Abstract

A problem of blind deconvolution arises when one attempts to restore a short-exposure image that has been degraded by random atmospheric turbulence. We attack the problem by using two short-exposure images as data inputs. The Fourier transform of each is taken, and the two are divided. The result is the quotient of the two unknown transfer functions. The latter are expressed, by means of the sampling theorem, as Fourier series in corresponding point-spread functions, the unknowns of the problem. Cross multiplying the division equation gives an equation that is linear in the unknowns. However, the problem has, initially, a multiplicity of solutions. This deficiency is overcome by use of the prior knowledge that the object and the point-spread functions have finite (albeit unknown) support extensions and also are positive. The result is a fixed-length, linear algorithm that is regularized to the presence of 4–15% additive noise of detection.

© 1999 Optical Society of America

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References

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  1. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  2. F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
    [CrossRef]
  3. J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1984), pp. 255–320.
  4. J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.
  5. R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1544 (1970).
    [CrossRef]
  6. T. J. Holmes, “Blind deconvolution of quantum-limited incoherent imagery: maximum-likelihood approach,” J. Opt. Soc. Am. A 9, 1052–1061 (1992).
    [CrossRef] [PubMed]
  7. M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).
  8. G. Ayers, J. C. Dainty, “Iterative blind deconvolution method and its applications,” Opt. Lett. 13, 547–549 (1988).
    [CrossRef]
  9. R. G. Lane, “Blind deconvolution of speckle images,” J. Opt. Soc. Am. A 9, 1508–1524 (1992).
    [CrossRef]
  10. P. A. Jansson, Deconvolution of Images and Spectra, 2nd ed. (Academic, San Diego, Calif., 1997).
  11. B. R. Frieden, “An exact, linear solution to the problem of imaging through turbulence,” Opt. Commun. 150, 15–21 (1998).
    [CrossRef]
  12. D. Fowley, M. Horton, J. Scordato, eds., matlab, Student Edition, Version 4 (Prentice-Hall, Englewood Cliffs, N. J.1995).
  13. B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed., Vol. 6 of Topics in Applied Physics (Springer-Verlag, New York, 1975), p. 221.

1998 (1)

B. R. Frieden, “An exact, linear solution to the problem of imaging through turbulence,” Opt. Commun. 150, 15–21 (1998).
[CrossRef]

1992 (2)

1988 (1)

1970 (1)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1544 (1970).
[CrossRef]

Ayers, G.

Dainty, J. C.

G. 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., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1984), pp. 255–320.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Fienup, J. R.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

Frieden, B. R.

B. R. Frieden, “An exact, linear solution to the problem of imaging through turbulence,” Opt. Commun. 150, 15–21 (1998).
[CrossRef]

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed., Vol. 6 of Topics in Applied Physics (Springer-Verlag, New York, 1975), p. 221.

Goodman, J. W.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

Holmes, T. J.

Jansson, P. A.

P. A. Jansson, Deconvolution of Images and Spectra, 2nd ed. (Academic, San Diego, Calif., 1997).

Lane, R. G.

Lawrence, R. S.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1544 (1970).
[CrossRef]

Roddier, F.

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
[CrossRef]

Roggemann, M. C.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

Strohbehn, J. W.

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1544 (1970).
[CrossRef]

Welsh, B.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

J. Opt. Soc. Am. A (2)

Opt. Commun. (1)

B. R. Frieden, “An exact, linear solution to the problem of imaging through turbulence,” Opt. Commun. 150, 15–21 (1998).
[CrossRef]

Opt. Lett. (1)

Proc. IEEE (1)

R. S. Lawrence, J. W. Strohbehn, “A survey of clear-air propagation effects relevant to optical communications,” Proc. IEEE 58, 1523–1544 (1970).
[CrossRef]

Other (8)

P. A. Jansson, Deconvolution of Images and Spectra, 2nd ed. (Academic, San Diego, Calif., 1997).

D. Fowley, M. Horton, J. Scordato, eds., matlab, Student Edition, Version 4 (Prentice-Hall, Englewood Cliffs, N. J.1995).

B. R. Frieden, “Image enhancement and restoration,” in Picture Processing and Digital Filtering, T. S. Huang, ed., Vol. 6 of Topics in Applied Physics (Springer-Verlag, New York, 1975), p. 221.

M. C. Roggemann, B. Welsh, Imaging through Turbulence (CRC Press, Boca Raton, Fla., 1996).

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).

F. Roddier, “The effects of atmospheric turbulence in optical astronomy,” in Progress in Optics, E. Wolf, ed. (North-Holland, Amsterdam, 1981), Vol. XIX, pp. 281–376.
[CrossRef]

J. C. Dainty, “Stellar speckle interferometry,” in Laser Speckle and Related Phenomena, 2nd ed., J. C. Dainty, ed., Vol. 9 of Topics in Applied Physics (Springer-Verlag, New York, 1984), pp. 255–320.

J. C. Dainty, J. R. Fienup, “Phase retrieval and image reconstruction for astronomy,” in Image Recovery, Theory and Application, H. Stark, ed. (Academic, New York, 1987), pp. 231–275.

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Figures (6)

Fig. 1
Fig. 1

Flow chart for the image division program.

Fig. 2
Fig. 2

Object reconstructions at given levels of detector noise (the indicated coordinates are pixels): (a) letter C object, (b) reconstruction with 0% detection noise, (c) reconstruction with 2% noise, (d) reconstruction with 4% noise.

Fig. 3
Fig. 3

Data PSF’s and their reconstructions for 4% noise: (a) PSF 1, (b) PSF 2, (c) reconstruction of PSF 1, (d) reconstruction of PSF 2.

Fig. 4
Fig. 4

Data images for 4% noise: (a) image 1, (b) image 2.

Fig. 5
Fig. 5

Impulse object case studies: (a) object, (b) reconstruction with 5% detection noise, (c) reconstruction with 10% noise, (d) reconstruction with 15% noise.

Fig. 6
Fig. 6

Does data inconsistency e attain a grand minimum at the correct support levels? Inconsistency e is plotted versus support component x for various values of component y : y = 9 (pluses), y = 8 (open circles), y = 7 (asterisks). The correct support levels (K x , K y ) = (8, 8) are indeed attained at the grand minimum (see the open-circle curve).

Equations (28)

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Iω-ΩωΩ=0Ldx ixexp-jωx.
Iω-ΩωΩ=m=0M-1 Δx imΔxexp-jωmΔx.
Δx  π/Ω
Iω0ω2Ω=m=0M-1 Δx im exp-jωmΔx,im  imΔx
ωn  2n/NΩ,  n=0, 1,, N-1.
jωnmΔx=j 2nN Ω mπΩ=2πjmnN.
Iωn  In=Δx m=0M-1 im exp-2πjmn/N,n=0, 1,, N-1.
Ini=τniOn,  i=1, 2; n=0, 1,, N-1.
Dn  In1In2=τn1Onτn2On=τn1τn2,  n=0, 1,, N-1.
τni=Δx m=0M-1 smi exp-2πjmn/N,  i=1, 2.
Dn=m=0M-1 sm1 exp-2πjmn/Nm=0M-1 sm2 exp-2πjmn/N,n=0, 1,, N-1.
m=0M-1sm1-Dnsm2exp-2πjmn/N=0,n=0, 1,, N-1.
Dn  Mn exp2πjϕn/N,
Re part:m sm1 cos2πmnN-m Mnsm2× cos2πNϕn-mn=0,Im part:m sm1 sin2πmnN+m Mnsm2× sin2πNϕn-mn=0.
Ei  m smi,  i=1, 2.
sk1=E1-mk sm1.
mk sm1cos2πmnN-cos2πknN-m Mmsm2×cos2πNϕn-mn=-E1 cos2πknN,mk sm1sin2πmnN-sin2πknN+m Mmsm2×sin2πNϕn-mn=-E1 sin2πknN.
Hx=b,
om0,  sm10,  sm20.
HKˆxKˆ=bKˆ.
HKˆxKˆ-bKˆ=min.,
Oˆi  Ii/τi,  i=1, 2
Kˆobx  Kimx-Kˆx+1.
iˆ+i  sˆ+i  oˆ+i,  i=1, 2.
e  e1+e2,  ei  |ii-iˆ+i||ii|.
oˆ  ½oˆ+1+oˆ+2.
ĩ1  a1i1+a2i2,ĩ2  b1i1+b2i2,  a1+a2=1, b1+b2=1.
s˜1=a1s1+a2s2,s˜2=b1s1+b2s2.

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