Abstract

A priori knowledge of the image support, often called the support constraint, when applied to a raw image formed by an imaging system leads, by means of convolution with the support spectrum, to a mixing of the spatial frequencies in the image. For a noisy raw image, such mixing causes motion of noise in its spatial-frequency spectrum. A simple model for describing the motion of atmospheric-turbulence-induced noise in the spectrum of a star image formed by a ground-based system under the application of a support constraint is presented. The transport of noise occurs in this model by a combination of ballistic motion, or drift, and diffusive spreading. For an inversion-symmetric support, the drift is absent, and noise transport is exclusively diffusive. An analytical expression for the reduction of noise that such diffusive spreading in the spatial-frequency plane can facilitate, when a circular support of arbitrary size is employed, is derived.

© 1999 Optical Society of America

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References

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  1. A. Lohmann, G. Weigelt, B. Wirnitzer, “Speckle masking in astronomy: triple correlation theory and applications,” Appl. Opt. 22, 4028–4037 (1983).
    [CrossRef] [PubMed]
  2. H. Bartelt, A. Lohmann, B. Wirnitzer, “Phase and amplitude recovery from bispectra,” Appl. Opt. 23, 3121–3129 (1984).
    [CrossRef] [PubMed]
  3. K. Knox, B. Thompson, “Recovery of images from atmospherically degraded short-exposure photographs,” Astrophys. J. 193, L45–L48 (1974).
    [CrossRef]
  4. See, e.g., R. Narayan, R. Nityananda, “Maximum-entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
    [CrossRef]
  5. C. Matson, “Variance reduction in Fourier spectra and their corresponding images with the use of support constraints,” J. Opt. Soc. Am. A 11, 97–106 (1994).
    [CrossRef]
  6. C. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Process. 42, 156–163 (1994).
    [CrossRef]
  7. C. Matson, M. Roggemann, “Noise reduction in adaptive optics imagery with the use of support constraints,” Appl. Opt. 34, 767–780 (1995).
    [CrossRef] [PubMed]
  8. D. Tyler, C. Matson, “Reduction of nonstationary noise in telescope imagery using a support constraint,” Opt. Express 1, 347–354 (1997).
    [CrossRef] [PubMed]
  9. Papers in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).
  10. C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
    [CrossRef]
  11. P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
    [CrossRef]
  12. This already introduces some superresolution, since the image support chosen in this way has exactly the same spatial-frequency content as the object support even beyond the system cutoff frequency.
  13. While this assumption is in general incorrect when adaptive optics are employed to obtain partial compensation, under certain conditions it may still be more or less valid.
  14. D. Tyler, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1998).
  15. A. Layberie, “Attainment of diffraction limited resolution in large telescopes by Fourier analyzing speckle patterns in star images,” Astron. Astrophys. 6, 85–87 (1970).
  16. D. Korff, “Analysis of a method for obtaining near-diffraction-limited information in the presence of atmospheric turbulence,” J. Opt. Soc. Am. 63, 971–980 (1973).
    [CrossRef]
  17. H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin1984).
  18. I. Gradshteyn, I. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (6.631(4)), p. 717.
  19. D. Tyler, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1999).

1997 (1)

1995 (1)

1994 (2)

C. Matson, “Variance reduction in Fourier spectra and their corresponding images with the use of support constraints,” J. Opt. Soc. Am. A 11, 97–106 (1994).
[CrossRef]

C. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Process. 42, 156–163 (1994).
[CrossRef]

1986 (1)

See, e.g., R. Narayan, R. Nityananda, “Maximum-entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

1984 (1)

1983 (1)

1974 (1)

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

1973 (1)

1970 (1)

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

1955 (1)

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

1948 (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
[CrossRef]

Bartelt, H.

Fellgett, P.

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

Gradshteyn, I.

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (6.631(4)), p. 717.

Knox, K.

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

Korff, D.

Layberie, A.

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

Linfoot, E.

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

Lohmann, A.

Matson, C.

Narayan, R.

See, e.g., R. Narayan, R. Nityananda, “Maximum-entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

Nityananda, R.

See, e.g., R. Narayan, R. Nityananda, “Maximum-entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

Risken, H.

H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin1984).

Roggemann, M.

Ryzhik, I.

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (6.631(4)), p. 717.

Shannon, C.

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
[CrossRef]

Thompson, B.

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

Tyler, D.

D. Tyler, C. Matson, “Reduction of nonstationary noise in telescope imagery using a support constraint,” Opt. Express 1, 347–354 (1997).
[CrossRef] [PubMed]

D. Tyler, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1998).

D. Tyler, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1999).

Weigelt, G.

Wirnitzer, B.

Annu. Rev. Astron. Astrophys. (1)

See, e.g., R. Narayan, R. Nityananda, “Maximum-entropy image restoration in astronomy,” Annu. Rev. Astron. Astrophys. 24, 127–170 (1986).
[CrossRef]

Appl. Opt. (3)

Astron. Astrophys. (1)

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

Astrophys. J. (1)

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

Bell Syst. Tech. J. (1)

C. Shannon, “A mathematical theory of communication,” Bell Syst. Tech. J. 27, 379–423 and 623–656 (1948).
[CrossRef]

IEEE Trans. Signal Process. (1)

C. Matson, “Fourier spectrum extrapolation and enhancement using support constraints,” IEEE Trans. Signal Process. 42, 156–163 (1994).
[CrossRef]

J. Opt. Soc. Am. (1)

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

Opt. Express (1)

Philos. Trans. R. Soc. London Ser. A (1)

P. Fellgett, E. Linfoot, “On the assessment of optical images,” Philos. Trans. R. Soc. London Ser. A 247, 369–407 (1955).
[CrossRef]

Other (7)

This already introduces some superresolution, since the image support chosen in this way has exactly the same spatial-frequency content as the object support even beyond the system cutoff frequency.

While this assumption is in general incorrect when adaptive optics are employed to obtain partial compensation, under certain conditions it may still be more or less valid.

D. Tyler, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1998).

Papers in Adaptive Optics for Large Telescopes, Vol. 19 of 1992 OSA Technical Digest Series (Optical Society of America, Washington, D.C., 1992).

H. Risken, The Fokker–Planck Equation (Springer-Verlag, Berlin1984).

I. Gradshteyn, I. Ryzhik, Table of Integrals, Series, and Products (Academic, New York, 1965), Eq. (6.631(4)), p. 717.

D. Tyler, University of New Mexico, Albuquerque, N.M. 87131 (personal communication, 1999).

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Figures (3)

Fig. 1
Fig. 1

(a) Elliptical support function in the physical plane, and (b) its spatial spectrum shown schematically by contour curves. The support and its spectrum both have the same elliptical symmetry, except that the spectrum is stretched out in the principal direction along which the support is physically narrow, and vice versa. As a result, diffusion is more pronounced along the u2 direction than along the u1 direction.

Fig. 2
Fig. 2

Diffusion coefficient D, in arbitrary units, for support-induced diffusion as a function of the normalized support diameter σ˜ (see text for definition). The solid and the dashed curves refer to the cases in which the 5/3-power-law and square-law structure functions (SF’s), respectively, were used in relation (41) to compute D.

Fig. 3
Fig. 3

Fractional noise reduction as a function of the normalized support diameter σ˜. The solid and the dashed curves show the theoretically computed asymptotic results without and with a regularizing filter, respectively, while the triangles represent the results of simulation for the case of D/r0=10.

Equations (73)

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is(x)=i(x)s(x)
Is(u)=I(u-u)S(u)d2u
F(u)=d2xf(x)exp(-i2πux).
δIs(u)=δI(u-u)S(u)d2u,
δf=f-f.
Ns(x)(u){Re[δIs(u)]}2=12 {|δIs(u)|2+Re[δIs(u)]2},
Ns(y)(u){Im[δIs(u)]}2=12 {|δIs(u)|2-Re[δIs(u)]2},
Cs(x, y)(u){Re[δIs(u)]Im[δIs(u)]}=12 Im[δIs(u)]2.
I(u)=O(u)H(u).
H(u)=1A P(ρ)P*(ρ-λfu)×exp{i[ϕ(ρ)-ϕ(ρ-λfu)]}d2ρ.
H(u)=H0(u)Hϕ(u),
H0(u)=1A P(ρ)P*(ρ-λfu)d2ρ
Hϕ(u)=exp{i[ϕ(ρ)-ϕ(ρ-λfu)]}
Ch(u1, u2)δH(u1)δH*(u2)=1A2 d2r1d2r2P(r1)P*(r1-λfu1)×P*(r2)P(r2-λfu2)(exp{i[ϕ(r1)-ϕ(r1-λfu1)-ϕ(r2)+ϕ(r2-λfu2)]}-exp{i[ϕ(r1)-ϕ(r1-λfu1)]}×exp{-i[ϕ(r2)-ϕ(r2-λfu2)]}).
(a-b-c+d)2
=(a-b)2+(c-d)2+(a-c)2+(b-d)2
-(a-d)2-(b-c)2,
Dϕ(|r1-r2|)=[ϕ(r1)-ϕ(r2)]2.
Ch(u1, u2)=1A2 exp{-(1/2)[Dϕ(λfu1)+Dϕ(λfu2)]}×d2r1d2r2P(r1)×P*(r1-λfu1)P*(r2)P(r2-λfu2)×[exp(-(1/2){Dϕ(|r1-r2|)+Dϕ[|r1-r2-λf(u1-u2)|]-Dϕ(|r1-r2-λfu1|)-Dϕ(|r1-r2+λfu2|)})-1].
Dϕ(r)=Crr02
0D/(λf )duuH0(u)exp[-(1/2)Dϕ(λfu)],
Ch(u1, u2)
=1A2 P(r1)P*(r1-λfu1)P*(r2)P(r2-λfu2)
×(exp[-(1/2)Dϕ(λf|u1-u2|)]-exp{-(1/2)[Dϕ(λfu1)+Dϕ(λfu2)]}).
Ch(u1, u2)=H0(u1)H0*(u2)(exp[-(1/2)Dϕ(λf|u1-u2|)]-exp{-(1/2)[Dϕ(λfu1)+Dϕ(λfu2)]}).
Ch/h(u1, u2)δH(u1)δH(u2)H(u1)H(u2),
Ch/h(u1, u2)=(exp{(1/2)[Dϕ(λfu1)+Dϕ(λfu2)-Dϕ(λf|u1-u2|)]}-1),
u1, u2<D/(λf ).
CI(d)(u1, u2)δI(d)(u1)δI(d)*(u2)=O(u1)O*(u2)Ch/h(u1, u2)=O(u1)O*(u2)(exp{(1/2)[Dϕ(λfu1)+Dϕ(λfu2)-Dϕ(λf|u1-u2|)]}-1).
CI(u1, u2)δI(u1)δI*(u2)=O(u1)O*(u2)H0(u1)H0*(u2)×(exp[-(1/2)Dϕ(λf|u1-u2|)]-exp{-(1/2)[Dϕ(λfu1)+Dϕ(λfu2)]}).
|δIs(u)|2=d2u1d2u2CI(u-u1,
u-u2)×S(u1)S*(u2),
CI(u1, u2)|O(u1)|2|H0(u1)|2×exp[-(1/2)Dϕ(λf|u1-u2|)]=|δI(u1)|2exp[-(1/2)Dϕ(λf|u1-u2|)],
|δIs(u)|2d2u1S(u1)|δI(u-u1)|2d2U×S*(u1+U)exp[-(1/2)Dϕ(λfU)].
|δIs(u)|2d2u1|δI(u-u1)|2S(u1)×exp[-(1/2)Dϕ(λfu1)].
O(u)=O*(-u),I(u)=I*(-u),H(u)=H*(-u),
[δIs(u)]2=d2u1d2u2CI(u-u1,u2-u)× S(u1)S(u2).
Im[δIs(u)]2=0.
[δIs(u)]2d2u1|δI(u-u1)|2S(u1)×exp[-(1/2)Dϕ(λf|2u-u1|)].
[δI(u)]2|δI(u)|2exp[-(1/2)Dϕ(2λfu)].
Ns(u)AN(u)+BN(u)+(1/2)Dˆ : N(u),
N(u)|δI(u)|2,Ns(u)|δIs(u)|2,
A=d2uS(u)exp[-(1/2)Dϕ(λfu)],
B=-d2uS(u)exp[-(1/2)Dϕ(λfu)]u,
Dˆ=d2uS(u)exp[-(1/2)Dϕ(λfu)]uu.
A1.
ΔN(u)Ns(u)-N(u)BN(u)+(1/2)Dˆ : N(u).
Ns(u)N(u)+BN(u)N(u+B),
Dˆ=D1ˆ,
D=12 d2uS(u)exp[-(1/2)Dϕ(λfu)]u2,
1ˆ=xˆxˆ+yˆyˆ,
ΔN(u)(1/2)D2N(u).
Ns(u)d2u
=N(u)d2uS(u1)exp[-(1/2)Dϕ(λfu1)]d2u1=Ad2uN(u).
A=|δis(x)|2d2x|δi(x)|2d2x,
ΔN(u)BN(u)+12 D1 2N(u)u12+D2 2N(u)u22,
Di=d2uS(u)exp[-(1/2)Dϕ(λfu)]ui2,i=1, 2.
S(u)=|x|<σ/2d2x exp(-i2πux)=πσ24 2J1(πσu)πσu.
Dπσ2 0u2J1(πσu)exp[-(1/2)Dϕ(λfu)]du,
D2π2σ2 π2σ22C(λf/r0)22 exp-π2σ22C(λf/r0)2.
N(d)(u)|δI(d)(u)|2{exp[Dϕ(λfu)]-1}
ΔW(d)u<ucΔN(d)(u)d2u.
ΔW(d)=12 D dduc exp[Dϕ(λfuc)]2πuc,ucua=r0λf.
W(d)=u<ucN(d)(u)d2u.
W2(d)=πC(λf/r0)2 {exp[C(λf/r0)2uc2]-1},
F2(d)(σ)ΔW2(d)W2(d)2C2(λf/r0)4uc2D×exp[C(λf/r0)2uc2]exp[C(λf/r0)2uc2]-1.
|δIs(u)|22πC(λf/r0)2 d2u1|S(u1)|2|δI(u-u1)|2.
Dˆ=2πC(λf/r0)2 d2u|S(u)|2uu,
I(x)exp[-3.44(λfu/r0)5/3]exp(i2πxu)d2u.
exp(i2πxu)d2x=δ(2)(u),
|x|R exp(i2πxu)d2x=Ru J1(2πuR)
fR=0 exp[-0.51(v/d˜)5/3]J1(v)dv,
d˜2R(λf/r0).

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