Abstract

The use of joint probability densities are examined as a method for the complete description of background. The requirements for the complete description leads to an infinite dimensional analysis of background. The relationships between joint probability densities, the autocorrelation function and the Wiener spectra are explored to obtain some insight into the nature of background from measured Wiener spectra. Techniques for theoretical and experimental determination of probability of detection and probability of false alarm are considered.

© 1961 Optical Society of America

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References

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  1. M. Born and E. Wolf, Principles of Optics (Permagon Press, New York, 1959), pp. 541–552.
  2. D. Z. Robinson, Proc. I. R. E. 47, 1554 (1959).
    [Crossref]
  3. A. I. Khinchin, Mathematical Foundations of Information Theory (Dover Publications, Inc., New York, 1957), Chap. II, p. 49.
  4. See reference 3, p. 34.
  5. W. R. Bennett, Proc. I. R. E. 44, 609 (1956).
    [Crossref]
  6. W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, 1950), 2nd ed., Vol. I, Chap. V.
  7. “Wire electrode in CRT for waveform sampling,” Electrical Design News, February1961, pp. 14–16.
  8. W. F. Schreiber, IRE Trans. on Information Theory IT-II, 94 (September, 1956).
    [Crossref]
  9. D. A. Bell, Information Theory and its Engineering Applications (Sir Isaac Pitman and Sons, Ltd., London, 1953), p. 111.
  10. A. I. Khinchin, Mathematical Foundations of Information Theory (Dover Publications, Inc., New York, 1957), Chap. I, p. 34.
  11. G. A. Barnard, IRE Trans. on Information Theory IT-1, 49 (March, 1955).
    [Crossref]
  12. G. A. Morton and S. V. Forgue, Proc. I. R. E. 47, 1607 (1959).
    [Crossref]
  13. R. E. Eisele, Proc. IRIS V, No. 3,  199 (1960).
  14. L. J. Free, J. Opt. Soc. Am. 49, 1007 (1959).
    [Crossref]
  15. J. Capon, Bounds to the Entropy of Television Signals (M.I.T., Research Laboratory Electronics, Cambridge, Massachusetts) Tech. Rept. 296 (May25, 1955).
  16. G. F. Aroyan, Proc. I. R. E. 47, 1561 (1959).
    [Crossref]
  17. H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1957), p. 213.
  18. J. A. Jamieson, Infrared Physics I, 133 (1961).
    [Crossref]
  19. J. A. Jamieson, Proc. IRIS IV, No. 4,  281 (1959).

1961 (2)

“Wire electrode in CRT for waveform sampling,” Electrical Design News, February1961, pp. 14–16.

J. A. Jamieson, Infrared Physics I, 133 (1961).
[Crossref]

1960 (1)

R. E. Eisele, Proc. IRIS V, No. 3,  199 (1960).

1959 (5)

L. J. Free, J. Opt. Soc. Am. 49, 1007 (1959).
[Crossref]

G. F. Aroyan, Proc. I. R. E. 47, 1561 (1959).
[Crossref]

J. A. Jamieson, Proc. IRIS IV, No. 4,  281 (1959).

D. Z. Robinson, Proc. I. R. E. 47, 1554 (1959).
[Crossref]

G. A. Morton and S. V. Forgue, Proc. I. R. E. 47, 1607 (1959).
[Crossref]

1956 (2)

W. R. Bennett, Proc. I. R. E. 44, 609 (1956).
[Crossref]

W. F. Schreiber, IRE Trans. on Information Theory IT-II, 94 (September, 1956).
[Crossref]

1955 (1)

G. A. Barnard, IRE Trans. on Information Theory IT-1, 49 (March, 1955).
[Crossref]

Aroyan, G. F.

G. F. Aroyan, Proc. I. R. E. 47, 1561 (1959).
[Crossref]

Barnard, G. A.

G. A. Barnard, IRE Trans. on Information Theory IT-1, 49 (March, 1955).
[Crossref]

Bell, D. A.

D. A. Bell, Information Theory and its Engineering Applications (Sir Isaac Pitman and Sons, Ltd., London, 1953), p. 111.

Bennett, W. R.

W. R. Bennett, Proc. I. R. E. 44, 609 (1956).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Permagon Press, New York, 1959), pp. 541–552.

Capon, J.

J. Capon, Bounds to the Entropy of Television Signals (M.I.T., Research Laboratory Electronics, Cambridge, Massachusetts) Tech. Rept. 296 (May25, 1955).

Cramer, H.

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1957), p. 213.

Eisele, R. E.

R. E. Eisele, Proc. IRIS V, No. 3,  199 (1960).

Feller, W.

W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, 1950), 2nd ed., Vol. I, Chap. V.

Forgue, S. V.

G. A. Morton and S. V. Forgue, Proc. I. R. E. 47, 1607 (1959).
[Crossref]

Free, L. J.

Jamieson, J. A.

J. A. Jamieson, Infrared Physics I, 133 (1961).
[Crossref]

J. A. Jamieson, Proc. IRIS IV, No. 4,  281 (1959).

Khinchin, A. I.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover Publications, Inc., New York, 1957), Chap. I, p. 34.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover Publications, Inc., New York, 1957), Chap. II, p. 49.

Morton, G. A.

G. A. Morton and S. V. Forgue, Proc. I. R. E. 47, 1607 (1959).
[Crossref]

Robinson, D. Z.

D. Z. Robinson, Proc. I. R. E. 47, 1554 (1959).
[Crossref]

Schreiber, W. F.

W. F. Schreiber, IRE Trans. on Information Theory IT-II, 94 (September, 1956).
[Crossref]

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Permagon Press, New York, 1959), pp. 541–552.

Electrical Design News (1)

“Wire electrode in CRT for waveform sampling,” Electrical Design News, February1961, pp. 14–16.

Infrared Physics (1)

J. A. Jamieson, Infrared Physics I, 133 (1961).
[Crossref]

IRE Trans. on Information Theory (1)

W. F. Schreiber, IRE Trans. on Information Theory IT-II, 94 (September, 1956).
[Crossref]

IRE Trans. on Information Theory IT-1 (1)

G. A. Barnard, IRE Trans. on Information Theory IT-1, 49 (March, 1955).
[Crossref]

J. Opt. Soc. Am. (1)

Proc. I. R. E. (4)

G. A. Morton and S. V. Forgue, Proc. I. R. E. 47, 1607 (1959).
[Crossref]

G. F. Aroyan, Proc. I. R. E. 47, 1561 (1959).
[Crossref]

D. Z. Robinson, Proc. I. R. E. 47, 1554 (1959).
[Crossref]

W. R. Bennett, Proc. I. R. E. 44, 609 (1956).
[Crossref]

Proc. IRIS IV (1)

J. A. Jamieson, Proc. IRIS IV, No. 4,  281 (1959).

Proc. IRIS V (1)

R. E. Eisele, Proc. IRIS V, No. 3,  199 (1960).

Other (8)

J. Capon, Bounds to the Entropy of Television Signals (M.I.T., Research Laboratory Electronics, Cambridge, Massachusetts) Tech. Rept. 296 (May25, 1955).

H. Cramer, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1957), p. 213.

W. Feller, An Introduction to Probability Theory and Its Applications (John Wiley & Sons, Inc., New York, 1950), 2nd ed., Vol. I, Chap. V.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover Publications, Inc., New York, 1957), Chap. II, p. 49.

See reference 3, p. 34.

D. A. Bell, Information Theory and its Engineering Applications (Sir Isaac Pitman and Sons, Ltd., London, 1953), p. 111.

A. I. Khinchin, Mathematical Foundations of Information Theory (Dover Publications, Inc., New York, 1957), Chap. I, p. 34.

M. Born and E. Wolf, Principles of Optics (Permagon Press, New York, 1959), pp. 541–552.

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Figures (7)

Fig. 1
Fig. 1

A possible first-order radiance probability density.

Fig. 2
Fig. 2

A possible second-order radiance probability density.

Fig. 3
Fig. 3

Effect of phase shift on probability density.

Fig. 4
Fig. 4

Effect of system nonlinearity on probability density.

Fig. 5
Fig. 5

Hypothetical sky patterns.

Fig. 6
Fig. 6

Point signal identifier diagram.

Fig. 7
Fig. 7

Point signal identifier in probability space.

Equations (5)

Equations on this page are rendered with MathJax. Learn more.

R ( A , B ) = ( N A - N ¯ A ) ( N B - N ¯ B ) av ,
R ( Δ ) = 0 0 ( N A - N ¯ A ) × ( N B - N ¯ B ) P ( N A , N B ) d N A d N B av ,
H ( A ) = - 0 P ( N A ) log P ( N A ) d N A .
H A ( B ) = H ( A B ) - H ( A )
H ( A B ) = - 0 0 P ( N A , N B ) log P ( N A , N B ) d N A d N B .