Abstract

The fundamental resolving power of optical systems is defined by use of decision theory. For binary decisions in the presence of Gaussian noise, the probability of a correct decision is determined by the quadratic content of the difference image and the noise variance per unit of area. In the spatial frequency domain the probability of correct decision involves the integral of the power spectrum of the difference image and the noise variance per unit of area. The merit of a particular optical system for the performance of a specific binary decision task may be evaluated by use of a formula involving its modulation transfer function. The resolution of two monochromatic, incoherent point sources under conditions of high background radiation is discussed, as an illustrative example. The resolution of two point sources is limited only by the precision with which the flux density at all points in the image plane can be determined. The manner in which this increased precision can be obtained by increasing the period of observation is discussed, for the case of a photon-counting detector.

© 1964 Optical Society of America

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References

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  1. J. L. Harris, Scripps Inst. Oceanog. Ref.58–56 (1958).
  2. J. L. Harris, Scripps Inst. Oceanog. Ref.59–65 (1959).
  3. Hoel, Introduction to Mathematical Statistics (John Wiley & Sons, Inc., New York, 1955), p. 40.
  4. S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1954), pp. 67, 81.
  5. G. Toraldo di Francia, J. Opt. Soc. Am. 45, 497 (1955).
    [Crossref]
  6. J. L. Harris, Scripps Inst. Oceanog. Ref.63–10 (1963).
  7. F. E. Washer and F. W. Rosberry, J. Opt. Soc. Am. 41, 597 (1951).
    [Crossref]

1963 (1)

J. L. Harris, Scripps Inst. Oceanog. Ref.63–10 (1963).

1959 (1)

J. L. Harris, Scripps Inst. Oceanog. Ref.59–65 (1959).

1958 (1)

J. L. Harris, Scripps Inst. Oceanog. Ref.58–56 (1958).

1955 (1)

1951 (1)

Goldman, S.

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1954), pp. 67, 81.

Harris, J. L.

J. L. Harris, Scripps Inst. Oceanog. Ref.63–10 (1963).

J. L. Harris, Scripps Inst. Oceanog. Ref.59–65 (1959).

J. L. Harris, Scripps Inst. Oceanog. Ref.58–56 (1958).

Rosberry, F. W.

Toraldo di Francia, G.

Washer, F. E.

J. Opt. Soc. Am. (2)

Scripps Inst. Oceanog. Ref. (3)

J. L. Harris, Scripps Inst. Oceanog. Ref.63–10 (1963).

J. L. Harris, Scripps Inst. Oceanog. Ref.58–56 (1958).

J. L. Harris, Scripps Inst. Oceanog. Ref.59–65 (1959).

Other (2)

Hoel, Introduction to Mathematical Statistics (John Wiley & Sons, Inc., New York, 1955), p. 40.

S. Goldman, Information Theory (Prentice-Hall, Inc., Englewood Cliffs, New Jersey, 1954), pp. 67, 81.

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Figures (2)

F. 1
F. 1

An image orthicon television camera was set up to view a standard test pattern. The video noise level was increased to a value approximately 128 times the normal operating noise level. A camera was used to record the monitor presentation, resulting in the photographs displayed above. Exposure periods corresponding to 1, 4, 16, 64, 256, and 1024 frame periods (1/30 second time interval) were used with neutral density filters selected to maintain constant flux-time product for all exposures. The result is a clear demonstration of the improvement of precision by increased observation time.

F. 2
F. 2

Curve showing the relationship between the Rayleigh function ϕ(r) and the Rayleigh factor, r. The Rayleigh factor is a measure of the separation of the two point sources normalized to the separation as specified by the Rayleigh resolving power criterion. The dashed straight line, having a slope of 4, is the asymptotic form of the Rayleigh function for small values of the Rayleigh factor.

Equations (46)

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L ( H 1 ) = i = 1 n 1 ( 2 π ) 1 2 σ exp [ ( R i H 1 i Δ x Δ y ) 2 2 σ 2 ] ,
L ( H 2 ) = i = 1 n 1 ( 2 π ) 1 2 σ exp [ ( R i H 2 i Δ x Δ y ) 2 2 σ 2 ] .
γ 12 = L ( H 1 ) L ( H 2 ) = i = 1 n exp [ ( R i H 1 i Δ x Δ y ) 2 + ( R i H 2 i Δ x Δ y ) 2 2 σ 2 ] .
ψ 12 = 2 σ 2 ln γ 12 = i = 1 n ( R i H 2 i Δ x Δ y ) 2 ( R i H 1 i Δ x Δ y ) 2 .
ψ 12 = i = 1 n [ 2 R i ( H 2 i H 1 i ) Δ x Δ y + H 2 i 2 ( Δ x Δ y ) 2 H 1 i 2 ( Δ x Δ y ) 2 ] .
R i = H 1 i Δ x Δ y + n i ,
ψ 12 ( H 1 ) = i = 1 n ( H 2 i Δ x Δ y H 1 i Δ x Δ y ) 2 2 n i ( H 2 i Δ x Δ y H 1 i Δ x Δ y ) .
μ = i = 1 n ( H 2 i Δ x Δ y H 1 i Δ x Δ y ) 2 ,
σ 1 2 = 4 σ 2 i = 1 n ( H 2 i Δ x Δ y H 1 i Δ x Δ y ) 2 .
P = 1 ( 2 π ) 1 2 σ 1 0 exp [ ( μ x ) 2 2 σ 1 2 ] d x .
P = 1 ( 2 π ) 1 2 μ / σ 1 e z 2 / 2 d z .
μ σ 1 = [ i = 1 n ( H 2 i Δ x Δ y H 1 i Δ x Δ y ) 2 ] 1 2 / 2 σ .
υ = σ 2 / Δ x Δ y .
μ σ 1 = 1 2 υ 1 2 { + + [ H 2 ( x , y ) H 1 ( x , y ) ] 2 dxdy } 1 2 .
μ σ 1 = 1 2 υ 1 2 [ 0 0 { F [ H 2 ( x , y ) H 1 ( x , y ) ] } 2 d f x d f y ] 1 2 ,
F [ H 2 ( x , y ) H 1 ( x , y ) ] = F [ H 2 ( x , y ) H 1 ( x , y ) ] T ( f x , f y ) ,
μ σ 1 = 1 2 υ 1 2 [ 0 0 { F [ H 2 ( x , y ) H 1 ( x , y ) ] } 2 × [ T ( f x , f y ) ] 2 d f x d f y ] 1 2 .
H ( x , y ) = 0 0 C cos 2 π f x x cos 2 π f y y d f x d f y + 0 0 D cos 2 π f x x sin 2 π f y y d f x d f y + 0 0 E sin 2 π f x x cos 2 π f y y d f x d f y + 0 0 F sin 2 π f x x sin 2 π f y y d f x d f y .
C = H ( x , y ) cos 2 π f x x cos 2 π f y ydxdy ,
D = H ( x , y ) cos 2 π f x x sin 2 π f y ydxdy ,
E = H ( x , y ) sin 2 π f x x cos 2 π f y y dxdy ,
F = H ( x , y ) sin 2 π f x x sin 2 π f y y dxdy .
D = E = F = 0 ,
C = A ,
H 1 ( x , y ) = A 0 0 cos 2 π f x x cos 2 π f y y d f x d f y .
H 2 ( x , y ) = A 2 { 0 0 [ cos 2 π f x ( x + Δ x 2 ) + cos 2 π f x ( x Δ x 2 ) ] cos 2 π f y y d f x d f y } .
H 1 ( x , y ) H 2 ( x , y ) = A 0 0 cos 2 π f y y × cos 2 π f x x ( 1 cos π f x Δ x ) d f x d f y .
F [ H 1 ( x , y ) H 2 ( x , y ) ] = A ( 1 cos π f x Δ x ) .
T ( f x , f y ) = [ 1 ( f x / f x c ) ] [ 1 ( f y / f y c ) ] ,
μ σ 1 = 1 2 υ 1 2 [ 0 0 A 2 ( 1 cos π f x Δ x ) 2 × ( 1 f y f y c ) 2 ( 1 f x f x c ) 2 d f x d f y ] 1 2 .
μ σ 1 = A f y c 1 2 f x c 1 2 2 3 υ 1 2 [ 1 2 + 4 ( sin π f x c Δ x π f x c Δ x ) ( π f x c Δ x ) 3 ( sin 2 π f x c Δ x 2 π f x c Δ x ) ( 2 π f x c Δ x ) 3 ] 1 2 .
g ( x , y ) = C sin 2 ( π x / x 0 ) ( π x / x 0 ) 2 sin 2 ( π y / y 0 ) ( π y / y 0 ) 2 .
x 0 = λ F / D x
y 0 = λ F / D y ,
G ( f x , f y ) = [ 1 ( f x / f x c ) ] [ 1 ( f y / f y c ) ] ; f x < f x c , f y < f y c ,
G ( f x , f y ) = 0 0 > f x > f x c ; 0 > f y > f y c ,
f x c = 1 / x 0 ,
f y c = 1 / y 0 .
Δ x = r x 0 ,
μ σ 1 = A D y 1 2 D x 1 2 2 3 υ 1 2 λ F [ 1 2 + 4 ( sin π r π r ) ( π r ) 3 sin 2 π r 2 π r ( 2 π r ) 3 ] 1 2 .
A = ρ T D y D x ,
m = η T Δ x Δ y D x D y / F 2 ,
υ = η T D x D y / F 2 .
μ σ 1 = ρ T 1 2 D y D x 2 3 η 1 2 λ [ 1 2 + 4 ( sin π r π r ) ( π r ) 3 sin 2 π r 2 π r ( 2 π r ) 3 ] 1 2 .
ϕ ( r ) = ( 12 π 4 / 7 ! ) r 4 ,
μ σ 1 = π 2 ( 7 ! ) 1 2 D y D x T 1 2 r 2 λ η 1 2 .