Abstract

The values of the radiance at points of an incoherently radiating object are considered as parameters of the statistical description of the field at the aperture of an observing optical instrument. By means of the Cramér–Rao inequality, a lower bound is set to the mean-square errors of unbiased estimates of the radiance values. The errors are shown to increase rapidly when the object is sampled at points separated by less than a conventional resolution interval.

© 1970 Optical Society of America

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References

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  1. H. Wolter, in Progress in Optics. I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 157.
  2. Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
    [CrossRef]
  3. C. K. Rushforth and R. W. Harris, J. Opt. Soc. Am. 58, 539 (1968).
    [CrossRef]
  4. L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
    [CrossRef]
  5. C. W. Helstrom, J. Opt. Soc. Am. 59, 164 (1969).
    [CrossRef]
  6. C. W. Helstrom, J. Opt. Soc. Am. 60, 521 (1970).
    [CrossRef]
  7. D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 109.
    [CrossRef]
  8. C. W. Helstrom, Statistical Theory of Signal Detection, 2nd ed. (Pergamon Press, Ltd., Oxford, 1968), pp. 69–72.
  9. Reference 5, Eqs. (1.8) and (1.9).
  10. H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), pp. 473 ff.
  11. C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).
  12. L. Schmetterer, Mathematische Statistik (Springer–Verlag, New York, 1966), p. 63.
  13. Reference 5, Appendix B.
  14. C. W. Helstrom, J. Opt. Soc. Am. 60, 233 (1970).
    [CrossRef]
  15. Reference 1, Eq. (A4).
  16. C. W. Helstrom, J. Opt. Soc. Am. 59, 924 (1969), Eq. (5.10).
  17. Reference 5, Eq. (6.3), in which F=M-12.
  18. Reference 6, Sec. I, especially Eqs. (1.15) and (1.21).

1970 (2)

1969 (2)

1968 (1)

1965 (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

1961 (1)

Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
[CrossRef]

1945 (1)

C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).

Cramér, H.

H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), pp. 473 ff.

Gabor, D.

D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 109.
[CrossRef]

Harris, R. W.

Helstrom, C. W.

Lo, Y. T.

Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
[CrossRef]

Mandel, L.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Rao, C. R.

C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).

Rushforth, C. K.

Schmetterer, L.

L. Schmetterer, Mathematische Statistik (Springer–Verlag, New York, 1966), p. 63.

Wolf, E.

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Wolter, H.

H. Wolter, in Progress in Optics. I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 157.

Bull. Calcutta Math. Soc. (1)

C. R. Rao, Bull. Calcutta Math. Soc. 37, 81 (1945).

J. Appl. Phys. (1)

Y. T. Lo, J. Appl. Phys. 32, 2052 (1961).
[CrossRef]

J. Opt. Soc. Am. (5)

Rev. Mod. Phys. (1)

L. Mandel and E. Wolf, Rev. Mod. Phys. 37, 231 (1965).
[CrossRef]

Other (10)

Reference 1, Eq. (A4).

Reference 5, Eq. (6.3), in which F=M-12.

Reference 6, Sec. I, especially Eqs. (1.15) and (1.21).

L. Schmetterer, Mathematische Statistik (Springer–Verlag, New York, 1966), p. 63.

Reference 5, Appendix B.

H. Wolter, in Progress in Optics. I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 157.

D. Gabor, in Progress in Optics I, E. Wolf, Ed. (North-Holland Publ. Co., Amsterdam, 1961), p. 109.
[CrossRef]

C. W. Helstrom, Statistical Theory of Signal Detection, 2nd ed. (Pergamon Press, Ltd., Oxford, 1968), pp. 69–72.

Reference 5, Eqs. (1.8) and (1.9).

H. Cramér, Mathematical Methods of Statistics (Princeton University Press, Princeton, New Jersey, 1946), pp. 473 ff.

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Figures (1)

Fig. 1
Fig. 1

Factor G0′ in lower bound to mean-square error of an unbiased estimate of the sample value of the radiance function of the object plane. Solid curves: sinc-function sampling; dashed curve: indicator-function sampling. Curves are labeled with the number of sample points. δR/a, Δ is the sampling interval, a is the width of aperture, λ is the wavelength, and R is the distance to object plane.

Tables (1)

Tables Icon

Table I Correlation coefficient of radiance estimates.

Equations (71)

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1 2 E ψ 0 ( u 1 , t 1 ) ψ 0 * ( u 2 , t 2 ) = C B ( u 1 ) δ ( u 1 - u 2 ) χ ( t 1 - t 2 ) exp [ - i Ω ( t 1 - t 2 ) ] ,
O F m ( u ) F n * ( u ) d 2 u = C m δ mn ,
B ( u ) = m B m F m ( u ) ,
F m ( u ) = 1 , { ( m x - 1 2 ) Δ x < u x < ( m x + 1 2 ) Δ x , ( m y - 1 2 ) Δ y < u y < ( m y + 1 2 ) Δ y , F m ( u ) = 0 , u elsewhere .
F m ( u ) = sinc ( u x Δ x - 1 - m x ) sinc ( u y Δ y - 1 - m y ) ,
B m = B ( m x Δ x , m y Δ y ) ,
F m ( u ) = exp [ 2 π i ( m x u x b x - 1 + m y u y b y - 1 ) ] , - 1 2 b x < u x < 1 2 b x , - 1 2 b y < u y < 1 2 b y .
φ s ( r 1 , t 1 ; r 2 , t 2 ) = 1 2 E [ ψ s ( r 1 , t 1 ) ψ s * ( r 2 , t 2 ) ] .
ψ + ( r , t ) = ψ s ( r , t ) + ψ n ( r , t )
φ s ( r 1 , t 1 ; r 2 , t 2 ) = φ s ( r 1 , r 2 ) χ ( t 1 - t 2 ) × exp [ - i Ω ( t 1 - t 2 ) ] ,
X ( ω ) = - χ ( τ ) e i ω τ d τ ,
W = [ - X ( ω ) d ω / 2 π ] 2 / - [ X ( ω ) ] 2 d ω / 2 π = χ ( 0 ) 2 / - χ ( τ ) 2 d τ .
E s = 2 Ω 2 c T A φ s ( r , r ) d 2 r ,
φ s ( r 1 , r 2 ) = ( 8 π R 2 Ω 2 c ) - 1 O B ( u ) E ( r 1 , u ) E * ( r 2 , u ) d 2 u , E ( r , u ) = exp ( i k 2 R r - u 2 ) ,
φ s ( r 1 , r 2 ; B ) = ( 8 π R 2 Ω 2 c ) - 1 × m B m O F m ( u ) E ( r 1 , u ) E * ( r 2 , u ) d 2 u .
m = E ( B ˆ m - B m ) 2 .
E ( B ˆ m ) = B m ,
m = E ( B ˆ m - B m ) 2 L mm ,
H mn = - E [ 2 B m B n ln p ( ψ ; B ) ] .
X ˜ VX X ˜ LX ,
V mn = E ( B ˆ m - B m ) ( B ˆ n - B n ) .
Y ˜ V - 1 Y Y ˜ HY ,
Y ˜ HY = m + 2.
H mn = B m ( 1 ) B n ( 2 ) H ( B ( 1 ) , B ( 2 ) ) | B ( 1 ) = B ( 2 ) = B ,
H ( B ( 1 ) , B ( 2 ) ) = ( E s N ) 2 ( W T ) - 1 A A φ s ( r 1 , r 2 ; B ( 1 ) ) × φ s ( r 2 , r 1 ; B ( 2 ) ) d 2 r 1 d 2 r 2 | A φ s ( r , r ; B ) d 2 r | - 2 .
H mn = ( E s / N ) 2 ( W T ) - 1 B T - 2 J mn ,
B T = O B ( u ) d 2 u
J mn = O O F m ( u 1 ) F n ( u 2 ) I ( u 1 - u 2 ) 2 d 2 u 1 d 2 u 2
I ( u ) = A - 1 A I A ( r ) exp ( - i k r · u R ) d 2 r
J mn = A - 2 A A K m ( r 1 - r 2 ) K n * ( r 1 - r 2 ) d 2 r 1 d 2 r 2 ,
K m ( r ) = O F m ( u ) exp ( i k u · r R ) d 2 u
f 1 ( D ) = D W - [ X ( ω ) ] 2 [ 1 + D W X ( ω ) ] - 1 d ω 2 π ,
N = [ exp ( Ω / K T ) - 1 ] - 1 .
M = [ A φ s ( r , r ; B ) d 2 r ] 2 × [ A A φ s ( r 1 , r 2 ; B ) 2 d 2 r 1 d 2 r 2 ] - 1
F m ( u ) = F 0 ( u x + m x Δ x , u y + m y Δ y ) .
J p = O O F 0 ( v 1 ) F 0 * ( v 2 ) I ( v 1 - v 2 - Δ p ) 2 d 2 v 1 d 2 v 2 = A - 2 A A K 0 ( r 1 - r 2 ) 2 × exp ( - i k Δ p · ( r 1 - r 2 ) R ) d 2 r 1 d 2 r 2 , Δ p = ( p x Δ x , p y Δ y ) .
q G p - q J p - m = δ pm .
g ( ω ) = p G p exp ( i p · ω ) ,
j ( ω ) = p J p exp ( i p · ω ) ,
g ( ω ) j ( ω ) = 1 ,
G 0 = - π π - π π [ j ( ω ) ] - 1 d 2 ω / ( 2 π ) 2 .
m ( N / E s ) 2 W T B T 2 G 0 .
j ( ω ) = A γ A - 2 A A m K 0 ( r 1 - r 2 ) 2 × δ ( x 1 - x 2 + γ x ( m x - ω x / 2 π ) ) × δ ( y 1 - y 2 + γ y ( m y - ω y / 2 π ) ) d x 1 d y 1 d x 2 d y 2 , γ x = λ R / Δ x ,             γ y = λ R / Δ y ,             A γ = γ x γ y ,
j ( ω x , ω y ) = A γ A - 2 m [ a x - γ x m x - ω x / 2 π ] × [ a y - γ y m y - ω y / 2 π ] × K 0 ( γ x ( m x - ω x / 2 π ) γ y ( m y - ω y / 2 π ) ) 2 ,
γ x = λ R / Δ x > 2 a x
γ y = λ R / Δ y > 2 a y ,
K 0 ( r ) = A Δ = Δ x Δ y ,             - 1 2 γ x < x < 1 2 γ x ,             - 1 2 γ y < y < 1 2 γ y , K 0 ( r ) = 0 elsewhere .
j ( ω ) = A δ A Δ [ 1 - δ x ω x / 2 π Δ x ] [ 1 - δ y ω y / 2 π Δ y ] , A δ = δ x δ y ,
G 0 = 4 A δ - 2 ln ( 1 - δ x 2 Δ x ) ln ( 1 - δ y 2 Δ y ) ,             Δ x > 1 2 δ x ,             Δ y > 1 2 δ y .
m B m 2 ( N E Δ ) 2 W T ( A Δ 2 A δ 2 ) ln ( 1 - δ x 2 Δ x ) ln ( 1 - δ y 2 Δ y ) = ( N / E Δ ) 2 W T ( A Δ / A δ ) G 0 ,
G 0 = 4 ( A Δ A δ ) ln ( 1 - δ x 2 Δ x ) ln ( 1 - δ y 2 Δ y ) .
δ x = λ R / a x ,             δ y = λ R / a y
m / B m 2 ( N / E s ) 2 ( M W T ) M Δ ,
M = A 0 / A δ = M Δ A Δ / A δ
m B m ,
E ( B ˆ T - B T ) 2 / B T 2 ( N / E s ) 2 M W T ,
J p = A δ A Δ J p
J p = ( A Δ / A δ ) sinc 2 ( p x Δ x / δ x ) sinc 2 ( p y Δ y / δ y ) ,             Δ x 1 2 δ x ,             Δ y 1 2 δ y ,
J p = ( 2 π 2 ) - 2 ( A δ / A Δ ) [ 1 - ( - 1 ) p x ] [ 1 - ( - 1 ) p y ] / p x 2 p y 2 , J 0 = ( 1 - δ x 4 Δ x ) ( 1 - δ y 4 Δ y ) ,             Δ x > 1 2 δ x ,             Δ y > 1 2 δ y .
J p = ( 2 π 2 ) - 1 ( δ / Δ ) [ 1 - ( - 1 ) p ] / p 2 J 0 = ( 1 - δ / 4 Δ ) ,             Δ > 1 2 δ ,
J p = ( Δ / δ ) sinc 2 ( p Δ / δ ) ,             Δ < 1 2 δ .
G 0 = 2 ( Δ / δ ) ln ( 1 - δ / 2 Δ ) ,             δ / Δ < 2.
G p = - π π [ j ( ω ) ] - 1 exp ( - i p ω ) d ω 2 π .
K 0 ( r ) = A Δ sinc ( x / γ x ) sinc ( y / γ y ) ,
G 0 = ( π 2 / δ ) - 1 2 1 2 csc 2 ( π u ) × { m = - ( u - m ) - 2 [ 1 - u - m δ / Δ ] } - 1 d u ,             Δ < 2 δ ,
m ( N / E s ) 2 W T B T 2 G mm ,
λ m F m ( u ) = O I ( u - v ) 2 F m ( v ) d 2 v ,
Φ ( ω ) = - - I ( u ) 2 exp ( i ω · u ) d 2 u = ( λ R A ) 2 A I A ( r ) I A ( k - 1 R ω - r ) d 2 r = ( λ / R A ) 2 [ a x - k - 1 R ω x ] [ a y - k - 1 R ω y ] , ω x < k a x / R ,             ω y < k a y / R ,
λ m = ( λ R ) 2 [ 1 - m x δ x / b x ] [ 1 - m y δ y / b y ] / A , δ x = λ R / a x b x ,             δ y = λ R / a x b y ,
B m = ( b x b y ) - 1 O B ( x , y ) × exp [ 2 π i ( m x x b x - 1 + m y y b y - 1 ) ] d x d y
m / B 0 2 ( N / E s ) 2 M W T [ 1 - m x δ x / b x ] - 1 × [ 1 - m y δ y / b y ] - 1 ,