Abstract

A background-limited optical system is to view an object through a turbulent medium. The threshold systems for detecting and resolving specific details of the object are derived and their probabilities of correct detection and resolution evaluated. The variance of a maximum-likelihood estimate of the position of the object is calculated for large signal-to-noise ratio. Specific numerical results are given for turbulence that produces a gaussian mutual-coherence function in light from a point object. A model of such turbulence, depicting it as concentrated in one or more thin, homogeneous phase screens, is proposed.

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  1. A bibliography of early work in this field is given by W. C. Hoffman, Proc. Symp. Appl. Math. 16, 117 (1964).
  2. J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964).
  3. R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).
  4. M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).
  5. D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).
  6. G. R. Heidbreder, J. Opt. Soc. Am. 57, 1477 (1967).
  7. W. P. Brown, Jr., J. Opt. Soc. Am. 57, 1539 (1967).
  8. C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969). Herein referred to as J.
  9. E denotes the expected value.
  10. D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).
  11. S. A. Bowhill, J. Res. Natl. Bur. Std. (U. S.) 65D, 275 (1961).
  12. R. P. Mercier, Proc. Cambridge Phil. Soc. 58, 382 (1962).
  13. E. E. Salpeter, Astrophys. J. 147, 433 (1967).
  14. W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 81
  15. D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Chs. 18–22.
  16. C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon Press, Ltd., Oxford, 1968), 2nd ed.
  17. The statistic U as written here differs from that in I by a term C that is independent of the data, but not of the auto-covariance of the signal field. In detection it is absorbed into the decision level with which U is compared.
  18. Reference 16, Ch. V, §1, p. 148.
  19. D. Middleton, Trans. IEEE IT-12, 230 (1966).
  20. G. N. Watson, Theory of Bessel Functions (Cambridge Univ. Press, 1944), 2nd ed. p. 395, Eq. (1).
  21. This section was prompted by a reviewer's remark.
  22. D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).
  23. Reference 16, Ch. IX, §1, pp. 290–295.
  24. Reference 16, Ch. IX, §2, pp. 295–312.
  25. Reference 16, p. 256.
  26. R. A. Fisher, Phil. Trans. Roy. Soc. (London) 222A, 309 (1922).
  27. Reference 16, p. 258.
  28. Reference 16, p. 265.
  29. Reference 15, p. 1076, Eq. A. l.31d.

Beran, M. J.

M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).

Bowhill, S. A.

S. A. Bowhill, J. Res. Natl. Bur. Std. (U. S.) 65D, 275 (1961).

Brown, Jr., W. P.

W. P. Brown, Jr., J. Opt. Soc. Am. 57, 1539 (1967).

Chase, D. M.

D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).

Davenport, W. B.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 81

Fisher, R. A.

R. A. Fisher, Phil. Trans. Roy. Soc. (London) 222A, 309 (1922).

Fried, D. L.

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).

Heidbreder, G. R.

G. R. Heidbreder, J. Opt. Soc. Am. 57, 1477 (1967).

Helstrom, C. W.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon Press, Ltd., Oxford, 1968), 2nd ed.

C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969). Herein referred to as J.

Hoffman, W. C.

A bibliography of early work in this field is given by W. C. Hoffman, Proc. Symp. Appl. Math. 16, 117 (1964).

Hufnagel, R. E.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

Keller, J. B.

J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964).

Mercier, R. P.

R. P. Mercier, Proc. Cambridge Phil. Soc. 58, 382 (1962).

Middleton, D.

D. Middleton, Trans. IEEE IT-12, 230 (1966).

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Chs. 18–22.

Root, W. L.

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 81

Salpeter, E. E.

E. E. Salpeter, Astrophys. J. 147, 433 (1967).

Stanley, N. R.

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

Watson, G. N.

G. N. Watson, Theory of Bessel Functions (Cambridge Univ. Press, 1944), 2nd ed. p. 395, Eq. (1).

Other (29)

A bibliography of early work in this field is given by W. C. Hoffman, Proc. Symp. Appl. Math. 16, 117 (1964).

J. B. Keller, Proc. Symp. Appl. Math. 16, 145 (1964).

R. E. Hufnagel and N. R. Stanley, J. Opt. Soc. Am. 54, 52 (1964).

M. J. Beran, J. Opt. Soc. Am. 56, 1475 (1966).

D. L. Fried, J. Opt. Soc. Am. 56, 1372 (1966).

G. R. Heidbreder, J. Opt. Soc. Am. 57, 1477 (1967).

W. P. Brown, Jr., J. Opt. Soc. Am. 57, 1539 (1967).

C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969). Herein referred to as J.

E denotes the expected value.

D. M. Chase, J. Opt. Soc. Am. 55, 1559 (1965).

S. A. Bowhill, J. Res. Natl. Bur. Std. (U. S.) 65D, 275 (1961).

R. P. Mercier, Proc. Cambridge Phil. Soc. 58, 382 (1962).

E. E. Salpeter, Astrophys. J. 147, 433 (1967).

W. B. Davenport and W. L. Root, An Introduction to the Theory of Random Signals and Noise (McGraw-Hill Book Co., New York, 1958), p. 81

D. Middleton, An Introduction to Statistical Communication Theory (McGraw-Hill Book Co., New York, 1960), Chs. 18–22.

C. W. Helstrom, Statistical Theory of Signal Detection (Pergamon Press, Ltd., Oxford, 1968), 2nd ed.

The statistic U as written here differs from that in I by a term C that is independent of the data, but not of the auto-covariance of the signal field. In detection it is absorbed into the decision level with which U is compared.

Reference 16, Ch. V, §1, p. 148.

D. Middleton, Trans. IEEE IT-12, 230 (1966).

G. N. Watson, Theory of Bessel Functions (Cambridge Univ. Press, 1944), 2nd ed. p. 395, Eq. (1).

This section was prompted by a reviewer's remark.

D. L. Fried, J. Opt. Soc. Am. 55, 1427 (1965).

Reference 16, Ch. IX, §1, pp. 290–295.

Reference 16, Ch. IX, §2, pp. 295–312.

Reference 16, p. 256.

R. A. Fisher, Phil. Trans. Roy. Soc. (London) 222A, 309 (1922).

Reference 16, p. 258.

Reference 16, p. 265.

Reference 15, p. 1076, Eq. A. l.31d.

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