Abstract

The Knox–Thompson, or cross-spectrum, method provides two two-dimensional difference equations for the phase of the object spectrum. We demonstrate that, in general, the object spectrum phase can be decomposed into a regular, single-valued function determined by the divergence of the phase gradient, as well as a multivalued function determined by the circulation of the phase gradient; this second function has been called the hidden phase. The standard least-squares solution to the two-dimensional difference equations will always miss this hidden phase. We present a solution method that gives both the regular and the hidden parts of the object spectrum phase. Finally, we illustrate several examples of imaging through turbulence and postprocessing with the Knox–Thompson method, including the hidden phase.

© 2000 Optical Society of America

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References

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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).
  2. K. T. Knox, B. J. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astron. J. 193, L45–L48 (1974).
    [CrossRef]
  3. 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]
  4. D. L. Fried, “Branch point problem in adaptive optics,” J. Opt. Soc. Am. A 15, 2759–2768 (1998).
    [CrossRef]
  5. J. C. Fontanella, A. Seve, “Reconstruction of turbulence-degraded images using the Knox–Thompson algorithm,” J. Opt. Soc. Am. A 4, 438–448 (1987).
    [CrossRef]
  6. H. Takajo, T. Takahashi, “Least-squares phase estimation from the phase difference,” J. Opt. Soc. Am. A 5, 416–425 (1988).
    [CrossRef]
  7. W. W. Arrasmith, “Branch-point tolerant least-squares phase reconstructor,” J. Opt. Soc. Am. A 16, 1864–1872 (1999).
    [CrossRef]
  8. E. Le Bigot, W. J. Wild, “Theory of branch-point detection and its implementation,” J. Opt. Soc. Am. A 16, 1724–1729 (1999).
    [CrossRef]
  9. T. W. Lawrence, D. M. Goodman, E. M. Johansson, J. P. Fitch, “Speckle imaging of satellites at the U.S. Air Force Maui Optical Station,” Appl. Opt. 31, 6307–6321 (1992).
    [CrossRef] [PubMed]
  10. A. Sommerfeld, Mechanics of Deformable Bodies: Lectures on Theoretical Physics (Academic, New York, 1959), Vol. II, Sec. IV.20.
  11. W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, New York, 1992), p. 116.

1999 (2)

1998 (1)

1992 (1)

1988 (2)

1987 (1)

1974 (1)

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

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).

Arrasmith, W. W.

Ayers, G. R.

Dainty, J. C.

Fitch, J. P.

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, New York, 1992), p. 116.

Fontanella, J. C.

Fried, D. L.

Goodman, D. M.

Johansson, E. M.

Knox, K. T.

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

Lawrence, T. W.

Le Bigot, E.

Northcott, M. J.

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, New York, 1992), p. 116.

Seve, A.

Sommerfeld, A.

A. Sommerfeld, Mechanics of Deformable Bodies: Lectures on Theoretical Physics (Academic, New York, 1959), Vol. II, Sec. IV.20.

Takahashi, T.

Takajo, H.

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, New York, 1992), p. 116.

Thompson, B. J.

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

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, New York, 1992), p. 116.

Wild, W. J.

Appl. Opt. (1)

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)

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

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

Other (2)

A. Sommerfeld, Mechanics of Deformable Bodies: Lectures on Theoretical Physics (Academic, New York, 1959), Vol. II, Sec. IV.20.

W. H. Press, B. P. Flannery, S. A. Teukolsky, W. T. Vetterling, Numerical Recipes in Fortran 77, 2nd ed. (Cambridge U. Press, New York, 1992), p. 116.

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Figures (2)

Fig. 1
Fig. 1

(a) Diffraction-limited four-star image. (b) Long-time average with D/ r 0 = 10. (c) Reconstruction with the hidden phase. (d) Reconstruction without the hidden phase.

Fig. 2
Fig. 2

(a) Diffraction-limited binary image. (b) Long-time average with D/ r 0 = 10. (c) Reconstruction with the hidden phase. (d) Reconstruction without the hidden phase.

Equations (25)

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dl; i, j.
d˜l; m, n=FFTdl; i, j.
Cxm, n=1Rl=1Rd˜l; m, nd˜*l; m+1, n-d˜*l; m¯+1, n¯, Cym, n=1Rl=1Rd˜l; m, nd˜*l; m, n+1-d˜*l; m¯, n¯+1,
Cxm, n=|Cxm, n|expiVx, Cym, n=|Cym, n|expiVy,
Vxm, nWm, n-Wm+1, n, Vym, nWm, n-Wm, n+1.
Vxm, n-ΔxWm, n, Vym, n-ΔyWm, n.
V˜xK, L=-Δ˜xW˜, V˜yK, L=-Δ˜yW˜,
Δ˜xK, L=exp-i2πK-1/N-1, Δ˜yK, L=exp-i2πL-1/N-1.
gradW˜Δ˜xW˜, Δ˜yW˜, divV˜Δ˜x*V˜x+Δ˜y*V˜y, curlV˜Δ˜xV˜y-Δ˜yV˜x.
VSΔ˜x, Δ˜yΔ˜xΔ˜x*+Δ˜yΔ˜y*Δ˜x*V˜x+Δ˜y*V˜y,
V˜H-Δ˜y*Δ˜xV˜y-Δ˜yV˜x, Δ˜x*Δ˜xV˜y-Δ˜yV˜xΔ˜xΔ˜x*+Δ˜yΔ˜y*.
curlV˜S=0, divV˜H=0.
VS=-Δx, Δy * WS, VH=-Δx, Δy * WH.
W˜S=-Δ˜x*V˜x+Δ˜y*V˜yΔ˜xΔ˜x*+Δ˜yΔ˜y*.
Wˆm, n=m,nFxm-m, n-nVxm, n+Fym-m, n-nVym, n.
Error=K,L |W˜K, L+F˜xK, LΔ˜x+F˜yK, LΔ˜yW˜K, L|2.
Wˆ˜K, L=-Δ˜x*V˜x+Δ˜y*V˜yΔ˜xΔ˜x*+Δ˜yΔ˜y*.
ΔyΔx-ΔxΔyWH=ΔxVy-ΔyVxω.
ΔyΔx-ΔxΔy12πarctann-n-0.5m-m-0.5=δn,nδm,m.
WHm, n=12πm,narctann-n-0.5m-m-0.5ωm, n.
Wm, n=FFT-1Wˆ˜K, L+WHm, n.
ΔxVy-ΔyVx=ωm, n.
ωm, n=0, ±2π.
ωm, n0 if -πωm, nπ, ωm, n2π if  πωm, n, ωm, n -2π if -πωm, n.
Ii, j=FFT-1|Ĩm, n|expiWSm, n+WHm, n.

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