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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 62, Iss. 4 — Apr. 1, 1972
  • pp: 511–518

Restoring with Maximum Likelihood and Maximum Entropy

B. Roy FRIEDEN  »View Author Affiliations

JOSA, Vol. 62, Issue 4, pp. 511-518 (1972)

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Given M sampled image values of an incoherent object, what can be deduced as the most likely object? Using a communication-theory model for the process of image formation, we find that the most likely object has a maximum entropy and is represented by a restoring formula that is positive and not band limited. The derivation is an adaptation to optics of a formulation by Jaynes for unbiased estimates of positive probability functions. The restoring formula is tested, via computer simulation, upon noisy images of objects consisting of random impulses. These are found to be well restored, with resolution often exceeding the Rayleigh limit and with a complete absence of spurious detail. The proviso is that the noise in each image input must not exceed about 40% of the signal image. The restoring method is applied to experimental data consisting of line spectra. Results are consistent with those of the computer simulations.

B. Roy FRIEDEN, "Restoring with Maximum Likelihood and Maximum Entropy," J. Opt. Soc. Am. 62, 511-518 (1972)

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  1. Y. Biraud, Astron. Astrophys. 1, 124 (1969).
  2. P. A. Jansson, R. H. Hunt, and E. K. Plyler, J. Opt. Soc. Am. 60, 596 (1970).
  3. R. S. Hershel, J. Opt. Soc. Am. 60, 1546A (1970).
  4. D. P. MacAdam, J. Opt. Soc. Am. 60, 1617 (1970).
  5. E. T. Jaynes, Trans. IEEE SSC-4, 227 (1968).
  6. S. Goldman, Information Theory (Prentice-Hall, New York, 1955), Appendix II.
  7. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968).
  8. B . L. Phillips, J. ACM 9, 84 (1962).
  9. E. H. Linfoot, Fourier Methods in Optical Image Evaluation (Focal, New York, 1964), p. 41.
  10. Reference 9, p. 57.
  11. See, e.g.,L. A. Pipes, Mathematics for Engineers and Physicists (McGraw-Hill, New York, 1958), p. 116.
  12. A. Wragg and D. C. Dowson, Trans. IEEE IT-16, 226 (1970).
  13. A. van den Bos, Trans. IEEE IT-17, 494 (1971).
  14. See, e.g.,R. Nathan, J. Opt. Soc. Am. 57, 578A (1967).
  15. The data in question cover the first three maxima of curve A in Fig. 2 of Ref. 2. Jansson's corresponding restoration is Fig. 2 (c).

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