Abstract

The recovery of Fourier phases from measurements of the bispectrum occupies a vital role in many astronomical speckle imaging schemes. In a recent paper [ J. Opt. Soc. Am. A 7, 14 ( 1990)] it was suggested that a least-squares solution to this problem must fail if the bispectrum phase is known only modulo 2π. Here an alternative nonlinear least-squares algorithm is presented that differs from the linear method discussed in the aforementioned paper and that permits the fitting of Fourier phases directly to modulo 2π measurements of the bispectrum phase, thus eliminating any need for phase unwrapping. Numerical simulations of this method confirm that it is reliable and robust in the presence of noise and verify its enhanced performance when compared with a linear least-squares method that includes the unwrapping of the bispectral phase before Fourier phase retrieval.

© 1991 Optical Society of America

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Corrections

Christopher A. Haniff, "Least-squares Fourier phase estimation from the modulo 2π bispectrum phase: erratum," J. Opt. Soc. Am. A 8, 1517-1517 (1991)
https://www.osapublishing.org/josaa/abstract.cfm?uri=josaa-8-9-1517

References

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  1. D. H. Rogstad, “A technique for measuring visibility phase with an optical interferometer in the presence of atmospheric seeing,” Appl. Opt. 7, 585–588 (1968).
    [CrossRef] [PubMed]
  2. H. Bartelt, A. W. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from the bispectrum,” Appl. Opt. 23, 3121–3129 (1984).
    [CrossRef] [PubMed]
  3. 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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
    [CrossRef]
  4. J. C. Marron, P. P. Sanchez, R. C. Sullivan, “Unwrapping algorithm for least-squares phase recovery from the modulo 2πbispectrum,” J. Opt. Soc. Am. A 7, 14–20 (1990).
    [CrossRef]
  5. A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
    [CrossRef]
  6. D. F. Buscher, “Getting the most out of C.O.A.S.T.,” Ph.D. dissertation (Cambridge University, Cambridge, 1988).
  7. See, e.g., J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.
  8. C. C. Paige, M. A. Saunders, “Algorithm 583 LSQR: sparse linear equations and least squares problems,”ACM Trans. Math. Software 8, 195–209 (1982).
    [CrossRef]
  9. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

1990 (1)

1989 (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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

1984 (1)

1982 (1)

C. C. Paige, M. A. Saunders, “Algorithm 583 LSQR: sparse linear equations and least squares problems,”ACM Trans. Math. Software 8, 195–209 (1982).
[CrossRef]

1968 (1)

Bartelt, H.

Buscher, D. F.

D. F. Buscher, “Getting the most out of C.O.A.S.T.,” Ph.D. dissertation (Cambridge University, Cambridge, 1988).

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
[CrossRef]

Goodman, J. W.

See, e.g., J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.

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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
[CrossRef]

Haniff, C. A.

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
[CrossRef]

Lohmann, A. W.

Marron, J. C.

Matthews, K.

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
[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 5m 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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
[CrossRef]

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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Paige, C. C.

C. C. Paige, M. A. Saunders, “Algorithm 583 LSQR: sparse linear equations and least squares problems,”ACM Trans. Math. Software 8, 195–209 (1982).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

Rogstad, D. H.

Sanchez, P. P.

Saunders, M. A.

C. C. Paige, M. A. Saunders, “Algorithm 583 LSQR: sparse linear equations and least squares problems,”ACM Trans. Math. Software 8, 195–209 (1982).
[CrossRef]

Sullivan, R. C.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

Weir, N.

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
[CrossRef]

Wirnitzer, B.

ACM Trans. Math. Software (1)

C. C. Paige, M. A. Saunders, “Algorithm 583 LSQR: sparse linear equations and least squares problems,”ACM Trans. Math. Software 8, 195–209 (1982).
[CrossRef]

Appl. Opt. (2)

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 5m telescope,” Astron. J. 98, 1783–1799 (1989).
[CrossRef]

J. Opt. Soc. Am. A (1)

Other (4)

A. M. Ghez, P. W. Gorham, C. A. Haniff, S. R. Kulkarni, K. Matthews, G. Neugebauer, N. Weir, “Infrared speckle imaging at Palomar,” in Amplitude and Intensity Spatial Interferometry, J. Breckinridge, ed., Proc. Soc. Photo-Opt. Instrum. Eng.1237, 249–258 (1990).
[CrossRef]

D. F. Buscher, “Getting the most out of C.O.A.S.T.,” Ph.D. dissertation (Cambridge University, Cambridge, 1988).

See, e.g., J. W. Goodman, Statistical Optics (Wiley, New York, 1985), Chap. 2.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes (Cambridge U. Press, Cambridge, 1986).

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Figures (11)

Fig. 1
Fig. 1

Argand diagram representations of the misfit criteria Δi,j and Θi,j corresponding to the objective function 2 and 3 described in the text. The two vectors O P and O Q , ending on the unit circle, represent a pair of measured and model unit bivectors, respectively.

Fig. 2
Fig. 2

Reconstructed hybrid image derived from slightly noisy bispectrum phases (γ = 2.5) by using the unwrapping and weighted least-squares method of Ref. 4. In this, and all subsequent plots, the solid curve shows the original test source and the dotted curve the reconstructed image, shifted so as to facilitate comparison with the original.

Fig. 3
Fig. 3

Image reconstruction from the same data set as Fig. 2 but using the weighted least-squares minimization of the objective function 3.

Fig. 4
Fig. 4

Unwrapped hybrid image of Ref. 4 derived from bispectral phases of low SNR (γ = 1.25).

Fig. 5
Fig. 5

3 image obtained by using the same data set as that of Fig. 4.

Fig. 6
Fig. 6

Unwrapped hybrid image of Ref. 4 derived from bispectral phases of low SNR (γ = 1.0). At these noise levels the unwrapping procedure of Ref. 4 has become unreliable, and so accurate image restoration is no longer possible.

Fig. 7
Fig. 7

3 image obtained by using the same data set as that of Fig. 6. Note the considerable improvement in image quality.

Fig. 8
Fig. 8

Hybrid image reconstruction obitaned by using the unwrapping algorithm of Ref. 4. In this example 20% of the bispectrum phases have been corrupted with errors uniformly distributed on ±π. The remainder are characterized by a γ value of 2.5.

Fig. 9
Fig. 9

3 hybrid image recovered from the same data set used to produce Fig. 8.

Fig. 10
Fig. 10

Hybrid image of Ref. 4 recovered from a γ = 2.5 data set, in which 75% of bispectrum phases have been corrupted with errors uniformly distributed on ±π.

Fig. 11
Fig. 11

3 hybrid image recovered from the same data set used to produce Fig. 10.

Equations (10)

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I ˜ 3 ( u , v ) = I ˜ ( u ) I ˜ ( v ) I ˜ * ( u + v ) = I ˜ 3 ( u , v ) exp [ i ψ ( u , v ) ] .
ψ ( u , v ) = ϕ ( u ) + ϕ ( v ) - ϕ ( u + v )             ( mod 2 π )
ψ i , j = ϕ i + ϕ j - ϕ i + j             ( mod 2 π ) ,
ψ i , j = ϕ i + ϕ j - ϕ i + j ,
A Φ = Ψ ,
F 1 = i , j P N [ ψ i , j - ( ϕ ^ i + ϕ ^ j - ϕ ^ i + j ) w i , j ] 2 ,
F 2 = i , j P N [ Re ( Δ i , j ) w i , j ] 2 + [ Im ( Δ i , j ) w i , j ] 2 .
Δ i , j = exp ( i ψ i , j ) - exp [ i ( ϕ ^ i + ϕ ^ j - ϕ ^ i + j ) ] .
F 3 = i , j P N { Mod [ ψ i , j - ( ϕ ^ i + ϕ ^ j - ϕ ^ i + j ) ] w i , j } 2 = i , j P N [ Mod ( Θ i , j ) w i , j ] 2 ,
Var [ ψ i , j ] = w i , j 2 = V [ I ˜ i , j 3 ] N I ˜ i , j 3 ,

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