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 103, detection is based on image intensity. With a signal-to-noise ratio less than 103, detection is based on image shape and size, as well as intensity. The intrinsic relative errors in measuring 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.

© 1966 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  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).
    [Crossref]
  3. W. H. Beall, J. Opt. Soc. Am. 54, 992 (1964).
    [Crossref]
  4. E. J. Farrell and C. D. Zimmerman, in Optical and Electro-Optical Information Processing, edited by J. T. Tippett and et al. (MIT Press, Cambridge, Massachusetts, 1965).
  5. C. W. Helstrom, IEEE Trans. Information Theory IT-10, 275 (1964).
    [Crossref]
  6. P. Swerling, IRE Trans. Information Theory IT-8, 315 (1962).
    [Crossref]
  7. For the intensity and spectral characteristics of stellar radiation, these assumptions are physically reasonable. The characteristics of photoelectric emissions are discussed by L. Mandel, Proc. Phys. Soc. 72, 1037 (1958); Proc. Phys. Soc. 74, 233 (1959).
    [Crossref]
  8. The derivation of joint probability-density function fN is similar to the derivation for arrival times of a Poisson process. See E. 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).
    [Crossref]
  10. W. D. Montgomery and P. W. Broome, J. Opt. Soc. Am. 52, 1259 (1962).
    [Crossref]
  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. See S. 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 are 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 Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1958), p. 477. The bound on the product (Varxˆ0) · (Varŷ0) is obtained from the Cramer–Rao bound on the generalized variance of (xˆ0,ŷ0). Similar results have been obtained for waveform parameter estimation by P. Swerling, IEEE Trans. Information Theory IT-10, 302 (1964).
    [Crossref]
  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.

1964 (3)

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

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

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

1962 (2)

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

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

1960 (1)

1958 (1)

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

Allen, C. W.

These data are 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.

Broome, P. W.

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 Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1958), p. 477. The bound on the product (Varxˆ0) · (Varŷ0) is obtained from the Cramer–Rao bound on the generalized variance of (xˆ0,ŷ0). Similar results have been obtained for waveform parameter estimation by P. Swerling, IEEE Trans. Information Theory IT-10, 302 (1964).
[Crossref]

Farrell, E. J.

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

Helstrom, C. W.

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

Jones, R. C.

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 by L. Mandel, Proc. Phys. Soc. 72, 1037 (1958); Proc. Phys. Soc. 74, 233 (1959).
[Crossref]

Montgomery, W. D.

Parzen, E.

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

Swerling, P.

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

Van der Lugt, A.

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

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. See S. 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 Information Processing, edited by J. T. Tippett and et al. (MIT Press, Cambridge, Massachusetts, 1965).

IEEE Trans. Information Theory (2)

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

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

IRE Trans. Information Theory (1)

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

J. Opt. Soc. Am. (3)

Proc. Phys. Soc. (1)

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

Other (9)

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

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

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. See S. 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 are 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 Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1958), p. 477. The bound on the product (Varxˆ0) · (Varŷ0) is obtained from the Cramer–Rao bound on the generalized variance of (xˆ0,ŷ0). Similar results have been obtained for waveform parameter estimation by P. Swerling, IEEE Trans. Information Theory IT-10, 302 (1964).
[Crossref]

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.

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

Cited By

OSA participates in Crossref's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (12)

Fig. 1
Fig. 1

Basic elements of sensor. L, lens; P, photoemissive surface; S, scanning mechanism; SA, signal amplification.

Fig. 2
Fig. 2

Illustration of photoelectric emissions. D, distribution of radiant flux in star image; V, field of view; P, photoemissive surface. D and V are not drawn to scale.

Fig. 3
Fig. 3

Graph of functions F1, F2, F3, F4.

Fig. 4
Fig. 4

Critical radii for approximating detection method. S, region in which the simplified detection method (4) can be used. The stellar background is from the galactic equator. Image area is one square minute of arc.

Fig. 5
Fig. 5

Comparison of impulse-response functions.

Fig. 6
Fig. 6

Bounds on Var x ˆ 0 and Vary0.

Fig. 7
Fig. 7

Graphs of functions H1 and H2.

Fig. 8
Fig. 8

Illustration of optical aberration. V, field of view. Image is not in scale relative to field of view.

Fig. 9
Fig. 9

Signal-to-noise ratio.

Fig. 10
Fig. 10

Probability of detection.

Fig. 11
Fig. 11

Accuracy of intensity measurement.

Fig. 12
Fig. 12

Lines of constant error in the x-coordinate measurement, standard deviations of 5 and 10 seconds of arc. V, field of view. A fourth-magnitude star is observed.

Equations (47)

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

I s ( λ ) T G s ( x - x 0 , y - y 0 ) A T 0 ( λ ) Δ λ Δ x Δ y ,
V G s ( x , y ) d x d y = 1.
I b ( λ ) T G b ( x , y ) A T 0 ( λ ) Δ λ Δ x Δ y .
V G b ( x , y ) d x d y = 1.
I s * = 0 I s ( λ ) T 0 ( λ ) Q ( λ ) d λ , I b * = 0 I b ( λ ) T 0 ( λ ) Q ( λ ) d λ .
T A R [ I s * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) ] d x d y .
N ¯ = T A [ I s * + I b * ] ,
f N ( x 1 , y 1 , , x N , y N ) = j = 1 N [ I s * G s ( x j - x 0 , y j - y 0 ) + I b * G b ( x j , y j ) I s * + I b * ] .
I s * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I s * + I b * .
= j = 1 N ln [ I 1 * G s ( x j - x 0 , y j - y 0 ) + I b * G b ( x j , y j ) I b * G b ( x j , y j ) ] .
= - h ( x , y ) j = 1 N δ ( x j - x , y j - y ) d x d y ,
h ( x , y ) = ln [ I 1 * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I b * G b ( x , y ) ] .
N ¯ - ( I s * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I s * + I b * ) · [ ( I 1 * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I b * G b ( x , y ) ) i u - 1 ] d x d y ,
E = N ¯ - ( I s * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I s * + I b * ) · ln [ I 1 * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I b * G b ( x , y ) ] d x d y Var = N ¯ - ( I s * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I s * + I b * ) · ln 2 [ I 1 * G s ( x - x 0 , y - y 0 ) + I b * G b ( x , y ) I b * G b ( x , y ) ] d x d y .
[ α / Γ ( β ) ] ( α x ) β - 1 e - α x ,
α = E / Var β = ( E ) 2 / Var .
Σ = ( σ x 2 σ x σ y ρ σ x σ y ρ σ y 2 ) .
R 2 = 1 1 - ρ 2 [ ( x - x 0 ) 2 σ x 2 - 2 ρ ( x - x 0 ) ( y - y 0 ) σ x σ y + ( y - y 0 ) 2 σ y 2 ] .
r 1 = I 1 * / π Σ 1 2 I b * G b ,
I 1 * G s ( x j - x 0 , y j - y 0 ) I b * G b ,
j = 1 N [ ln r 1 2 e - R j 2 / 2 + 1 ] ,
I s * A T [ F 1 ( r 1 ) + F 2 ( r 1 ) / r ]
I 1 * A T F 2 ( r 1 ) / r 1
F 1 ( r 1 ) = ( 1 + 2 / r 1 ) ln ( 1 + r 1 / 2 ) - 1 F 2 ( r 1 ) = 2 0 ln [ r 1 2 e - z + 1 ] d z .
I s * A T [ F 3 ( r 1 ) + F 4 ( r 1 ) / r ]
I 1 * A T F 4 ( r 1 ) / r 1
F 3 ( r 1 ) = ( 1 + 2 / r 1 ) [ ln 2 ( 1 + r 1 / 2 ) - 2 ln ( 1 + r 1 / 2 ) ] + 2 F 4 ( r 1 ) = 2 0 ln 2 [ r 1 2 e - z + 1 ] d z .
j = 1 N ln [ 9.80 × 10 4 e - 0.921 M - R j 2 / 2 + 1 ] ,
9.80 × 10 4 e - 0.921 M - R j 2 / 2 10 - 2
N ln ( 9.80 × 10 4 e - 0.921 M ) - 1 2 j = 1 N R j 2 .
9.80 × 10 4 e - 0.921 M - R j 2 / 2 10 2 ,
{ If             2 N ln ( r 1 / 2 ) - j = 1 N R j 2 > C p a star is present . Only photoemissions with R j < R 0 are considered ; N is the number of these photoemissions .
f = R R 0 G s ( x , y ) d x d y .
h ( x , y ) = { 2 ln ( r 1 / 2 ) - R 2 ( x , y )             for             R R 0 0             otherwise ,
R 2 ( x , y ) = 1 1 + ρ 2 [ ( x - x 0 ) 2 σ x 2 - 2 ρ ( x - x 0 ) ( y - y 0 ) σ x σ y + ( y - y 0 ) 2 σ y 2 ] .
j = 1 N r 1 2 e - R j 2 / 2 .
h ( x , y ) = { ( r 1 / 2 ) exp ( - R 2 ( x , y ) / 2 )             for             R ( x , y ) R 0 0             otherwise .
* = max ( x 0 , y 0 ) V { j = 1 N [ I 1 * G s ( x j - x 0 , y j - y 0 ) + I b * G b ( x j , y j ) I b * G b ( x j , y j ) ] } .
max ( x 0 , y 0 ) V { 2 N ln ( r 1 / 2 ) - j = 1 N R j 2 } .
Var x ˆ 0 B x - 1 ,             Var ŷ 0 B y - 1 Var x ˆ 0 Var ŷ 0 ( B x B y - B x y 2 ) - 1 Var ( Î s * / I s * ) B I - 1 ,
B x = I s * A T - [ 1 G s ( x , y ) ] 2 G s ( x , y ) + I b * G b ( x , y ) / I s * d x d y B y = I s * A T - [ 2 G s ( x , y ) ] 2 G s ( x , y ) + I b * G b ( x , y ) / I s * d x d y B x y = I s * A T - 1 G s ( x , y ) 2 G s ( x , y ) G s ( x , y ) + I b * G b ( x , y ) / I s * d x d y B I = I s * A T - G s 2 ( x , y ) G s ( x , y ) + I b * G b ( x , y ) / I s d x d y ,
1 G s ( x , y ) = G s ( x , y ) / x ,             2 G s ( x , y ) = G s ( x , y ) / y .
Σ = ( σ x 2 σ x σ y ρ σ x σ y ρ σ y 2 ) .
B x - 1 = σ x 2 ( 1 - ρ 2 ) / I s * A T H 1 ( r ) B y - 1 = σ y 2 ( 1 - ρ 2 ) / I s * A T H 1 ( r ) ( B x B y - B x y 2 ) - 1 = σ x 2 σ y 2 ( 1 - ρ 2 ) / [ I s * A T H 1 ( r ) ] 2 B I - 1 = 1 / I s * A T H 2 ( r ) ,
H 1 ( r ) = - x 2 2 π exp ( - x 2 - y 2 ) exp ( - x 2 / 2 - y 2 / 2 ) + 2 / r d x d y , H 2 ( r ) = - 1 2 π exp ( - x 2 - y 2 ) exp ( - x 2 / 2 - y 2 / 2 ) + 2 / r d x d y .
σ r = 50 ( d / 23 ) 2 + 10 σ t = 20 ( d / 23 ) 2 + 10 ,
r = 3.48 × 10 4 [ 5 ( d / 23 ) 2 + 1 ] [ 2 ( d / 23 ) 2 + 1 ] ,