Abstract

A new weighted-least-squares algorithm for phase estimation from the bispectrum is presented. The algorithm uses phasors instead of phases to avoid modulo-2π ambiguities present in the bispectrum phases. The performance of this algorithm is compared with the performances of several other previously proposed algorithms from both simulated and field data. It is shown that the new algorithm results in more accurate phase-spectrum estimates than the other approaches.

© 1991 Optical Society of America

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  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).
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    [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. T. Matsuoka, T. J. Ulrych, “Phase estimation using the bispectrum,” Proc. IEEE 72, 1403–1411 (1984).
    [CrossRef]
  5. 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).
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  6. 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]
  7. D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johnson, “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. Intrum. Eng.1351, 546–560 (1990).
    [CrossRef]
  8. M. Rangoussi, G. B. Giannakis, “FIR modeling using logbispectra: weighted least-squares algorithms and performance analysis,” IEEE Trans. Circuits Syst. 38, 281–296 (1991).
    [CrossRef]
  9. A. Lannes, “On a new class of iterative algorithms for phaseclosure 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. O. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas closure phase imaging of six binary stars with the Hale 5 m telescope,” Astron. J. 98, 1783–1799 (1989).
    [CrossRef]
  11. J. Meng, G. J. M. Aitken, E. K. Hege, J. S. Morgan, “Triple-correlation subplane reconstruction of photon-address stellar images,” J. Opt. Soc. Am. A 7, 1243–1250 (1990).
    [CrossRef]
  12. T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bi-spectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
    [CrossRef]
  13. J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).
  14. J. D. Freeman, “Some statistical properties of the bispectrum of one-dimensional infrared astronomical speckle data,” in Amplitude and Intensity Spatial Interferometry, J. Breckenridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 436–447 (1990).
    [CrossRef]
  15. R. H. Hudgin, “Wavefront reconstruction for compensated imaging,” J. Opt. Soc. Am. 67, 375–378 (1977).
    [CrossRef]

1991

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

1990

1989

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

1988

1984

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

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

1983

1977

1970

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

Aitken, G. J. M.

Ayers, G. R.

Bartelt, H.

Cohn, D. L.

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Dainty, J. C.

Fitch, J. P.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bi-spectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
[CrossRef]

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johnson, “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. Intrum. Eng.1351, 546–560 (1990).
[CrossRef]

Freeman, J. D.

J. D. Freeman, “Some statistical properties of the bispectrum of one-dimensional infrared astronomical speckle data,” in Amplitude and Intensity Spatial Interferometry, J. Breckenridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 436–447 (1990).
[CrossRef]

Ghez, A. M.

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. O. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas 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 logbispectra: weighted least-squares algorithms and performance analysis,” IEEE Trans. Circuits Syst. 38, 281–296 (1991).
[CrossRef]

Goodman, D. M.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bi-spectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
[CrossRef]

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johnson, “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. Intrum. Eng.1351, 546–560 (1990).
[CrossRef]

Gorham, P. W.

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

Hege, E. K.

Hudgin, R. H.

Johnson, E. M.

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johnson, “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. Intrum. Eng.1351, 546–560 (1990).
[CrossRef]

Kulkarni, S. R.

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. O. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas 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 phaseclosure 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. Johnson, “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. Intrum. Eng.1351, 546–560 (1990).
[CrossRef]

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bi-spectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
[CrossRef]

Lohmann, A. W.

Marron, J. C.

Massie, N. A.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bi-spectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
[CrossRef]

Matsuoka, T.

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

Melsa, J. L.

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

Meng, J.

Morgan, J. S.

Nakajima, T.

P. W. Gorham, A. M. Ghez, S. R. Kulkarni, T. Nakajima, G. Neugebauer, J. O. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas 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. O. Oke, T. A. Prince, “Diffraction-limited imaging. III. 30 mas 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. O.

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

Prince, T. A.

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

Rangoussi, M.

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

Sanchez, P. P.

Sherwood, R. J.

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bi-spectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
[CrossRef]

Sullivan, R. C.

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.

Astron. Astrophys.

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.

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

IEEE Trans. Circuits Syst.

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

J. Opt. Soc. Am.

J. Opt. Soc. Am. A

Proc. IEEE

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

Other

D. M. Goodman, T. W. Lawrence, J. P. Fitch, E. M. Johnson, “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. Intrum. Eng.1351, 546–560 (1990).
[CrossRef]

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

T. W. Lawrence, J. P. Fitch, D. M. Goodman, N. A. Massie, R. J. Sherwood, “Experimental validation of extended image reconstruction using bi-spectral speckle interferometry,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 522–537 (1990).
[CrossRef]

J. L. Melsa, D. L. Cohn, Decision and Estimation Theory (McGraw-Hill, New York, 1978).

J. D. Freeman, “Some statistical properties of the bispectrum of one-dimensional infrared astronomical speckle data,” in Amplitude and Intensity Spatial Interferometry, J. Breckenridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 436–447 (1990).
[CrossRef]

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Figures (6)

Fig. 1
Fig. 1

Mean-square phasor error (in square radians) of the five estimated phase spectra (phasor WLS, classical WLS, WLS1, WLS2, and recursive) for a 2:1 binary star versus spatial-frequency magnitude normalized to 1 at the diffraction limit.

Fig. 2
Fig. 2

Reconstructions of a 2:1 binary star with (a) the truth Fourier spectrum and (b) the phasor WLS, (c) the classical WLS, (d) the WLS2, (e) the recursive, and (f) the WLS1 algorithms. The contour levels are at 5%, and the Fourier spectrum was truncated at 70% of the diffraction limit before inverse Fourier transformation.

Fig. 3
Fig. 3

Truth model of a satellite used for simulation results.

Fig. 4
Fig. 4

Mean-square phase error (in square radians) of the five estimated phase spectra (phasor WLS, classical WLS, WLS1, WLS2, and recursive) for the satellite pictured in Fig. 3 versus spatial-frequency magnitude normalized to 1 at the diffraction limit.

Fig. 5
Fig. 5

Reconstructions of the satellite shown in Fig. 3 with (a) the truth Fourier spectrum and (b) the phasor WLS, (c) the classical WLS, (d) the WLS2, (e) the recursive, and (f) the WLS1 algorithms. The Fourier spectrum was truncated at 40% of the diffraction limit before inverse Fourier transformation.

Fig. 6
Fig. 6

Reconstructions of 126 Tauri from phase spectra reconstructed by five different algorithms: (a) phasor WLS, (b) classical WLS, (c) WLS2, (d) recursive, (e) WLS1. The contour levels are at 5%, and the Fourier spectrum was truncated at 60% of the diffraction limit before inverse Fourier transformation.

Tables (1)

Tables Icon

Table 1 Summary of Phase Reconstruction Algorithms

Equations (22)

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B ( u , υ ) = O ( u ) O ( υ ) O * ( u + υ ) ,
β ( u , υ ) = ϕ ( u ) + ϕ ( υ ) ϕ ( u + υ ) ,
β = H ϕ ,
J ( Θ ) = ( β ) T W ( β ) ,
ϕ wls = ( H T WH ) 1 H T W β .
H T W H i ϕ wls = H T W i β .
ϕ i = ( u υ w ( u , υ ) [ β ( u , υ ) ϕ ( u ) + ϕ ( u + υ ) ] + u = υ 4 w ( u , υ ) { [ β ( u , υ ) + ϕ ( u + υ ) ] / 2 } + w ( u , υ u ) [ β ( u , υ u ) + ϕ ( u ) + ϕ ( υ u ) ] ) / [ u υ w ( u , υ ) + u = υ 4 w ( u , υ ) + w ( u , υ u ) ] ,
exp ( j ϕ i ) = u υ w ( u , υ ) [ B ( u , υ ) O ( u ) O * ( u + υ ) ] + u = υ 4 w ( u , υ ) × [ B ( u , υ ) O * ( u + υ ) ] 1 / 2 + w ( u , υ u ) [ B ( u , υ u ) O ( u ) O ( υ u ) ] * ,
O ( u + υ ) = w ( u , υ ) [ B ( u , υ ) O ( u ) O ( υ ) ] * ,
β = β + 2 π k ,
H ϕ = β + 2 π k .
CH = 0 .
Ck = C β / 2 π
k = ( H ϕ rec β ) / 2 π .
W ( i , j ) = { { SNR [ B ( i ) ] } 2 i = j 0 i j ,
exp ( j ϕ i ) = u υ w ( u , υ ) [ B ( u , υ ) O ( u ) O * ( u + υ ) ] + u = υ 4 w ( u , υ ) × [ B ( u , υ ) O * ( u + υ ) ] 1 / 2 + w ( u , υ u ) [ B ( u , υ u ) O ( u ) O ( υ u ) ] * .
exp ( j ϕ i ) = k w ( k ) exp ( j Θ k ) ,
ϕ i = tan 1 { Im [ exp ( j ϕ i ) ] Re [ exp ( j ϕ i ) ] } = tan 1 { k w ( k ) Im [ exp ( j Θ k ) ] k w ( k ) Re [ exp ( j Θ k ) ] } .
Im [ exp ( j Θ k ) ] Θ k ,
Re [ exp ( j Θ k ) ] 1 ,
tan 1 ( x ) x .
ϕ i = k w ( k ) Θ k / k w ( k ) ,

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