Abstract

Traditional techniques in astronomical speckle imaging require simultaneous measurements on a reference star to estimate the object Fourier modulus. Although the phase recovery problem has been successfully solved with the use of the bispectrum, the unwanted requirement of measurements on a reference is still present. Two minimization approaches to recover the desired diffraction-limited object from the bispectral phase alone are suggested.

© 1998 Optical Society of America

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References

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  1. G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef]
  2. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
    [CrossRef]
  3. J. C. Christou, E. K. Hege, S. M. Jefferies, “Multiframe blind deconvolution for object and PSF recovery for astronomical imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 70–73.
  4. R. G. Paxman, J. H. Seldin, “Simulation validation of phase-diverse speckle imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 85–87.
  5. G. Weigelt, “Triple-correlation imaging in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. XXIX, pp. 293–319.
  6. J.-Y. Zhang, J. C. Dainty, “Effects of aberrations on transfer functions used in high angular resolution astronomical imaging,” J. Mod. Opt. 39, 2383–2404 (1992).
    [CrossRef]
  7. P. Negrete-Regagnon, “Phase recovery from the bispectrum aided by the error-reduction algorithm,” Opt. Lett. 21, 275–277 (1996).
    [CrossRef] [PubMed]
  8. A. Glindemann, J. C. Dainty, “Object fitting to the bispectral phase using least squares,” J. Opt. Soc. Am. A 10, 1056–1063 (1993).
    [CrossRef]
  9. P. Negrete-Regagnon, “Practical aspects of image recovery by means of the bispectrum,” J. Opt. Soc. Am. A 13, 1557–1576 (1996).
    [CrossRef]
  10. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J., Lett. Ed. 193, L45–L48 (1974).
    [CrossRef]
  11. A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  12. A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
    [CrossRef]
  13. G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
    [CrossRef]
  14. J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), p. 260.
  15. B. Wirnitzer, “Bispectral analysis at low light levels and astronomical speckle masking,” J. Opt. Soc. Am. A 2, 14–21 (1985).
    [CrossRef]
  16. NAG Fortran Library Manual, Mark 14, Chapter E04, Routine E04DGF. The NAG Fortran Library is available from NAG Inc., 1400 Opus Place, Suite 200, Downers Grove, Illinois 60515-5702.
  17. E. Thiébaut, J.-M. Conan, “Strict a priori constraints for maximum-likelihood blind deconvolution,” J. Opt. Soc. Am. A 12, 485–492 (1995).
    [CrossRef]

1996 (2)

1995 (1)

1993 (1)

1992 (2)

R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1514 (1992).
[CrossRef]

J.-Y. Zhang, J. C. Dainty, “Effects of aberrations on transfer functions used in high angular resolution astronomical imaging,” J. Mod. Opt. 39, 2383–2404 (1992).
[CrossRef]

1988 (2)

1985 (1)

1984 (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

1983 (1)

1974 (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J., Lett. Ed. 193, L45–L48 (1974).
[CrossRef]

Ayers, G. R.

Christou, J. C.

J. C. Christou, E. K. Hege, S. M. Jefferies, “Multiframe blind deconvolution for object and PSF recovery for astronomical imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 70–73.

Conan, J.-M.

Dainty, J. C.

A. Glindemann, J. C. Dainty, “Object fitting to the bispectral phase using least squares,” J. Opt. Soc. Am. A 10, 1056–1063 (1993).
[CrossRef]

J.-Y. Zhang, J. C. Dainty, “Effects of aberrations on transfer functions used in high angular resolution astronomical imaging,” J. Mod. Opt. 39, 2383–2404 (1992).
[CrossRef]

G. R. Ayers, M. J. Northcott, J. C. Dainty, “Knox–Thompson and triple-correlation imaging through atmospheric turbulence,” J. Opt. Soc. Am. A 5, 963–985 (1988).
[CrossRef]

G. R. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
[CrossRef]

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), p. 260.

Glindemann, A.

Hege, E. K.

J. C. Christou, E. K. Hege, S. M. Jefferies, “Multiframe blind deconvolution for object and PSF recovery for astronomical imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 70–73.

Jefferies, S. M.

J. C. Christou, E. K. Hege, S. M. Jefferies, “Multiframe blind deconvolution for object and PSF recovery for astronomical imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 70–73.

Knox, K. T.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J., Lett. Ed. 193, L45–L48 (1974).
[CrossRef]

Lane, R. G.

Lohmann, A. W.

Negrete-Regagnon, P.

Northcott, M. J.

Paxman, R. G.

R. G. Paxman, J. H. Seldin, “Simulation validation of phase-diverse speckle imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 85–87.

Seldin, J. H.

R. G. Paxman, J. H. Seldin, “Simulation validation of phase-diverse speckle imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 85–87.

Thiébaut, E.

Thompson, B. J.

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J., Lett. Ed. 193, L45–L48 (1974).
[CrossRef]

Weigelt, G.

A. W. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
[CrossRef] [PubMed]

G. Weigelt, “Triple-correlation imaging in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. XXIX, pp. 293–319.

Wirnitzer, B.

Zhang, J.-Y.

J.-Y. Zhang, J. C. Dainty, “Effects of aberrations on transfer functions used in high angular resolution astronomical imaging,” J. Mod. Opt. 39, 2383–2404 (1992).
[CrossRef]

Appl. Opt. (1)

Astrophys. J., Lett. Ed. (1)

K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J., Lett. Ed. 193, L45–L48 (1974).
[CrossRef]

J. Mod. Opt. (1)

J.-Y. Zhang, J. C. Dainty, “Effects of aberrations on transfer functions used in high angular resolution astronomical imaging,” J. Mod. Opt. 39, 2383–2404 (1992).
[CrossRef]

J. Opt. Soc. Am. A (6)

Opt. Lett. (2)

Proc. IEEE (1)

A. W. Lohmann, B. Wirnitzer, “Triple correlations,” Proc. IEEE 72, 889–901 (1984).
[CrossRef]

Other (5)

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed. (Springer-Verlag, Berlin, 1984), p. 260.

NAG Fortran Library Manual, Mark 14, Chapter E04, Routine E04DGF. The NAG Fortran Library is available from NAG Inc., 1400 Opus Place, Suite 200, Downers Grove, Illinois 60515-5702.

J. C. Christou, E. K. Hege, S. M. Jefferies, “Multiframe blind deconvolution for object and PSF recovery for astronomical imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 70–73.

R. G. Paxman, J. H. Seldin, “Simulation validation of phase-diverse speckle imaging,” in Signal Recovery and Synthesis, Vol. 11 of 1995 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1995), pp. 85–87.

G. Weigelt, “Triple-correlation imaging in optical astronomy,” in Progress in Optics, E. Wolf, ed. (Elsevier, Amsterdam, 1991), Vol. XXIX, pp. 293–319.

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Figures (6)

Fig. 1
Fig. 1

Five different nonnegative objects reconstructed from the bispectral phase of a simulated object. Sixteen subplanes (k=0, , 3, l=0, , 3) of the bispectral phase are illustrated in (a). The original object is shown in (b)(1). The minimization has been carried out over E2 by varying the real and imaginary image–Fourier components.

Fig. 2
Fig. 2

Five object estimates, at different levels of stagnation, reconstructed from the bispectral phase. The bispectral phase corresponds to the one shown in Fig. 1(a). The original object is shown in (1). The minimization has been carried out over E2 by varying image components.

Fig. 3
Fig. 3

Reconstructions of the binary star β Del based only on bispectral-phase information. The data correspond to the adaptively compensated speckle frames described in Subsection 8.B of Ref. 9.

Fig. 4
Fig. 4

Example showing a badly driven minimization that is due to a nonadequate initial estimate. The average bispectral phase (and its SNR) corresponds to the β Del object and is shown in Fig. 3(a). The minimization has been carried out over E2 by varying the image–Fourier components. For this particular example, the initial guess has been set to zero for both real and imaginary arrays. The reconstructed object is positive and possesses exactly the same bispectral phase as those shown in Figs. 3(b) and 3(c) but is clearly not the desired final image.

Fig. 5
Fig. 5

Reconstruction of a simulated star cluster: (a) object recovery with use of the conventional phase-only bispectral technique, (b) reconstruction based only on bispectral phase information. The original object and the speckle data set are described in Fig. 12 of Ref. 9. The object was recovered by using 16 bispectral subplanes (k=0, , 3, l=0, , 3), and the minimization was carried out over image–Fourier components for the results shown in (b). The associated Fourier moduli and phases are compared in Fig. 6 below. The contour lines correspond to the area indicated by the white rectangles in the images.

Fig. 6
Fig. 6

Normalized Fourier moduli and modulo 2π phase differences for the reconstructed objects of Fig. 5. The Fourier moduli are associated with (a) the diffraction-limited image, (b) the reconstructed object with use of the information from a reference star, and (c) the object recovered from the bispectral phase only. The difference between the Fourier phase associated with the diffraction-limited image and the phase recovered from the bispectral phase with use of the conventional algorithm is given in (d), and the difference between the Fourier phase and the phase estimated with use of the algorithm described in this paper is given in (e). The difference between both phases is shown in (f). The telescope diffraction limit is indicated by the circles. The gray scale of (a) applies also to (b) and (c), and the one in (f) applies also to (d) and (e).

Equations (37)

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in(x, y)=o(x, y)*hn(x, y),
O(3)(u1, v1, u2, v2)=O(u1, v1)O(u2, v2)O*(u1+u2, v1+v2)
=|O(3)(u1, v1, u2, v2)|exp[iβ(u1, v1, u2, v2)],
In(3)(u1, v1, u2, v2)=O(3)(u1, v1, u2, v2)Tn(3)(u1, v1, u2, v2),
β(u1, v1, u2, v2)=ϕ(u1, v1)+ϕ(u2, v2)-ϕ(u1+u2, v1+v2).
E2=l=lminlmaxk=kminkmaxj=1n coli=1n row{[Re(Δijkl)]2+[Im(Δijkl)]2}SNRijklβ,
Δijkl=exp(iβijkl)-exp[i(ϕij+ϕkl-ϕi+k,j+l)],
Re(Δijkl)=cos(βijkl)-cos(ϕij+ϕkl-ϕi+k,j+l),
Im(Δijkl)=sin(βijkl)-sin(ϕij+ϕkl-ϕi+k,j+l).
ϕ(u, v)=arctanIm[F{i(x, y)}]Re[F{i(x, y)}],
ϕmn=arctan(Imn/Rmn),
E2(Fmn),
E2Fmn=E2ϕmnϕmnFmn.
E2ϕmn=l=lminlmaxk=kminkmax[2 Re(Δmnkl)sin(ϕmn+ϕkl-ϕm+k,n+l)-2 Im(Δmnkl)cos(ϕmn+ϕkl-ϕm+k,n+l)]SNRmnklβ+j=1n coli=1n row[2 Re(Δijmn)sin(ϕij+ϕmn-ϕi+m,j+n)-2 Im(Δijmn)cos(ϕij+ϕmn-ϕi+m,j+n)]SNRijmnβ+l=lminlmaxk=kminkmax[2 Im(Δm-k,n-l,k,l)cos(ϕm-k,n-l+ϕkl-ϕmn)-2 Re(Δm-k,n-l,k,l)sin(ϕm-k,n-l+ϕkl-ϕmn)]*SNRm-k,n-l,k,lβ.
ϕmnFmn=ϕmnRmn=-ImnRmn2+Imn2,
ϕmnFmn=ϕmnImn=RmnRmn2+Imn2.
Ei=ij|iij|-iij22,
EiFmn=EiiijiijFmn,
Eiiij=0-(|iij|-iij)ifiij0ifiij<0.
iij=m=1Mn=1N(Rmn+iImn)×expi2π(i-1)(m-1)M+(j-1)(n-1)N,
iijFmn=iijRmn=cos(2πΘ)+i sin(2πΘ)=cos(2πΘ),
iijFmn=iijImn=-sin(2πΘ)+i cos(2πΘ)=-sin(2πΘ),
Θ=(i-1)(m-1)M+(j-1)(n-1)N.
E2(Fmn)+Ei(Fmn),
E2Rmn+EiRmn=E2ϕmnϕmnRmn+EiiijiijRmn,
E2Imn+EiImn=E2ϕmnϕmnImn+EiiijiijImn.
Es=ij(iij-iijsij)2,
Esiij=2(iij-iijsij)(1-sij).
E2(iij)+Ei(iij),
E2iij+Eiiij=E2ϕmnϕmniij+Eiiij.
ϕmniij=arctan(arg)arg argiij,
arg=ImnRmn,
arctan(arg)arg=11+arg2=Rmn2Rmn2+Imn2.
argiij=RmnImniij-ImnRmniijRmn2,
Rmniij=cos(2πΘ),
Imniij=-sin(2πΘ).
ϕmniij=-Rmn sin(2πΘ)+Imn cos(2πΘ)Rmn2+Imn2.

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