Abstract

The restoration of optical images, as well as the unfolding of spectroscopic and other data that have been convolved with a window function or an instrumental impulse response, can be viewed as the solution of an integral equation. Solution of such an integral equation when the data are corrupted by noise or experimental error is treated as the problem of finding an estimate that is a linear functional of the data and minimizes the mean squared error between the true solution and itself. The estimate depends on assumptions about the spectral densities of the images and the noise, the choice of which is discussed. Coherent optical processing and digital processing are described.

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  1. E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963), p. 20.
  2. P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. A247, 369 (1955).
  3. In general we shall use a lower-case letter to denote the Fourier transform of a spatial function designated by the corresponding upper-case letter.
  4. J. L. Harris, Sr., J. Opt. Soc. Am. 56, 569 (1966).
  5. R. J. Trumpler and H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, Calif., 1953), Ch. 1.4, p. 95 ff.
  6. F. D. Kahn, Proc. Cambridge Phil. Soc. 51, 519 (1955).
  7. D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).
  8. S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).
  9. S. Twomey, J. Franklin Inst. 279, 95 (1965).
  10. For continuous processes such as these, a probability density must be treated by a limiting procedure as samples of the process are taken closer and closer together in the plane. The assertions of the text remain valid.
  11. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, N. Y., 1960), Section 21.4, p. 994 ff.
  12. L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).
  13. A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).
  14. N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series (John Wiley & Sons, Inc., New York, N. Y., 1949), p. 84.
  15. H. W. Bode and C. E. Shannon, Proc. IRE 38, 417 (1950).
  16. G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press, New York, London, 1966), p. 79ff.
  17. R. Deutsch, Estimation Theory (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1965), p. 66.
  18. A. Reiz, Arkiv Mat. Astron. Fysik 29A, #29, 1 (1943).
  19. J R. Scarborough, Numerical Mathematical Analysis (The Johns Hopkins Press, Baltimore, Md., 1930), pp. 48, 114.

Bode, H. W.

H. W. Bode and C. E. Shannon, Proc. IRE 38, 417 (1950).

Cutrona, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Deutsch, R.

R. Deutsch, Estimation Theory (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1965), p. 66.

Fellgett, P. B.

P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. A247, 369 (1955).

Harris, Sr., J. L.

J. L. Harris, Sr., J. Opt. Soc. Am. 56, 569 (1966).

Kahn, F. D.

F. D. Kahn, Proc. Cambridge Phil. Soc. 51, 519 (1955).

Leith, E. N.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Linfoot, E. H.

P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. A247, 369 (1955).

Lugt, A. Vander

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

Middleton, D.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, N. Y., 1960), Section 21.4, p. 994 ff.

O’Neill, E. L.

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963), p. 20.

Palermo, C. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Phillips, D. L.

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).

Porcello, L. J.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

Reiz, A.

A. Reiz, Arkiv Mat. Astron. Fysik 29A, #29, 1 (1943).

Scarborough, J. R.

J R. Scarborough, Numerical Mathematical Analysis (The Johns Hopkins Press, Baltimore, Md., 1930), pp. 48, 114.

Shannon, C. E.

H. W. Bode and C. E. Shannon, Proc. IRE 38, 417 (1950).

Stroke, G. W.

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press, New York, London, 1966), p. 79ff.

Trumpler, R. J.

R. J. Trumpler and H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, Calif., 1953), Ch. 1.4, p. 95 ff.

Twomey, S.

S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).

S. Twomey, J. Franklin Inst. 279, 95 (1965).

Weaver, H. F.

R. J. Trumpler and H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, Calif., 1953), Ch. 1.4, p. 95 ff.

Wiener, N.

N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series (John Wiley & Sons, Inc., New York, N. Y., 1949), p. 84.

Other (19)

E. L. O'Neill, Introduction to Statistical Optics (Addison-Wesley Publishing Co., Reading, Mass., 1963), p. 20.

P. B. Fellgett and E. H. Linfoot, Phil. Trans. Roy. Soc. A247, 369 (1955).

In general we shall use a lower-case letter to denote the Fourier transform of a spatial function designated by the corresponding upper-case letter.

J. L. Harris, Sr., J. Opt. Soc. Am. 56, 569 (1966).

R. J. Trumpler and H. F. Weaver, Statistical Astronomy (University of California Press, Berkeley, Calif., 1953), Ch. 1.4, p. 95 ff.

F. D. Kahn, Proc. Cambridge Phil. Soc. 51, 519 (1955).

D. L. Phillips, J. Assoc. Comp. Mach. 9, 84 (1962).

S. Twomey, J. Assoc. Comp. Mach. 10, 97 (1963).

S. Twomey, J. Franklin Inst. 279, 95 (1965).

For continuous processes such as these, a probability density must be treated by a limiting procedure as samples of the process are taken closer and closer together in the plane. The assertions of the text remain valid.

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, N. Y., 1960), Section 21.4, p. 994 ff.

L. J. Cutrona, E. N. Leith, C. J. Palermo, and L. J. Porcello, Trans. IRE IT-6, 386 (1960).

A. Vander Lugt, Trans. IEEE IT-10, 139 (1964).

N. Wiener, The Extrapolation, Interpolation and Smoothing of Stationary Time Series (John Wiley & Sons, Inc., New York, N. Y., 1949), p. 84.

H. W. Bode and C. E. Shannon, Proc. IRE 38, 417 (1950).

G. W. Stroke, An Introduction to Coherent Optics and Holography (Academic Press, New York, London, 1966), p. 79ff.

R. Deutsch, Estimation Theory (Prentice-Hall, Inc., Englewood Cliffs, N. J., 1965), p. 66.

A. Reiz, Arkiv Mat. Astron. Fysik 29A, #29, 1 (1943).

J R. Scarborough, Numerical Mathematical Analysis (The Johns Hopkins Press, Baltimore, Md., 1930), pp. 48, 114.

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