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Journal of the Optical Society of America

Journal of the Optical Society of America

  • Vol. 59, Iss. 5 — May. 1, 1969
  • pp: 547–553

Bandlimited Image Restoration by Linear Mean-Square Estimation

CHARLES L. RINO  »View Author Affiliations


JOSA, Vol. 59, Issue 5, pp. 547-553 (1969)
http://dx.doi.org/10.1364/JOSA.59.000547


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Abstract

A linear mean-square estimator, optimum for image data available only on a finite interval, is derived for the restoration of images degraded by a system with a bandlimited spread function. The analysis is carried out in one dimension using a prolate-spheroidal-wavefunction expansion of the image data. When the noise is bandlimited to the same bandwidth as the spread function, the expansion represents the image data with zero mean-square error on the entire interval, and the mean-square reconstruction error is equal to that of the optimum linear estimate for image data on the infinite interval. The rate at which the series representation of the estimate converges is discussed and an example presented.

Citation
CHARLES L. RINO, "Bandlimited Image Restoration by Linear Mean-Square Estimation," J. Opt. Soc. Am. 59, 547-553 (1969)
http://www.opticsinfobase.org/josa/abstract.cfm?URI=josa-59-5-547


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References

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  2. See for example J. D. Kraus, Radio Astronomy (McGraw-Hill Book Co., New York, 1966), p. 69.
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  15. Expectation should be interpreted as the statistical average over the ensemble of process realizations.
  16. Note that Slepian and Pollak6 define the frequency interval (—Ω, Ω) rather than (-Ω/2, Ω/2) as in this paper. Hence in their formulas Ω should be replaced by Ω/2. With this slight change their definitions are used directly.
  17. This is derived in Ref. (8), Eq. (2.7).
  18. See Ref. 11, Eq. (2.5).
  19. See Ref. 1, Eq. (2.6).
  20. This follows from Ref. (6), Eqs. (27), (28), and (29).
  21. The expression [x] means the smallest integer larger than x.
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  26. See Ref. 11, Eq. (2.13).
  27. See Ref. 6, Sec. 4.1.
  28. H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).
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  31. D. Slepian, Trans. IRE P GIT-3, 68 (1954).

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