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 lowerfrequency 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

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  1. B. R. Frieden, "Restoring with maximum likelihood and maximum entropy," J. Opt. Soc. Am. 62, 511–518 (1972).
  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.
  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Meeting on Image Processing for 2-D and 3-D Reconstruction from Projections, Stanford, California, August 4–7, 1975 (unpublished).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).
  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).
  9. R. T. Lacoss, "Data Adaptive Spectral Analysis Methods," Geophysics 36, 661–675 (1971).
  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).
  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).
  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

B. R. Frieden and W. Swindell, "Restored Pictures of Ganymede, Moon of Jupiter," Science 191, 1237–1241 (1976).

1974

S. J. Wernecke and L. R. D'Addario, "Maximum Entropy Image Reconstruction," IEEE Trans. Comput. C-26, 351–364 (1974).

1972

1971

R. T. Lacoss, "Data Adaptive Spectral Analysis Methods," Geophysics 36, 661–675 (1971).

1968

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

1965

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).

1962

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).

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).

Frieden, B. R.

B. R. Frieden and W. Swindell, "Restored Pictures of Ganymede, Moon of Jupiter," Science 191, 1237–1241 (1976).

B. R. Frieden, "Restoring with maximum likelihood and maximum entropy," J. Opt. Soc. Am. 62, 511–518 (1972).

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.

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).

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).

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).

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).

Wernecke, S. J.

S. J. Wernecke and L. R. D'Addario, "Maximum Entropy Image Reconstruction," IEEE Trans. Comput. C-26, 351–364 (1974).

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).

Wolter, H.

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

Geophysics

R. T. Lacoss, "Data Adaptive Spectral Analysis Methods," Geophysics 36, 661–675 (1971).

IEEE Trans.

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

IEEE Trans. Comput.

S. J. Wernecke and L. R. D'Addario, "Maximum Entropy Image Reconstruction," IEEE Trans. Comput. C-26, 351–364 (1974).

J. Assoc. Comput. Mach.

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).

J. Opt. Soc. Am.

Numerische Mathematik

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).

Science

B. R. Frieden and W. Swindell, "Restored Pictures of Ganymede, Moon of Jupiter," Science 191, 1237–1241 (1976).

Other

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)].

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).

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.

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).

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