Abstract

Two new weighted-least-squares algorithms for magnitude-spectrum estimation from the bispectrum are presented. The performance of these algorithms is compared both with direct-magnitude-spectrum estimation and with several previously proposed algorithms for magnitude-spectrum estimation from the bispectrum. It is shown that the new algorithms result in more accurate magnitude-spectrum estimates than these other approaches.

© 1991 Optical Society of America

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  1. G. Sundaramoorthy, M. R. Raghuveer, S. A. Dianat, “Bispectral reconstruction of signals in noise: amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38,1297–1306 (1990).
    [CrossRef]
  2. A. W. Lohmann, G. P. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  3. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23,3121–3129 (1984).
    [CrossRef] [PubMed]
  4. C. L. Matson, “Weighted-least-squares phase reconstruction from the bispectrum,” J. Opt. Soc. Am. A 8, 1905–1913 (1991).
    [CrossRef]
  5. T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
    [CrossRef]
  6. M. J. Northcott, G. R. Ayers, J. C. Dainty, “Algorithms for image reconstruction from photon-limited data using the triple correlation,” J. Opt. Soc. Am. A 5, 986–992 (1988).
    [CrossRef]
  7. J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 7, 14–20 (1990).
    [CrossRef]
  8. D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 546–560 (1990).
    [CrossRef]
  9. A. Lannes, “On a new class of iterative algorithms for phase-closure imaging and bispectral analysis,” in High-Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 169–180.
  10. P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
    [CrossRef]
  11. G. B. Giannakis, “Signal reconstruction from multiple correlations: frequency- and time-domain approaches,” J. Opt. Soc. Am. A 6, 682–697 (1989).
    [CrossRef]
  12. M. Rangoussi, G. B. Giannakis, “FIR modeling using log-bispectra: weighted least-squares algorithms and performance analysis,” IEEE Trans. Circuits Syst. 38, 281–296 (1991).
    [CrossRef]
  13. S. A. Dianat, M. R. Raghuveer, “Fast algorithms for phase and magnitude reconstruction from bispectra,” Opt. Eng. 29, 504–512 (1990).
    [CrossRef]
  14. B. M. Sadler, G. B. Giannakis, “Image sequence analysis and reconstruction from the bispectrum,” in Proceedings of the 23rd Annual Conference on Information Science and Systems, H. L. Weinert, G. L. Meyer, eds. (Johns Hopkins U. Press, Baltimore, Md., 1989), pp. 242–246.
  15. A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  16. A. Papoulis, Probability, Random Variables, and Stochastic Processes (McGraw-Hill, New York, 1965).

1991 (2)

C. L. Matson, “Weighted-least-squares phase reconstruction from the bispectrum,” J. Opt. Soc. Am. A 8, 1905–1913 (1991).
[CrossRef]

M. Rangoussi, G. B. Giannakis, “FIR modeling using log-bispectra: weighted least-squares algorithms and performance analysis,” IEEE Trans. Circuits Syst. 38, 281–296 (1991).
[CrossRef]

1990 (3)

S. A. Dianat, M. R. Raghuveer, “Fast algorithms for phase and magnitude reconstruction from bispectra,” Opt. Eng. 29, 504–512 (1990).
[CrossRef]

G. Sundaramoorthy, M. R. Raghuveer, S. A. Dianat, “Bispectral reconstruction of signals in noise: amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38,1297–1306 (1990).
[CrossRef]

J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2π bispectrum phase,” J. Opt. Soc. Am. A 7, 14–20 (1990).
[CrossRef]

1989 (2)

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

G. B. Giannakis, “Signal reconstruction from multiple correlations: frequency- and time-domain approaches,” J. Opt. Soc. Am. A 6, 682–697 (1989).
[CrossRef]

1988 (1)

1984 (2)

H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23,3121–3129 (1984).
[CrossRef] [PubMed]

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

1983 (1)

1970 (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Ayers, G. R.

Bartelt, H.

Dainty, J. C.

Dianat, S. A.

G. Sundaramoorthy, M. R. Raghuveer, S. A. Dianat, “Bispectral reconstruction of signals in noise: amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38,1297–1306 (1990).
[CrossRef]

S. A. Dianat, M. R. Raghuveer, “Fast algorithms for phase and magnitude reconstruction from bispectra,” Opt. Eng. 29, 504–512 (1990).
[CrossRef]

Fitch, J. P.

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 546–560 (1990).
[CrossRef]

Ghez, A.M.

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Giannakis, G. B.

M. Rangoussi, G. B. Giannakis, “FIR modeling using log-bispectra: weighted least-squares algorithms and performance analysis,” IEEE Trans. Circuits Syst. 38, 281–296 (1991).
[CrossRef]

G. B. Giannakis, “Signal reconstruction from multiple correlations: frequency- and time-domain approaches,” J. Opt. Soc. Am. A 6, 682–697 (1989).
[CrossRef]

B. M. Sadler, G. B. Giannakis, “Image sequence analysis and reconstruction from the bispectrum,” in Proceedings of the 23rd Annual Conference on Information Science and Systems, H. L. Weinert, G. L. Meyer, eds. (Johns Hopkins U. Press, Baltimore, Md., 1989), pp. 242–246.

Goodman, D. M.

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 546–560 (1990).
[CrossRef]

Gorham, P. W.

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Johansson, E. M.

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 546–560 (1990).
[CrossRef]

Kulkarni, S.R.

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Labeyrie, A.

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Lannes, A.

A. Lannes, “On a new class of iterative algorithms for phase-closure imaging and bispectral analysis,” in High-Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 169–180.

Lawrence, T. W.

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 546–560 (1990).
[CrossRef]

Lohmann, A. W.

Marron, J. C.

Matson, C. L.

Matsuoka, T.

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

Nakajima, T.

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Neugebauer, G.

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Northcott, M. J.

Oke, J. B.

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Papoulis, A.

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

Prince, T.A.

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Raghuveer, M. R.

S. A. Dianat, M. R. Raghuveer, “Fast algorithms for phase and magnitude reconstruction from bispectra,” Opt. Eng. 29, 504–512 (1990).
[CrossRef]

G. Sundaramoorthy, M. R. Raghuveer, S. A. Dianat, “Bispectral reconstruction of signals in noise: amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38,1297–1306 (1990).
[CrossRef]

Rangoussi, M.

M. Rangoussi, G. B. Giannakis, “FIR modeling using log-bispectra: weighted least-squares algorithms and performance analysis,” IEEE Trans. Circuits Syst. 38, 281–296 (1991).
[CrossRef]

Sadler, B. M.

B. M. Sadler, G. B. Giannakis, “Image sequence analysis and reconstruction from the bispectrum,” in Proceedings of the 23rd Annual Conference on Information Science and Systems, H. L. Weinert, G. L. Meyer, eds. (Johns Hopkins U. Press, Baltimore, Md., 1989), pp. 242–246.

Sanchez, P. P.

Sullivan, R. C.

Sundaramoorthy, G.

G. Sundaramoorthy, M. R. Raghuveer, S. A. Dianat, “Bispectral reconstruction of signals in noise: amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38,1297–1306 (1990).
[CrossRef]

Ulrych, T. J.

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

Weigelt, G. P.

Wirnitzer, B.

Appl. Opt. (2)

Astron. Astrophys. (1)

A. Labeyrie, “Attainment of diffraction-limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).

Astron. J. (1)

P. W. Gorham, A.M. Ghez, S.R. Kulkarni, T. Nakajima, G. Neugebauer, J. B. Oke, T.A. Prince, “Diffraction-limited imaging. III. 30mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process. (1)

G. Sundaramoorthy, M. R. Raghuveer, S. A. Dianat, “Bispectral reconstruction of signals in noise: amplitude reconstruction issues,” IEEE Trans. Acoust. Speech Signal Process. 38,1297–1306 (1990).
[CrossRef]

IEEE Trans. Circuits Syst. (1)

M. Rangoussi, G. B. Giannakis, “FIR modeling using log-bispectra: weighted least-squares algorithms and performance analysis,” IEEE Trans. Circuits Syst. 38, 281–296 (1991).
[CrossRef]

J. Opt. Soc. Am. A (4)

Opt. Eng. (1)

S. A. Dianat, M. R. Raghuveer, “Fast algorithms for phase and magnitude reconstruction from bispectra,” Opt. Eng. 29, 504–512 (1990).
[CrossRef]

Proc. IEEE (1)

T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
[CrossRef]

Other (4)

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johansson, “Bispectral-based optimization algorithms for speckle imaging,” in Digital Image Synthesis and Inverse Optics, A. F. Gmitro, P. S. Idell, I. Lattaie, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1351, 546–560 (1990).
[CrossRef]

A. Lannes, “On a new class of iterative algorithms for phase-closure imaging and bispectral analysis,” in High-Resolution Imaging by Interferometry, F. Merkle, ed. (European Southern Observatory, Garching, Germany, 1988), pp. 169–180.

B. M. Sadler, G. B. Giannakis, “Image sequence analysis and reconstruction from the bispectrum,” in Proceedings of the 23rd Annual Conference on Information Science and Systems, H. L. Weinert, G. L. Meyer, eds. (Johns Hopkins U. Press, Baltimore, Md., 1989), pp. 242–246.

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

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Figures (4)

Fig. 1
Fig. 1

Satellite image used in the first computer-simulated data set.

Fig. 2
Fig. 2

Results from the first computer-simulated data set plotted as a function of spatial frequency normalized to 1 at the diffraction limit. (a) SNR of the direct power-spectrum estimate. (b)–(f) Slices of the reconstructed magnitude spectrum and the magnitude spectrum obtained by using the following algorithms: (b) direct magnitude, (c) cost-function WLS, (d) recursive WLS, (e) classical WLS, (f) recursive.

Fig. 3
Fig. 3

Satellite image used in the second computer-simulated data set.

Fig. 4
Fig. 4

Results from the second computer-simulated data set plotted as a function of spatial frequency normalized to 1 at the diffraction limit. (a) SNR of the direct power-spectrum estimate. (b)–(f) Slices of the reconstructed magnitude spectrum and the magnitude spectrum obtained by using the following algorithms: (b) direct magnitude, (c) cost-function WLS, (d) recursive WLS, (e) classical WLS (f) recursive.

Equations (19)

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B ( u , υ ) = O ( u ) O ( υ ) O * ( u + υ ) ,
B ( 0 , υ ) = O ( 0 ) O ( υ ) O * ( υ ) = O ( 0 ) | O ( υ ) | 2 .
| B ( u , υ ) | = | O ( u ) | | O ( υ ) | | O ( u + υ ) | .
J = [ | B ( u , υ ) | | O ( u ) | | O ( υ ) | | O ( u + υ ) | ] 2 σ B 2 ( u , υ ) ,
| O ( υ ) | = u υ w ( u , υ ) | B ( u , υ ) | | O ( u ) | | O ( u + υ ) | + u = υ w ( u , υ ) [ | B ( u , υ ) | | O ( u + υ ) | ] 1 / 2 + u υ w ( u , υ u ) | B ( u , υ u ) | | O ( u ) | | O ( υ u ) | / [ u υ w ( u , υ ) + u = υ w ( u , υ ) + w ( u , υ u ) ] ,
w ( u , υ ) = { ( { | B ( u , υ ) | / [ | O ( u ) | | O ( u + υ ) | ] } 2 × [ σ B 2 ( u , υ ) / | B ( u , υ ) | 2 + σ O 2 ( u ) / | O ( u ) | 2 + σ O 2 ( u + υ ) / | O ( u + υ ) | 2 ] ) 1 u υ , ( { | B ( u , υ ) | / [ 4 * | O ( u + υ ) | ] } × [ σ B 2 ( u , υ ) ] / | B ( u , υ ) | 2 + σ O 2 ( u + υ ) / | O ( u + υ ) | 2 } ) 1 u = υ
| O ( u + υ ) | = w ( u , υ ) | B ( u , υ ) | / | O ( u ) | | O ( υ ) | w ( u , υ ) ,
ln [ | B ( u , υ ) | ] = ln [ | O ( u ) | ] + ln [ | O ( υ ) | ] + ln [ | O ( u + υ ) | ] .
| O ( υ ) | = | B ( u , υ ) | / ( | O ( u ) | | O ( u + υ ) | ) ,
| O ( υ ) | = [ | B ( u , υ ) | / | O ( u + υ ) | ] 1 / 2 ,
g ( x , y ) = g ( x ¯ , y ¯ ) + ( x x ¯ ) g / x + ( y y ¯ ) g / y + .
σ g 2 ( x , y ) ( g / x ) 2 σ x 2 + ( g / y ) 2 σ y 2 .
σ w 2 = y 2 σ x 2 + x 2 σ y 2 .
σ g 2 ( z , w ) = σ z 2 / w 2 + ( z / w 2 ) 2 σ w 2 = ( z / w ) 2 ( σ z 2 / z 2 + σ w 2 / w 2 ) .
σ O 2 ( υ ) = { | B ( u , υ ) | [ | O ( u ) | | O ( u + υ ) | ] } 2 × [ σ B 2 ( u , υ ) | B ( u , υ ) | 2 + σ O 2 ( u ) | O ( u ) | 2 + σ O 2 ( u + υ ) | O ( u + υ ) | 2 ] .
σ g 2 ( x , y ) = ( x / 4 y ) ( σ x 2 / x 2 + σ y 2 / y 2 ) .
σ O 2 ( υ ) = { | B ( u , υ ) | [ 4 * | O ( u + υ ) | ] } × [ σ B 2 ( u , υ ) | B ( u , υ ) | 2 + σ O 2 ( u + υ ) | O ( u + υ ) | 2 ] .
σ g 2 ( x ) ( g / x ) 2 σ x 2 ) .
σ ln | B | 2 ( u , υ ) = σ B 2 ( u , υ ) / | B ( u , υ ) | 2 .

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