Abstract

The information that we can extract from a photoelectric image of a star is limited by (i) noise introduced in the signal amplification, (ii) conversion of the two-dimensional image into a temporal signal, and (iii) background radiation, optical aberrations, and photon noise. This third limitation is the primary concern in this paper; it determines the information content of the two-dimensional image. The information content of a photoelectric star image is measured by its probability of detection, and by the intrinsic error in measuring its position and intensity. The maximum achievable probability of detection is expressed in terms of the image characteristics. A detection method that maximizes the probability of detection is described; it depends on the signal-to-noise ratio. With a signal-to-noise ratio greater than 10<sup>3</sup>, detection is based on image intensity. With a signal-to-noise ratio less than 10<sup>3</sup>, detection is based on image shape and size, as well as intensity. The intrinsic relative errors in meas ring position and intensity are inversely proportional to the square root of the number of photoelectric emissions for a fixed signal-to-noise ratio. The errors are monotonic decreasing functions of the signal-to-noise ratio. Equations are derived that express the rms error in terms of the image shape, image intensity, and signal-to-noise ratio. Several of the basic results apply to arbitrary photoelectric images. In this paper, we are interested in the intrinsic detection and measurement limits of photoelectric images, as opposed to specific techniques or devices.

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  1. Astronomical Techniques, edited by W. A. Hiltner (University of Chicago Press, Chicago, Illinois, 1960).
  2. R. C. Jones, J. Opt. Soc. Am. 50, 1166, 883 (1960).
  3. W. H. Beall, J. Opt. Soc. Am. 54, 992 (1964).
  4. E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Informnation Processing, edited by J. T. Tippett et al. (MIT Press, Cambridge, Massachusetts, 1965).
  5. C. W. Helstrom, IEEE Trans. Information Theory IT-10, 275 (1964).
  6. P. Swerling, IRE Trans. Information Theory IT-8, 315 (1962).
  7. For the intensity and spectral characteristics of stellar radiation, these assumptions are physically reasonable. The characteristics of photoelectric emissions are discussed byL. Mandel, Proc. Phys. Soc. 72, 1037 (1958); 74, 233 (1959).
  8. The derivation of joint probability-density function ƒN is similar to the derivation for arrival times of a Poisson process. SeeE. Parzen, Stochastic Processes (Holden-Day, Inc., San Francisco, California, 1962), Chap. 4.
  9. A. Van der Lugt, IEEE Trans. Information Theory IT-10, 139 (1964).
  10. W. D. Montgomery and P. W. Broome, J. Opt. Soc. Am. 52, 1259 (1962).
  11. Detection is basically a statistical problem of testing the hypothesis that Is* = 0 as opposed to Is* = I1*. The optimality of the above detection method can be proven with the Neyman-Pearson lemma. SeeS. S. Wilks, Mathematical Statistics (John Wiley & Sons, Inc., New York, 1962), p. 398.
  12. The derivation of the characteristic function is similar to derivation of the characteristic function of shot noise. See J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956), p. 149.
  13. A gaussian function is a reasonable description of the flux distribution in a star image. An exact description is impractical for optical systems that are limited by optical aberrations. A gaussian-density function has the advantage of being functionally simple, and yet having three shape parameters σx, σy, ρ.
  14. These data arc from C. W. Allen, Astrophysical Quantities (University of London, The Athlone Press, London, England, 1963), p. 235.
  15. The bounds present here can be derived from the basic results of H. Cramer in Mathnatical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1958), p. 477. The bound on the product (Varx⌃0) · (Variŷ0) is obtained from the Cramer-Rao bound on the generalized variance of (x⌃00). Similar results have been obtained for waveform parameter estimation byP. Swerling IEEE Trans. Information Theory IT-10, 302 (1964).
  16. The emission rate Is* is obtained from data given by A. D. Code, in Stellar Atmospheres, edited by J. L. Greenstein (University of Chicago Press, Chicago, Illinois, 1960), p. 50.

Allen, C. W.

These data arc from C. W. Allen, Astrophysical Quantities (University of London, The Athlone Press, London, England, 1963), p. 235.

Battin, R. H.

The derivation of the characteristic function is similar to derivation of the characteristic function of shot noise. See J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956), p. 149.

Beall, W. H.

W. H. Beall, J. Opt. Soc. Am. 54, 992 (1964).

Broome, P. W.

W. D. Montgomery and P. W. Broome, J. Opt. Soc. Am. 52, 1259 (1962).

Code, A. D.

The emission rate Is* is obtained from data given by A. D. Code, in Stellar Atmospheres, edited by J. L. Greenstein (University of Chicago Press, Chicago, Illinois, 1960), p. 50.

Cramer, H.

The bounds present here can be derived from the basic results of H. Cramer in Mathnatical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1958), p. 477. The bound on the product (Varx⌃0) · (Variŷ0) is obtained from the Cramer-Rao bound on the generalized variance of (x⌃00). Similar results have been obtained for waveform parameter estimation byP. Swerling IEEE Trans. Information Theory IT-10, 302 (1964).

Farrell, E. J.

E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Informnation Processing, edited by J. T. Tippett et al. (MIT Press, Cambridge, Massachusetts, 1965).

Helstrom, C. W.

C. W. Helstrom, IEEE Trans. Information Theory IT-10, 275 (1964).

Jones, R. C.

R. C. Jones, J. Opt. Soc. Am. 50, 1166, 883 (1960).

Laning, J. H.

The derivation of the characteristic function is similar to derivation of the characteristic function of shot noise. See J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956), p. 149.

Mandel, L.

For the intensity and spectral characteristics of stellar radiation, these assumptions are physically reasonable. The characteristics of photoelectric emissions are discussed byL. Mandel, Proc. Phys. Soc. 72, 1037 (1958); 74, 233 (1959).

Montgomery, W. D.

W. D. Montgomery and P. W. Broome, J. Opt. Soc. Am. 52, 1259 (1962).

Parzen, E.

The derivation of joint probability-density function ƒN is similar to the derivation for arrival times of a Poisson process. SeeE. Parzen, Stochastic Processes (Holden-Day, Inc., San Francisco, California, 1962), Chap. 4.

Swerling, P.

P. Swerling, IRE Trans. Information Theory IT-8, 315 (1962).

Van der Lugt, A.

A. Van der Lugt, IEEE Trans. Information Theory IT-10, 139 (1964).

Wilks, S. S.

Detection is basically a statistical problem of testing the hypothesis that Is* = 0 as opposed to Is* = I1*. The optimality of the above detection method can be proven with the Neyman-Pearson lemma. SeeS. S. Wilks, Mathematical Statistics (John Wiley & Sons, Inc., New York, 1962), p. 398.

Zimmerman, C. D.

E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Informnation Processing, edited by J. T. Tippett et al. (MIT Press, Cambridge, Massachusetts, 1965).

Other

Astronomical Techniques, edited by W. A. Hiltner (University of Chicago Press, Chicago, Illinois, 1960).

R. C. Jones, J. Opt. Soc. Am. 50, 1166, 883 (1960).

W. H. Beall, J. Opt. Soc. Am. 54, 992 (1964).

E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Informnation Processing, edited by J. T. Tippett et al. (MIT Press, Cambridge, Massachusetts, 1965).

C. W. Helstrom, IEEE Trans. Information Theory IT-10, 275 (1964).

P. Swerling, IRE Trans. Information Theory IT-8, 315 (1962).

For the intensity and spectral characteristics of stellar radiation, these assumptions are physically reasonable. The characteristics of photoelectric emissions are discussed byL. Mandel, Proc. Phys. Soc. 72, 1037 (1958); 74, 233 (1959).

The derivation of joint probability-density function ƒN is similar to the derivation for arrival times of a Poisson process. SeeE. Parzen, Stochastic Processes (Holden-Day, Inc., San Francisco, California, 1962), Chap. 4.

A. Van der Lugt, IEEE Trans. Information Theory IT-10, 139 (1964).

W. D. Montgomery and P. W. Broome, J. Opt. Soc. Am. 52, 1259 (1962).

Detection is basically a statistical problem of testing the hypothesis that Is* = 0 as opposed to Is* = I1*. The optimality of the above detection method can be proven with the Neyman-Pearson lemma. SeeS. S. Wilks, Mathematical Statistics (John Wiley & Sons, Inc., New York, 1962), p. 398.

The derivation of the characteristic function is similar to derivation of the characteristic function of shot noise. See J. H. Laning and R. H. Battin, Random Processes in Automatic Control (McGraw-Hill Book Company, Inc., New York, 1956), p. 149.

A gaussian function is a reasonable description of the flux distribution in a star image. An exact description is impractical for optical systems that are limited by optical aberrations. A gaussian-density function has the advantage of being functionally simple, and yet having three shape parameters σx, σy, ρ.

These data arc from C. W. Allen, Astrophysical Quantities (University of London, The Athlone Press, London, England, 1963), p. 235.

The bounds present here can be derived from the basic results of H. Cramer in Mathnatical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1958), p. 477. The bound on the product (Varx⌃0) · (Variŷ0) is obtained from the Cramer-Rao bound on the generalized variance of (x⌃00). Similar results have been obtained for waveform parameter estimation byP. Swerling IEEE Trans. Information Theory IT-10, 302 (1964).

The emission rate Is* is obtained from data given by A. D. Code, in Stellar Atmospheres, edited by J. L. Greenstein (University of Chicago Press, Chicago, Illinois, 1960), p. 50.

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