–451 (1969).
    [CrossRef]
  12. J. W. Goodman, Statistical Optics (Wiley, New York, 1985).
  13. T. Sato, S. Wadaka, J. Yamamoto, J. Ishii, “Imaging System Using an Intensity Triple Correlator,” Appl. Opt. 17, 2047–2052 (1978).
    [CrossRef] [PubMed]
  14. A. S. Marathay, “Phase Function of Spatial Coherence from Multiple Intensity Correlations,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 273–276 (1986).
    [CrossRef]
  15. A. S. Marathay, Y. Hu, P. S. Idell, “Object Reconstruction Using Third and Fourth Order Intensity Correlations,” presented at the Workshop On Higher-Order Spectral Analysis, Vail, Colo., 1989.
  16. L. I. Goldfischer, “Autocorrelation Function and Power Spectral Density of Laser-Produced Speckle Patterns,” J. Opt. Soc. Am. 55, 247–253 (1965).
    [CrossRef]
  17. K.-S. Kim, D. Caballero, “Derivation and Simulation of an Imaging Correlography Algorithm: A Modification of the Idell–Fienup Algorithm,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 153–156 (1987).
  18. L. Fang, H. J. Tiziani, “A New Method for Determining the Modulation-Transfer Function from Edge Traces,” Optik 74, 17–21 (1986).
  19. K. A. O’Donnell, “Time-Varying Speckle Phenomena in Astronomical Imaging and in Laser Scattering,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1983).
  20. J. C. Marron, “Accuracy of Fourier-Magnitude Estimation from Speckle Intensity Correlation,” J. Opt. Soc. Am. A 5, 864–870 (1988).
    [CrossRef]
  21. The assumption leading to condition (A8) ignores certain double-frequency effects that result in terms whose magnitudes are smaller than the |μ(Δu)|2 term in Eq. (A9) but in total may be larger than the |μ(Δu)|4 term. For objects with significant detail (such as that used in this paper), the |μ(Δu)|2 term makes an insignificant contribution to the overall SNR expression [cf. Eq. (A12)] at detector separations Δu more than a few speckle widths. Hence the approximation leading to condition (A8) would seem justified for our experimental work.

1989 (1)

1988 (3)

P. D. Henshaw, D. E. B. Lees, “Electronically Agile Multiple Aperture Imager Receiver,” Opt. Eng. 27, 793–800 (1988).

J. R. Fienup, P. S. Idell, “Imaging Correlography with Sparse Arrays of Detectors,” Opt. Eng., 27, 778–784 (1988).

J. C. Marron, “Accuracy of Fourier-Magnitude Estimation from Speckle Intensity Correlation,” J. Opt. Soc. Am. A 5, 864–870 (1988).
[CrossRef]

1987 (1)

1986 (2)

J. R. Fienup, C. C. Wackerman, “Phase Retrieval Stagnation Problems and Solutions,” J. Opt. Soc. Am. A 3, 1897–1907 (1986).
[CrossRef]

L. Fang, H. J. Tiziani, “A New Method for Determining the Modulation-Transfer Function from Edge Traces,” Optik 74, 17–21 (1986).

1982 (1)

1978 (1)

1969 (1)

R. Q. Twiss, “Applications of Intensity Interferometry in Physics and Astronomy,” Opt. Acta 16, 423–451 (1969).
[CrossRef]

1965 (1)

1956 (1)

R. Hanbury Brown, R. Q. Twiss, “A Test of a New Type of Stellar Interferometer on Sirius,” Nature (London) 178, 1046–1048 (1956).
[CrossRef]

Caballero, D.

K.-S. Kim, D. Caballero, “Derivation and Simulation of an Imaging Correlography Algorithm: A Modification of the Idell–Fienup Algorithm,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 153–156 (1987).

Elbaum, M.

M. Elbaum, M. King, M. Greenebaum, Laser Correlography: Transmission of High-Resolution Object Signatures Through the Turbulent Atmosphere, Tech. Rep. T-1/306-3-11 (Riverside Research Institute, New York, 1974).

Fang, L.

L. Fang, H. J. Tiziani, “A New Method for Determining the Modulation-Transfer Function from Edge Traces,” Optik 74, 17–21 (1986).

Fienup, J. R.

Goldfischer, L. I.

Gonglewski, J. D.

Goodman, J. W.

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

Goodman, R. S.

Greenebaum, M.

M. Elbaum, M. King, M. Greenebaum, Laser Correlography: Transmission of High-Resolution Object Signatures Through the Turbulent Atmosphere, Tech. Rep. T-1/306-3-11 (Riverside Research Institute, New York, 1974).

Hanbury Brown, R.

R. Hanbury Brown, R. Q. Twiss, “A Test of a New Type of Stellar Interferometer on Sirius,” Nature (London) 178, 1046–1048 (1956).
[CrossRef]

R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).

Henshaw, P. D.

P. D. Henshaw, D. E. B. Lees, “Electronically Agile Multiple Aperture Imager Receiver,” Opt. Eng. 27, 793–800 (1988).

Hu, Y.

A. S. Marathay, Y. Hu, P. S. Idell, “Object Reconstruction Using Third and Fourth Order Intensity Correlations,” presented at the Workshop On Higher-Order Spectral Analysis, Vail, Colo., 1989.

Idell, P. S.

P. S. Idell, J. D. Gonglewski, D. G. Voelz, J. Knopp, “Image Synthesis from Nonimaged Laser-Speckle Patterns: Experimental Verification,” Opt. Lett. 14, 154–156 (1989).
[CrossRef] [PubMed]

J. R. Fienup, P. S. Idell, “Imaging Correlography with Sparse Arrays of Detectors,” Opt. Eng., 27, 778–784 (1988).

P. S. Idell, J. R. Fienup, R. S. Goodman, “Image Synthesis from Nonimaged Laser-Speckle Patterns,” Opt. Lett. 12, 858–860 (1987).
[CrossRef] [PubMed]

A. S. Marathay, Y. Hu, P. S. Idell, “Object Reconstruction Using Third and Fourth Order Intensity Correlations,” presented at the Workshop On Higher-Order Spectral Analysis, Vail, Colo., 1989.

Ishii, J.

Kim, K.-S.

K.-S. Kim, D. Caballero, “Derivation and Simulation of an Imaging Correlography Algorithm: A Modification of the Idell–Fienup Algorithm,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 153–156 (1987).

King, M.

M. Elbaum, M. King, M. Greenebaum, Laser Correlography: Transmission of High-Resolution Object Signatures Through the Turbulent Atmosphere, Tech. Rep. T-1/306-3-11 (Riverside Research Institute, New York, 1974).

Knopp, J.

Lees, D. E. B.

P. D. Henshaw, D. E. B. Lees, “Electronically Agile Multiple Aperture Imager Receiver,” Opt. Eng. 27, 793–800 (1988).

Marathay, A. S.

A. S. Marathay, “Phase Function of Spatial Coherence from Multiple Intensity Correlations,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 273–276 (1986).
[CrossRef]

A. S. Marathay, Y. Hu, P. S. Idell, “Object Reconstruction Using Third and Fourth Order Intensity Correlations,” presented at the Workshop On Higher-Order Spectral Analysis, Vail, Colo., 1989.

Marron, J. C.

Navarro, R.

M. Nieto-Vesperinas, M. J. Perez-Ilzarbe, R. Navarro, “Object Reconstruction from Experimental Far-Field Data Using Phase Retrieval Algorithms,” in Signal Recovery and Synthesis III, Vol. 15 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 146–149.

Nieto-Vesperinas, M.

M. Nieto-Vesperinas, M. J. Perez-Ilzarbe, R. Navarro, “Object Reconstruction from Experimental Far-Field Data Using Phase Retrieval Algorithms,” in Signal Recovery and Synthesis III, Vol. 15 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 146–149.

O’Donnell, K. A.

K. A. O’Donnell, “Time-Varying Speckle Phenomena in Astronomical Imaging and in Laser Scattering,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1983).

Perez-Ilzarbe, M. J.

M. Nieto-Vesperinas, M. J. Perez-Ilzarbe, R. Navarro, “Object Reconstruction from Experimental Far-Field Data Using Phase Retrieval Algorithms,” in Signal Recovery and Synthesis III, Vol. 15 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 146–149.

Sato, T.

Tiziani, H. J.

L. Fang, H. J. Tiziani, “A New Method for Determining the Modulation-Transfer Function from Edge Traces,” Optik 74, 17–21 (1986).

Twiss, R. Q.

R. Q. Twiss, “Applications of Intensity Interferometry in Physics and Astronomy,” Opt. Acta 16, 423–451 (1969).
[CrossRef]

R. Hanbury Brown, R. Q. Twiss, “A Test of a New Type of Stellar Interferometer on Sirius,” Nature (London) 178, 1046–1048 (1956).
[CrossRef]

Voelz, D. G.

Wackerman, C. C.

Wadaka, S.

Yamamoto, J.

Appl. Opt. (2)

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (2)

Nature (London) (1)

R. Hanbury Brown, R. Q. Twiss, “A Test of a New Type of Stellar Interferometer on Sirius,” Nature (London) 178, 1046–1048 (1956).
[CrossRef]

Opt. Acta (1)

R. Q. Twiss, “Applications of Intensity Interferometry in Physics and Astronomy,” Opt. Acta 16, 423–451 (1969).
[CrossRef]

Opt. Eng. (2)

J. R. Fienup, P. S. Idell, “Imaging Correlography with Sparse Arrays of Detectors,” Opt. Eng., 27, 778–784 (1988).

P. D. Henshaw, D. E. B. Lees, “Electronically Agile Multiple Aperture Imager Receiver,” Opt. Eng. 27, 793–800 (1988).

Opt. Lett. (2)

Optik (1)

L. Fang, H. J. Tiziani, “A New Method for Determining the Modulation-Transfer Function from Edge Traces,” Optik 74, 17–21 (1986).

Other (9)

K. A. O’Donnell, “Time-Varying Speckle Phenomena in Astronomical Imaging and in Laser Scattering,” Ph.D. dissertation (University of Rochester, Rochester, N.Y., 1983).

The assumption leading to condition (A8) ignores certain double-frequency effects that result in terms whose magnitudes are smaller than the |μ(Δu)|2 term in Eq. (A9) but in total may be larger than the |μ(Δu)|4 term. For objects with significant detail (such as that used in this paper), the |μ(Δu)|2 term makes an insignificant contribution to the overall SNR expression [cf. Eq. (A12)] at detector separations Δu more than a few speckle widths. Hence the approximation leading to condition (A8) would seem justified for our experimental work.

M. Nieto-Vesperinas, M. J. Perez-Ilzarbe, R. Navarro, “Object Reconstruction from Experimental Far-Field Data Using Phase Retrieval Algorithms,” in Signal Recovery and Synthesis III, Vol. 15 of OSA 1989 Technical Digest Series (Optical Society of America, Washington, D.C., 1989), pp. 146–149.

R. Hanbury Brown, The Intensity Interferometer (Taylor & Francis, London, 1974).

M. Elbaum, M. King, M. Greenebaum, Laser Correlography: Transmission of High-Resolution Object Signatures Through the Turbulent Atmosphere, Tech. Rep. T-1/306-3-11 (Riverside Research Institute, New York, 1974).

J. W. Goodman, Statistical Optics (Wiley, New York, 1985).

A. S. Marathay, “Phase Function of Spatial Coherence from Multiple Intensity Correlations,” in Advanced Technology Optical Telescopes III, L. D. Barr, ed., Proc. Soc. Photo-Opt. Instrum. Eng.628, 273–276 (1986).
[CrossRef]

A. S. Marathay, Y. Hu, P. S. Idell, “Object Reconstruction Using Third and Fourth Order Intensity Correlations,” presented at the Workshop On Higher-Order Spectral Analysis, Vail, Colo., 1989.

K.-S. Kim, D. Caballero, “Derivation and Simulation of an Imaging Correlography Algorithm: A Modification of the Idell–Fienup Algorithm,” in Digital Image Recovery and Synthesis, P. S. Idell, ed., Proc. Soc. Photo-Opt. Instrum. Eng.828, 153–156 (1987).

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Figures (12)

Fig. 1
Fig. 1

Schematic diagram of the imaging correlography technique.

Fig. 2
Fig. 2

Example of a laser-speckle imaging sequence using laboratory data: (a) portion of a measured speckle intensity pattern, (b) estimated object power spectrum obtained from 1000 speckle snapshots (for display purposes the log of the estimate is shown), (c) estimated Fourier modulus after filtering, and (d) image recovered after 500 phase-retrieval iterations.

Fig. 3
Fig. 3

Truth image of the painted, metal target.

Fig. 4
Fig. 4

Data processing flow chart.

Fig. 5
Fig. 5

Diagram of the laboratory setup.

Fig. 6
Fig. 6

Selected SNR values as functions of spatial frequency and number of speckle snapshots estimated for the metal target. The spatial-frequency axis has been normalized to 1.0 at the diffraction limit of the receiving aperture.

Fig. 7
Fig. 7

Radial average power spectra estimated from computer-simulated data (N is the number of speckle snapshots used for the estimate). The spatial-frequency axis has been normalized to 1.0 at the diffraction limit of the receiving aperture.

Fig. 8
Fig. 8

Radial average power spectra estimated from laboratory data (N is the number of speckle snapshots used for the estimate). The spatial-frequency axis has been normalized to 1.0 at the diffraction limit of the receiving aperture.

Fig. 9
Fig. 9

Comparison of truth image autocorrelation and power spectrum with the target autocorrelation and power spectrum estimated with the laboratory data: (a) truth image autocorrelation, (b) laboratory data autocorrelation estimate obtained from 1000 speckle snapshots, (c) magnitude of the difference between (a) and (b), (d) TRUTH image power spectrum, (e) laboratory data power-spectrum estimate, (f) magnitude of the difference between (d) and (e).

Fig. 10
Fig. 10

Radial average Wiener-filtered power spectra estimated from 1000 speckle snapshot laboratory and computer-simulated data. The spatial-frequency axis has been normalized to 1.0 at the diffraction limit of the receiving aperture.

Fig. 11
Fig. 11

Comparison of images recovered by using laboratory and computer-simulated data. Also shown are truth images that have been Wiener filtered such that they contain spatial frequencies for which SNR = 1 as determined with Eq. (6).

Fig. 12
Fig. 12

Radial average phase error as a function of spatial frequency for the images recovered from 1000-speckle snapshot laboratory and computer-simulated data. The spatial-frequency axis has been normalized to 1.0 at the diffraction limit of the receiving aperture.

Equations (23)

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I n ( u ) = F { f n ( x ) } 2 ,
f n ( x ) = f 0 ( x ) exp [ i ϕ n ( x ) ] ,
C ^ I ( Δ u , N ) = 1 N - P ( u + Δ u ) P ( u ) [ I n ( u + Δ u ) - I ¯ ] [ I n ( u ) - I ¯ ] d u = - P ( u + Δ u ) P ( u ) { 1 N n = 1 N [ I n ( u + Δ u ) - I ¯ ] × [ I n ( u ) - I ¯ ] } d 2 u ,
lim N 1 N n = 1 N [ I n ( u + Δ u ) - I ¯ ] [ I n ( u ) - I ¯ ] = Γ ( Δ u ) 2 ,
lim N C ^ I ( Δ u , N ) = OTF ( Δ u ) Γ ( Δ u ) 2 ,
SNR C ^ I = μ ( Δ u ) 2 [ N N s OTF ( Δ u ) 1 + 4 μ ( Δ u ) 2 + 3 μ ( Δ u ) 4 ] 1 / 2 ,
N s = ( W D λ z ) 2 ,
C ˜ I ( Δ u , N ) = F [ 1 N n = 1 N F - 1 { I n ( u ) - I ¯ n } 2 ] ,
W ( Δ u ) = OTF ( Δ u ) Φ ^ 0 ( Δ u ) 2 OTF ( Δ u ) 2 Φ ^ 0 ( Δ u ) 2 + Φ ^ n 0 ( Δ u ) 2 ,
( λ z d W ) 2 = 18.3 detectors / speckle lobe ,
SNR C ^ I ( Δ u , N ) = E C ^ I ( Δ u , N ) [ Var C ^ I ( Δ u , N ) ] 1 / 2 = N E C ^ I ( Δ u , 1 ) [ Var C ^ I ( Δ u , 1 ) ] 1 / 2 ,
C ^ I ( Δ u , 1 ) = P ( x ) P ( x + Δ u ) [ I ( x + Δ u ) - I ¯ ] [ I ( x ) - I ¯ ] d 2 x .
E C ^ I ( Δ u , 1 ) = E C ^ I ( Δ u , N ) = OTF ( Δ u ) Γ ( Δ u ) 2 ,
Var C ^ I ( Δ u , 1 ) = E C ^ I ( Δ u , 1 ) 2 - E C ^ I ( Δ u , 1 ) 2 ,
Var C ^ I ( Δ u , 1 ) = P ( x ) P ( x + Δ u ) P ( x ) P ( x + Δ u ) × K c ( Δ u ; x , x ) d 2 x d 2 x ,
K c ( Δ u ; x , x ) = E [ c I ( Δ u , x ) c I ( Δ u , x ) ] - E c I ( Δ u , x ) E c I ( Δ u , x ) ,
c I ( Δ u , x ) = [ I ( x ) - I ¯ ] [ I ( x + Δ u ) - I ¯ ] .
K c ( Δ u ; x , x ) Var c I ( Δ u ) A s δ ( x - x ) ,
Var c I ( Δ u ) = K c ( Δ u ; x , x ) = E [ c I ( Δ u , x ) ] 2 - [ E c I ( Δ u , x ) ] 2 = I ¯ 4 { 1 + 4 μ ( Δ u ) 2 + 3 μ ( Δ u ) 4 } .
Var C ^ I ( Δ u , 1 ) Var c I ( Δ u ) A s OTF ( Δ u ) ,
SNR C ^ I ( Δ u ) Γ ( Δ u ) 2 [ Var c I ( Δ u ) ] 1 / 2 [ N · N s · OTF ( Δ u ) ] 1 / 2
= μ ( Δ u ) 2 [ 1 + 4 μ ( Δ u ) 2 + 3 μ ( Δ u ) 4 ] 1 / 2 [ N · N s · OTF ( Δ u ) ] 1 / 2 ,
N s A p / A s

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