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.

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  1. R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).
  2. See for example J. D. Kraus, Radio Astronomy (McGraw-Hill Book Co., New York, 1966), p. 69.
  3. See Ref. 1, p. 627.
  4. D. S. Jones, Generalized Functions (McGraw-Hill Book Co., New York, 1966), p. 354.
  5. H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, Sec. 4.6.
  6. D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).
  7. G. J. Buck and J. J. Gustinic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).
  8. B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).
  9. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
  10. N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series (John Wiley & Sons, Inc., New York, 1949), Ch. III, p. 81.
  11. C. W. Helstrom. J. Opt. Soc. Am. 57, 297 (1967).
  12. D. Slepian, J. Opt. Soc. Am. 57, 918 (1967).
  13. See for example W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 224.
  14. A. V. Balakrishnan, Trans. IRE IT-3, 143 (1957).
  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.
  22. D. Slepian, J. Math. Phys. 44, No. 2, 107 (1965). Note that Flammer’s normalization is used in this reference.
  23. J. A. Stratton, P. M. Morre, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (M.I.T. Press, Cambridge, Mass., 1956), pp. 203ndash;229.
  24. D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).
  25. See Ref. 11, Sec. 2.
  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).
  29. E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963), p. 77.
  30. R. Courant and D. Helbert, Methods of Mathematical Physics (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 427.
  31. D. Slepian, Trans. IRE P GIT-3, 68 (1954).

Balakrishnan, A. V.

A. V. Balakrishnan, Trans. IRE IT-3, 143 (1957).

Bracewell, R. N.

R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).

Buck, G. J.

G. J. Buck and J. J. Gustinic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).

Chu, L. J.

J. A. Stratton, P. M. Morre, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (M.I.T. Press, Cambridge, Mass., 1956), pp. 203ndash;229.

Corbato, F. J.

J. A. Stratton, P. M. Morre, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (M.I.T. Press, Cambridge, Mass., 1956), pp. 203ndash;229.

Courant, R.

R. Courant and D. Helbert, Methods of Mathematical Physics (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 427.

Davenport, W. B.

See for example W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 224.

Frieden, B. R.

B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).

Gustinic, J. J.

G. J. Buck and J. J. Gustinic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).

Harris, R. W.

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

Helbert, D.

R. Courant and D. Helbert, Methods of Mathematical Physics (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 427.

Helstrom., C. W.

C. W. Helstrom. J. Opt. Soc. Am. 57, 297 (1967).

Jones, D. S.

D. S. Jones, Generalized Functions (McGraw-Hill Book Co., New York, 1966), p. 354.

Kraus, J. D.

See for example J. D. Kraus, Radio Astronomy (McGraw-Hill Book Co., New York, 1966), p. 69.

Landau, H. J.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).

Little, J. D. C.

J. A. Stratton, P. M. Morre, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (M.I.T. Press, Cambridge, Mass., 1956), pp. 203ndash;229.

Morre, P. M.

J. A. Stratton, P. M. Morre, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (M.I.T. Press, Cambridge, Mass., 1956), pp. 203ndash;229.

O’Neill, E. L.

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963), p. 77.

Pollak, H. O.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

Roberts, J. A.

R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).

Root, W. L.

See for example W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 224.

Rushforth, C. K.

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

Slepian, D.

D. Slepian, J. Opt. Soc. Am. 57, 918 (1967).

D. Slepian, J. Math. Phys. 44, No. 2, 107 (1965). Note that Flammer’s normalization is used in this reference.

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

D. Slepian, Trans. IRE P GIT-3, 68 (1954).

Sonnenblick, E.

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).

Stratton, J. A.

J. A. Stratton, P. M. Morre, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (M.I.T. Press, Cambridge, Mass., 1956), pp. 203ndash;229.

Wiener, N.

N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series (John Wiley & Sons, Inc., New York, 1949), Ch. III, p. 81.

Wolter, H.

H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, Sec. 4.6.

Other (31)

R. N. Bracewell and J. A. Roberts, Australian J. Phys. 7, 615 (1954).

See for example J. D. Kraus, Radio Astronomy (McGraw-Hill Book Co., New York, 1966), p. 69.

See Ref. 1, p. 627.

D. S. Jones, Generalized Functions (McGraw-Hill Book Co., New York, 1966), p. 354.

H. Wolter, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), Ch. V, Sec. 4.6.

D. Slepian and H. O. Pollak, Bell System Tech. J. 40, 43 (1961).

G. J. Buck and J. J. Gustinic, IEEE Trans. Antennas Propagation AP-15, 376 (1967).

B. R. Frieden, J. Opt. Soc. Am. 57, 1013 (1967).

C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).

N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series (John Wiley & Sons, Inc., New York, 1949), Ch. III, p. 81.

C. W. Helstrom. J. Opt. Soc. Am. 57, 297 (1967).

D. Slepian, J. Opt. Soc. Am. 57, 918 (1967).

See for example W. B. Davenport and W. L. Root, Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 224.

A. V. Balakrishnan, Trans. IRE IT-3, 143 (1957).

Expectation should be interpreted as the statistical average over the ensemble of process realizations.

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.

This is derived in Ref. (8), Eq. (2.7).

See Ref. 11, Eq. (2.5).

See Ref. 1, Eq. (2.6).

This follows from Ref. (6), Eqs. (27), (28), and (29).

The expression [x] means the smallest integer larger than x.

D. Slepian, J. Math. Phys. 44, No. 2, 107 (1965). Note that Flammer’s normalization is used in this reference.

J. A. Stratton, P. M. Morre, L. J. Chu, J. D. C. Little, F. J. Corbato, Spheroidal Wave Functions (M.I.T. Press, Cambridge, Mass., 1956), pp. 203ndash;229.

D. Slepian and E. Sonnenblick, Bell System Tech. J. 44, 1745 (1965).

See Ref. 11, Sec. 2.

See Ref. 11, Eq. (2.13).

See Ref. 6, Sec. 4.1.

H. J. Landau and H. O. Pollak, Bell System Tech. J. 41, 1295 (1962).

E. L. O’Neill, Introduction to Statistical Optics (Addison-Wesley Publ. Co., Reading, Mass., 1963), p. 77.

R. Courant and D. Helbert, Methods of Mathematical Physics (Interscience Publ., John Wiley & Sons, Inc., New York, 1966), p. 427.

D. Slepian, Trans. IRE P GIT-3, 68 (1954).

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