Abstract

The method of constrained iteration is used to deconvolve measured histograms of intensity fluctuations of a laser beam propagated through atmospheric turbulence. The histograms are blurred by noise and background contributions in consequence of the combination law of probability-density functions. The deconvolution is applied first without constraint and then with the physical condition of positivity of the counting values. A criterion for properly stopping the iteration is also presented. The results are checked by evaluating the moments and comparing them with those previously obtained from direct noise and background removal.

© 1993 Optical Society of America

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References

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  1. A. Consortini, G. Conforti, “Detector saturation effect on higher-order moments of intensity fluctuations in atmospheric laser propagation measurements,” J. Opt. Soc. Am. A 1, 1075–1077 (1984).
    [CrossRef]
  2. A. Consortini, E. Briccolani, G. Conforti, “Strong scintillation statistics deterioration due to detector saturation,” J. Opt. Soc. Am. A 3, 101–107 (1986).
    [CrossRef]
  3. A. Consortini, R. J. Hill, “Reduction of the moments of intensity fluctuations caused by amplifier saturation for both the Kand the log-normally modulated exponential probability densities,” Opt. Lett. 12, 304–306 (1987).
    [CrossRef] [PubMed]
  4. N. Ben-Yoseph, E. Goldner, “Sample size influence on optical scintillation analysis. 1: Analytical treatment of the higher-order irradiance moments,” Appl. Opt. 27, 2167–2171 (1988).
    [CrossRef]
  5. E. Goldner, N. Ben-Yoseph, “Sample size influence on optical scintillation analysis. 2: Simulation approach,” Appl. Opt. 27, 2772–2777 (1988).
    [CrossRef]
  6. R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
    [CrossRef]
  7. M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1989), Vol. 75, pp. 1–120.
    [CrossRef]
  8. M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.
    [CrossRef]
  9. P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitaetsverteilung in Spektrallinien II,” Z. Phys. 69, 298–308 (1931).
    [CrossRef]
  10. L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
    [CrossRef]
  11. H. Bialy, “Iterative Behandlung von linearen Funktionalgleichungen,” Arch. Rat. Mech. Anal. 4, 166–176 (1959).
    [CrossRef]
  12. R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
    [CrossRef]
  13. C. Youla, H. Webb, “Image reconstruction by the method of convex projections I,”IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
    [CrossRef]
  14. H. J. Trussell, M. R. Cinvalar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
    [CrossRef]
  15. The experiment was done during a joint measurement campaign of the National Oceanic and Atmospheric Administration/Environmental Research Laboratory/Wave Propagation Laboratory and the Department of Physics, University of Florence, Florence, Italy; it was also supported by the Consiglio Nazionale delle Ricerche, Italy.
  16. J. H. Churnside, R. J. Hill, G. Conforti, A. Consortini, “Aperture size and bandwidth requirements for measuring strong scintillation in the atmosphere,” Appl. Opt. 28, 4126–4132 (1989).
    [CrossRef] [PubMed]
  17. H. Stark, Image Recovery and Applications (Academic, New York, 1987).
  18. A. Consortini, F. Cochetti, “Removing noise from histograms of probability density in laser scintillation,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.
  19. C. Aime, E. Aristidi, H. Lanteri, G. Ricot, “Probability imaging of extended astronomical sources at low light levels,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

1989

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[CrossRef]

J. H. Churnside, R. J. Hill, G. Conforti, A. Consortini, “Aperture size and bandwidth requirements for measuring strong scintillation in the atmosphere,” Appl. Opt. 28, 4126–4132 (1989).
[CrossRef] [PubMed]

1988

N. Ben-Yoseph, E. Goldner, “Sample size influence on optical scintillation analysis. 1: Analytical treatment of the higher-order irradiance moments,” Appl. Opt. 27, 2167–2171 (1988).
[CrossRef]

E. Goldner, N. Ben-Yoseph, “Sample size influence on optical scintillation analysis. 2: Simulation approach,” Appl. Opt. 27, 2772–2777 (1988).
[CrossRef]

1987

1986

1984

A. Consortini, G. Conforti, “Detector saturation effect on higher-order moments of intensity fluctuations in atmospheric laser propagation measurements,” J. Opt. Soc. Am. A 1, 1075–1077 (1984).
[CrossRef]

H. J. Trussell, M. R. Cinvalar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

1982

C. Youla, H. Webb, “Image reconstruction by the method of convex projections I,”IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

1981

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

1959

H. Bialy, “Iterative Behandlung von linearen Funktionalgleichungen,” Arch. Rat. Mech. Anal. 4, 166–176 (1959).
[CrossRef]

1951

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

1931

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitaetsverteilung in Spektrallinien II,” Z. Phys. 69, 298–308 (1931).
[CrossRef]

Aime, C.

C. Aime, E. Aristidi, H. Lanteri, G. Ricot, “Probability imaging of extended astronomical sources at low light levels,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

Aristidi, E.

C. Aime, E. Aristidi, H. Lanteri, G. Ricot, “Probability imaging of extended astronomical sources at low light levels,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

Ben-Yoseph, N.

N. Ben-Yoseph, E. Goldner, “Sample size influence on optical scintillation analysis. 1: Analytical treatment of the higher-order irradiance moments,” Appl. Opt. 27, 2167–2171 (1988).
[CrossRef]

E. Goldner, N. Ben-Yoseph, “Sample size influence on optical scintillation analysis. 2: Simulation approach,” Appl. Opt. 27, 2772–2777 (1988).
[CrossRef]

Bertero, M.

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1989), Vol. 75, pp. 1–120.
[CrossRef]

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.
[CrossRef]

Bialy, H.

H. Bialy, “Iterative Behandlung von linearen Funktionalgleichungen,” Arch. Rat. Mech. Anal. 4, 166–176 (1959).
[CrossRef]

Briccolani, E.

Churnside, J. H.

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[CrossRef]

J. H. Churnside, R. J. Hill, G. Conforti, A. Consortini, “Aperture size and bandwidth requirements for measuring strong scintillation in the atmosphere,” Appl. Opt. 28, 4126–4132 (1989).
[CrossRef] [PubMed]

Cinvalar, M. R.

H. J. Trussell, M. R. Cinvalar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

Cochetti, F.

A. Consortini, F. Cochetti, “Removing noise from histograms of probability density in laser scintillation,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

Conforti, G.

Consortini, A.

De Mol, C.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.
[CrossRef]

Frehlich, R. G.

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[CrossRef]

Goldner, E.

N. Ben-Yoseph, E. Goldner, “Sample size influence on optical scintillation analysis. 1: Analytical treatment of the higher-order irradiance moments,” Appl. Opt. 27, 2167–2171 (1988).
[CrossRef]

E. Goldner, N. Ben-Yoseph, “Sample size influence on optical scintillation analysis. 2: Simulation approach,” Appl. Opt. 27, 2772–2777 (1988).
[CrossRef]

Hill, R. J.

Landweber, L.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Lanteri, H.

C. Aime, E. Aristidi, H. Lanteri, G. Ricot, “Probability imaging of extended astronomical sources at low light levels,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Ricot, G.

C. Aime, E. Aristidi, H. Lanteri, G. Ricot, “Probability imaging of extended astronomical sources at low light levels,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Stark, H.

H. Stark, Image Recovery and Applications (Academic, New York, 1987).

Trussell, H. J.

H. J. Trussell, M. R. Cinvalar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

Van Cittert, P. H.

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitaetsverteilung in Spektrallinien II,” Z. Phys. 69, 298–308 (1931).
[CrossRef]

Viano, G. A.

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.
[CrossRef]

Webb, H.

C. Youla, H. Webb, “Image reconstruction by the method of convex projections I,”IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Youla, C.

C. Youla, H. Webb, “Image reconstruction by the method of convex projections I,”IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

Am. J. Math.

L. Landweber, “An iteration formula for Fredholm integral equations of the first kind,” Am. J. Math. 73, 615–624 (1951).
[CrossRef]

Appl. Opt.

Arch. Rat. Mech. Anal.

H. Bialy, “Iterative Behandlung von linearen Funktionalgleichungen,” Arch. Rat. Mech. Anal. 4, 166–176 (1959).
[CrossRef]

IEEE Trans. Acoust. Speech Signal Process.

H. J. Trussell, M. R. Cinvalar, “The feasible solution in signal restoration,”IEEE Trans. Acoust. Speech Signal Process. ASSP-32, 201–212 (1984).
[CrossRef]

IEEE Trans. Med. Imaging

C. Youla, H. Webb, “Image reconstruction by the method of convex projections I,”IEEE Trans. Med. Imaging MI-1, 81–94 (1982).
[CrossRef]

J. Mod. Opt.

R. G. Frehlich, J. H. Churnside, “Statistical properties of estimates of the moments of laser scintillation,” J. Mod. Opt. 36, 1645–1659 (1989).
[CrossRef]

J. Opt. Soc. Am. A

Opt. Lett.

Proc. IEEE

R. W. Schafer, R. M. Mersereau, M. A. Richards, “Constrained iterative restoration algorithms,” Proc. IEEE 69, 432–450 (1981).
[CrossRef]

Z. Phys.

P. H. Van Cittert, “Zum Einfluss der Spaltbreite auf die Intensitaetsverteilung in Spektrallinien II,” Z. Phys. 69, 298–308 (1931).
[CrossRef]

Other

The experiment was done during a joint measurement campaign of the National Oceanic and Atmospheric Administration/Environmental Research Laboratory/Wave Propagation Laboratory and the Department of Physics, University of Florence, Florence, Italy; it was also supported by the Consiglio Nazionale delle Ricerche, Italy.

H. Stark, Image Recovery and Applications (Academic, New York, 1987).

A. Consortini, F. Cochetti, “Removing noise from histograms of probability density in laser scintillation,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

C. Aime, E. Aristidi, H. Lanteri, G. Ricot, “Probability imaging of extended astronomical sources at low light levels,” presented at the International Commission for Optics Topical Meeting on Atmospheric, Volume and Surface Scattering and Propagation, Florence, Italy, 1991.

M. Bertero, “Linear inverse and ill-posed problems,” in Advances in Electronics and Electron Physics, P. W. Hawkes, ed. (Academic, New York, 1989), Vol. 75, pp. 1–120.
[CrossRef]

M. Bertero, C. De Mol, G. A. Viano, “The stability of inverse problems,” in Inverse Scattering Problems in Optics, Vol. 20 of Topics in Current Physics, H. P. Baltes, ed. (Springer-Verlag, Berlin, 1980), pp. 161–214.
[CrossRef]

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Figures (5)

Fig. 1
Fig. 1

Probability-density histograms: curve a, total signal (lower and larger distribution); curve b, noise (thinner and higher distribution).

Fig. 2
Fig. 2

Gaussian noise histogram used as a blurring function in the deconvolution.

Fig. 3
Fig. 3

Fourier transform of the original data (dashed–dotted curve) and of the deconvoluted signal after n = 1, 10, and 100 iterations (solid, dashed, and dotted curves, respectively). Relaxation parameter τ = 1.

Fig. 4
Fig. 4

Restoration without constraint: probability-density histogram of the deconvoluted signal after (a) 10 iterations and (b) 100 iterations. Relaxation parameter τ = 1.

Fig. 5
Fig. 5

Restoration with positivity constraint: probability-density histogram of the deconvoluted signal after 100 iterations. Relaxation parameter τ = 1.

Equations (16)

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s = i + b ,
P ( s ) = P ( i ) P ( b ) ,
P ( s ) ( I ) = 0 P ( i ) ( I ) P ( b ) ( I - I ) d I ,             with 0 I < .
N k ( s ) N T ( s ) = [ P ( i ) P ( b ) ] k + δ N k .
g = H f ,
H m , i = N i - m ( g )             for i > m ,
d = H * g .
K = H * H ,
H - = ( H * H ) - 1 H * = K - 1 H * .
f ( n + 1 ) = H * g + ( I - τ K ) f ( n ) ,
I - τ K < 1 ,
f ( n ) = K - 1 [ 1 - ( 1 - τ K ) n ] f ( 0 ) = K - 1 [ 1 - ( 1 - τ K ) n ] H * g = L H * g ,
L ( n ) ( ω i ) = 1 κ ( ω i ) { 1 - [ 1 - τ κ ( ω i ) ] n } .
L ( n ) ( ω i ) = { 1 κ ( ω i ) if τ κ ~ 1 τ n if τ κ 1 .
f ( n + 1 ) = P [ H * g + ( I - τ K ) f ( n ) ] ,
P f = { f if f 0 0 if f < 0 .

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