Abstract

The maximum entropy (ME) restoring formalism has previously been derived under the assumptions of (i) zero background and (ii) additive noise in the image. However, the noise in the signals from many modern image detectors is actually Poisson, i.e., dominated by single-photon statistics. Hence, the noise is no longer additive. Particularly in astronomy, it is often accurate to model the image as being composed of two fundamental Poisson features: (i) a component due to a smoothly varying background image, such as caused by interstellar dust, plus (ii) a superimposed component due to an unknown array of point and line sources (stars, galactic arms, etc.). The latter is termed the “foreground image” since it contains the principal object information sought by the viewer. We include in the background all physical backgrounds, such as the night sky, as well as the mathematical background formed by lower-frequency components of the principal image structure. The role played by the background, which may be separately and easily estimated since it is smooth, is to pointwise modify the known noise statistics in the foreground image according to how strong the background is. Given the estimated background, a maximum-likelihood restoring formula was derived for the foreground image. We applied this approach to some one-dimensional simulations and to some real astronomical imagery. Results are consistent with the maximum-likelihood and Poisson hypotheses: i.e., where the background is high and consequently contributes much noise to the observed image, a restored star is broader and smoother than where the background is low. This nonisoplanatic behavior is desirable since it permits extra resolution only where the noise is sufficiently low to reliably permit it.

© 1978 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
    [Crossref] [PubMed]
  2. B. R. Frieden, “Image Enhancement and Restoration,” in Topics in Applied Physics, edited by T. S. Huang, (Springer-Verlag, New York, 1975), Vol. 6, pp. 177–248.
    [Crossref]
  3. H. Wolter, in Progress in Optics I, edited by E. Wolf (North-Holland, Amsterdam, 1961).
  4. L. D’Addario “Maximum Entropy and Maximum a Posteriori Probability Reconstruction,” in the Proceedings of the Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections, Stanford, California, August 4–7, 1975 (unpublished).
  5. S. J. Wernecke and L. R. D’Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1974).
    [Crossref]
  6. L. L. Smith, “The Use of the Maximum Entropy Transform in the Analyses of IR Astronomical Spectra,” in SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 114–117; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).
  7. J. E. Brolley, R. B. Lazarus, and B. R. Suydam, “Maximum Entropy Restoration of Laser-Fusion Target X-Ray Photographs,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 106–109; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).
  8. B. R. Frieden and W. Swindell, “Restored Pictures of Ganymede, Moon of Jupiter,” Science 191, 1237–1241 (1976).
    [Crossref] [PubMed]
  9. R. T. Lacoss, “Data Adaptive Spectral Analysis Methods,” Geophysics 36, 661–675 (1971).
    [Crossref]
  10. D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
    [Crossref]
  11. E. T. Jaynes, “Prior Probabilities,” IEEE Trans. SSC-4, 227–241 (1968).
  12. R. Kikuchi and B. H. Soffer, “Maximum Entropy Image Restoration, I: The Entropy Expression,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 95–98; July 19–23, 1976, Toronto, Canada [J. Opt. Soc. Am. 67, 1656–1665 (1977)].
  13. A useful summary of this technique is provided by R. S. Martin and J. H. Wilkinson, “Symmetric Decomposition of Positive Definite Band Matrices,” Numerische Mathematik 7, 355–361 (1965).
    [Crossref]
  14. R. Hershel, Optical Sciences Center (private communication).
  15. R. Nathan, in Pictorial Pattern Recognition, edited by G. C. Cheng (Thompson, Washington, D.C., 1968).

1976 (1)

B. R. Frieden and W. Swindell, “Restored Pictures of Ganymede, Moon of Jupiter,” Science 191, 1237–1241 (1976).
[Crossref] [PubMed]

1974 (1)

S. J. Wernecke and L. R. D’Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1974).
[Crossref]

1972 (1)

1971 (1)

R. T. Lacoss, “Data Adaptive Spectral Analysis Methods,” Geophysics 36, 661–675 (1971).
[Crossref]

1968 (1)

E. T. Jaynes, “Prior Probabilities,” IEEE Trans. SSC-4, 227–241 (1968).

1965 (1)

A useful summary of this technique is provided by R. S. Martin and J. H. Wilkinson, “Symmetric Decomposition of Positive Definite Band Matrices,” Numerische Mathematik 7, 355–361 (1965).
[Crossref]

1962 (1)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

Brolley, J. E.

J. E. Brolley, R. B. Lazarus, and B. R. Suydam, “Maximum Entropy Restoration of Laser-Fusion Target X-Ray Photographs,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 106–109; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).

D’Addario, L.

L. D’Addario “Maximum Entropy and Maximum a Posteriori Probability Reconstruction,” in the Proceedings of the Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections, Stanford, California, August 4–7, 1975 (unpublished).

D’Addario, L. R.

S. J. Wernecke and L. R. D’Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1974).
[Crossref]

Frieden, B. R.

B. R. Frieden and W. Swindell, “Restored Pictures of Ganymede, Moon of Jupiter,” Science 191, 1237–1241 (1976).
[Crossref] [PubMed]

B. R. Frieden, “Restoring with maximum likelihood and maximum entropy,” J. Opt. Soc. Am. 62, 511–518 (1972).
[Crossref] [PubMed]

B. R. Frieden, “Image Enhancement and Restoration,” in Topics in Applied Physics, edited by T. S. Huang, (Springer-Verlag, New York, 1975), Vol. 6, pp. 177–248.
[Crossref]

Hershel, R.

R. Hershel, Optical Sciences Center (private communication).

Jaynes, E. T.

E. T. Jaynes, “Prior Probabilities,” IEEE Trans. SSC-4, 227–241 (1968).

Kikuchi, R.

R. Kikuchi and B. H. Soffer, “Maximum Entropy Image Restoration, I: The Entropy Expression,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 95–98; July 19–23, 1976, Toronto, Canada [J. Opt. Soc. Am. 67, 1656–1665 (1977)].

Lacoss, R. T.

R. T. Lacoss, “Data Adaptive Spectral Analysis Methods,” Geophysics 36, 661–675 (1971).
[Crossref]

Lazarus, R. B.

J. E. Brolley, R. B. Lazarus, and B. R. Suydam, “Maximum Entropy Restoration of Laser-Fusion Target X-Ray Photographs,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 106–109; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).

Martin, R. S.

A useful summary of this technique is provided by R. S. Martin and J. H. Wilkinson, “Symmetric Decomposition of Positive Definite Band Matrices,” Numerische Mathematik 7, 355–361 (1965).
[Crossref]

Nathan, R.

R. Nathan, in Pictorial Pattern Recognition, edited by G. C. Cheng (Thompson, Washington, D.C., 1968).

Phillips, D. L.

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

Smith, L. L.

L. L. Smith, “The Use of the Maximum Entropy Transform in the Analyses of IR Astronomical Spectra,” in SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 114–117; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).

Soffer, B. H.

R. Kikuchi and B. H. Soffer, “Maximum Entropy Image Restoration, I: The Entropy Expression,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 95–98; July 19–23, 1976, Toronto, Canada [J. Opt. Soc. Am. 67, 1656–1665 (1977)].

Suydam, B. R.

J. E. Brolley, R. B. Lazarus, and B. R. Suydam, “Maximum Entropy Restoration of Laser-Fusion Target X-Ray Photographs,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 106–109; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).

Swindell, W.

B. R. Frieden and W. Swindell, “Restored Pictures of Ganymede, Moon of Jupiter,” Science 191, 1237–1241 (1976).
[Crossref] [PubMed]

Wernecke, S. J.

S. J. Wernecke and L. R. D’Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1974).
[Crossref]

Wilkinson, J. H.

A useful summary of this technique is provided by R. S. Martin and J. H. Wilkinson, “Symmetric Decomposition of Positive Definite Band Matrices,” Numerische Mathematik 7, 355–361 (1965).
[Crossref]

Wolter, H.

H. Wolter, in Progress in Optics I, edited by E. Wolf (North-Holland, Amsterdam, 1961).

Geophysics (1)

R. T. Lacoss, “Data Adaptive Spectral Analysis Methods,” Geophysics 36, 661–675 (1971).
[Crossref]

IEEE Trans. (1)

E. T. Jaynes, “Prior Probabilities,” IEEE Trans. SSC-4, 227–241 (1968).

IEEE Trans. Comput. (1)

S. J. Wernecke and L. R. D’Addario, “Maximum Entropy Image Reconstruction,” IEEE Trans. Comput. C-26, 351–364 (1974).
[Crossref]

J. Assoc. Comput. Mach. (1)

D. L. Phillips, “A Technique for the Numerical Solution of Certain Integral Equations of the First Kind,” J. Assoc. Comput. Mach. 9, 84–97 (1962).
[Crossref]

J. Opt. Soc. Am. (1)

Numerische Mathematik (1)

A useful summary of this technique is provided by R. S. Martin and J. H. Wilkinson, “Symmetric Decomposition of Positive Definite Band Matrices,” Numerische Mathematik 7, 355–361 (1965).
[Crossref]

Science (1)

B. R. Frieden and W. Swindell, “Restored Pictures of Ganymede, Moon of Jupiter,” Science 191, 1237–1241 (1976).
[Crossref] [PubMed]

Other (8)

R. Hershel, Optical Sciences Center (private communication).

R. Nathan, in Pictorial Pattern Recognition, edited by G. C. Cheng (Thompson, Washington, D.C., 1968).

R. Kikuchi and B. H. Soffer, “Maximum Entropy Image Restoration, I: The Entropy Expression,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 95–98; July 19–23, 1976, Toronto, Canada [J. Opt. Soc. Am. 67, 1656–1665 (1977)].

B. R. Frieden, “Image Enhancement and Restoration,” in Topics in Applied Physics, edited by T. S. Huang, (Springer-Verlag, New York, 1975), Vol. 6, pp. 177–248.
[Crossref]

H. Wolter, in Progress in Optics I, edited by E. Wolf (North-Holland, Amsterdam, 1961).

L. D’Addario “Maximum Entropy and Maximum a Posteriori Probability Reconstruction,” in the Proceedings of the Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections, Stanford, California, August 4–7, 1975 (unpublished).

L. L. Smith, “The Use of the Maximum Entropy Transform in the Analyses of IR Astronomical Spectra,” in SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 114–117; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).

J. E. Brolley, R. B. Lazarus, and B. R. Suydam, “Maximum Entropy Restoration of Laser-Fusion Target X-Ray Photographs,” SPSE Conference on Image Analysis and Evaluation, Technical Digest, pp. 106–109; July 19–23, 1976, Toronto, Canada (Soc. Phot. Scientists and Engrs., Washington, D.C., 1976).

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (7)

FIG. 1
FIG. 1

A one-dimensional test of the overall approach. Dotted lines define the true object; dashed lines are its image, without noise; the thin, solid lines are the background estimate; and the heavy, solid lines are the restored foreground object. In the region between foreground features, all curves are coincident with the dotted ramp.

FIG. 2
FIG. 2

Point spread functions for the ME approach. Each is the ME restoration of a fixed spike sitting atop a plateau of height B. Spread function shape, and hence resolution, are seen to vary markedly with B.

FIG. 3
FIG. 3

Background B vs resolution σ, from the curves of Fig. 2 (and others not shown). The original image spread function had a σ of 0.9881. Hence the dashed line in the figure delineates those cases where there was a gain in resolution from those where there was a net loss. The background value B ≃ 250 separates the two regions.

FIG. 4
FIG. 4

Application to restoration of the Bird nebula. (a) the given image, blurred by long-term atmospheric turbulence; (b) the background estimate, formed by further blurring (a); (c) the restored foreground object, using ρ = 25 in the ME step; (d) the total restoration, formed by adding (b) and (c).

FIG. 5
FIG. 5

Restorations of the Bird at higher levels of resolution. (a) use of ρ = 200; (b) use of ρ = 400. Resolution is now saturating with ρ, being ultimately limited by the pixel subdivision in use.

FIG. 6
FIG. 6

Attempted restoration by a linear method, which was constrained to produce comparable resolution with those in Fig. 5. Gibbs oscillations come on very strongly.

FIG. 7
FIG. 7

An intensity map of the estimated SNR for the image in Fig. 4a. Black regions have SNR = 0, and all visible regions have 130 ≳ SNR ≳ 20. Comparison with the ME restorations in Figs. 4c and 5 shows that high resolution regions are also high SNR regions. This suggests good reliability for the enhanced stars there.

Equations (40)

Equations on this page are rendered with MathJax. Learn more.

Entropy H - field d x d y ô ( x , y ) ln ô ( x , y ) = maximum ,
i m = n = 1 N o n s ( x m - x n ) + B m + n m .
P ( { o n } , { n m } ) = maximum .
P ( { o n } , { n m } ) = P 1 ( { o n } ) P 2 ( { n m } { o n } ) .
P 1 ( { o n } ) P 2 ( { i m } { o n } ) = maximum
P 1 ( { o n } ) = ( E / Δ o ) ! n = 1 N ( o n / Δ o ) ! .
P 2 ( { i m } { o n } ) = m = 1 M a m i m / Δ i e - a m ( i m / Δ i ) ! , { o n } fixed .
a m i m s / Δ i = Δ i - 1 ( n = 1 N o n s ( x m - x n ) + B m ) .
ln P 1 ( { o n } ) + ln P 2 ( { i m } { o n } ) = maximum .
ln P 1 ( { o n } ) + ln P 2 ( { i m } { o n } ) - m = 1 M λ m ( i m - i m data ) - μ ( n = 1 N o n - E ) = maximum .
- n = 1 N ( o n / Δ o ) ln ( o n / Δ o ) + m = 1 M [ ( i m / Δ i ) ln a m - a m - ( i m / Δ i ) ln ( i m / Δ i ) ] - m = 1 M λ m ( i m - i m data ) - μ ( n = 1 N o n - E ) = maximum .
/ n m | { o n } fixed = / i m | { o n } fixed ,
i m / o n | { n m } fixed = s ( x m - x n ) .
m = 1 M [ Δ i - 1 ln a m - Δ i - 1 ln ( i m / Δ i ) - Δ i - 1 - λ m ] = 0.
i m = ( n = 1 N o n s ( x m - x n ) + B m ) e - 1 - Δ i λ m .
P 2 ( { i m } { o n } ) = m = 1 M 1 ( 2 π i m s ) 1 / 2 e - ( i m - i m s ) 2 / 2 i m s .
P 2 ( { i m } { o n } ) = ( 2 π i s ) - N / 2 m = 1 M e - ( i m - i s ) 2 / 2 i s .
- 2 - 1 M ln ( 2 π i s ) - m = 1 M ( i m - i s ) 2 / 2 i s .
- Δ o - 1 - Δ o - 1 ln ( o n / Δ o ) - m = 1 M λ m s ( x m - x n ) - μ = 0.
o n = Δ o exp ( - 1 - μ Δ o - Δ o m = 1 M λ m s ( x m - x n ) ) .
i m = i m data
n = 1 N o n = E .
ρ = Δ o / Δ i
Λ m Δ o λ m Γ Δ o μ - ln Δ o .
ô n = exp ( - 1 - Γ - m = 1 M Λ m s ( x m - x n ) ) .
i m data = ( n = 1 N ô n s ( x m - x n ) + B ˆ m ) e - 1 - Λ m / ρ , m = 1 , 2 , , M .
E data = n = 1 N ô n ,
Δ i = i max / ( SNR ) 2 .
ρ = ( Δ o / i max ) ( SNR ) 2 .
ρ = ( SNR ) 2 / Q .
i m data = f m + B m ,
f m signal = o s m ,
i m data B n .
exp [ - ( r / α ) p ] ,
T N 2 .
T N 2 log N 2 ,
SNR = f / σ tot ,
σ tot 2 = σ B 2 + σ f 2 ,
σ tot 2 = ( B + f ) Δ i .
SNR = ( i data - B ) / ( Δ i i data ) 1 / 2 .