Abstract

A transfer theory is developed that determines the image space, and three-dimensional image spectrum, of a 3–D object. For both incoherent and coherent illumination, the image is found to obey convolution, transfer, and sampling theorems that resemble the familiar results of ordinary 2-D theory. A 3-D transfer function is related to the pupil function of the image-forming optical system. One result of the theory is that with incoherent illumination, the object image space contains no more than 1/(λ3f/no.4) degrees of freedom/unit volume, where λ is the wavelength of light. The transfer theory is based on the existence of volumes of stationarity, termed “isotomes;” into which the object must be partitioned. Isotomicity is shown to be approximated, over sufficiently small volumes, in the diffraction-limited case.

© 1967 Optical Society of America

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References

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  1. B. R. Frieden, J. Opt. Soc. Am. 56, 1495 (1966).
    [Crossref]
  2. P. M. Duffieux, L’integrale de Fourier et ses Applications a l’Optique, Besançon (1947), privately printed.
  3. See, e.g., H. H. Hopkins, in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (John Wiley & Sons, Inc., New York, 1963), pp. 483, 484.
  4. M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.
  5. From the Greek words “iso” and “tomos,” meaning “equal” and “volume section,” respectively.
  6. P. Dumontet, Opt. Acta 2, 53 (1955).
    [Crossref]
  7. Ref. 4, p. 752.
  8. R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), p. 7.
  9. Reference 4, p. 382.
  10. The orientation of axes α′β′γ′ is as described by B. R. A. Nijboer, Thesis, Groningen, 1948. The approximate coincidence of axes α, β, γ and α′, β′, γ′, is implicitly assumed in, e.g., Ref. 4, p. 480.
  11. Reference 4, p. 484.
  12. Reference 4; p. 754, Eq. (10) and p. 756, Eq. (23).
  13. H. H. Hopkins, Opt. Acta 2, 23 (1955).
    [Crossref]
  14. The corresponding one-dimensional formulae are derived by S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1955), p. 82.
  15. A very similar concept was previously used by C. W. Mc-Cutchen, J. Opt. Soc. Am. 54, 240 (1964).
    [Crossref]
  16. Reference 8, pp. 11, 91.

1966 (1)

1964 (1)

1955 (2)

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[Crossref]

Born, M.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.

Bracewell, R. M.

R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), p. 7.

Duffieux, P. M.

P. M. Duffieux, L’integrale de Fourier et ses Applications a l’Optique, Besançon (1947), privately printed.

Dumontet, P.

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

Frieden, B. R.

Goldman, S.

The corresponding one-dimensional formulae are derived by S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1955), p. 82.

Hopkins, H. H.

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[Crossref]

See, e.g., H. H. Hopkins, in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (John Wiley & Sons, Inc., New York, 1963), pp. 483, 484.

Mc-Cutchen, C. W.

Nijboer, B. R. A.

The orientation of axes α′β′γ′ is as described by B. R. A. Nijboer, Thesis, Groningen, 1948. The approximate coincidence of axes α, β, γ and α′, β′, γ′, is implicitly assumed in, e.g., Ref. 4, p. 480.

Wolf, E.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.

J. Opt. Soc. Am. (2)

Opt. Acta (2)

H. H. Hopkins, Opt. Acta 2, 23 (1955).
[Crossref]

P. Dumontet, Opt. Acta 2, 53 (1955).
[Crossref]

Other (12)

Ref. 4, p. 752.

R. M. Bracewell, The Fourier Transform and its Applications (McGraw-Hill Book Co., New York, 1965), p. 7.

Reference 4, p. 382.

The orientation of axes α′β′γ′ is as described by B. R. A. Nijboer, Thesis, Groningen, 1948. The approximate coincidence of axes α, β, γ and α′, β′, γ′, is implicitly assumed in, e.g., Ref. 4, p. 480.

Reference 4, p. 484.

Reference 4; p. 754, Eq. (10) and p. 756, Eq. (23).

P. M. Duffieux, L’integrale de Fourier et ses Applications a l’Optique, Besançon (1947), privately printed.

See, e.g., H. H. Hopkins, in Proceedings of the Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (John Wiley & Sons, Inc., New York, 1963), pp. 483, 484.

M. Born and E. Wolf, Principles of Optics (Pergamon Press, New York, 1959), p. 481.

From the Greek words “iso” and “tomos,” meaning “equal” and “volume section,” respectively.

The corresponding one-dimensional formulae are derived by S. Goldman, Information Theory (Prentice-Hall, Inc., New York, 1955), p. 82.

Reference 8, pp. 11, 91.

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Figures (5)

Fig. 1
Fig. 1

Object and image spaces for three-dimensional image formation. The region of isotomicity is exaggerated for pictorial emphasis. Image irradiance i(r) is shown to arise from overlap of the point spread functions from differential areas i of the object.

Fig. 2
Fig. 2

Parameters of the pupil and of the spread function space.

Fig. 3
Fig. 3

Transfer function F represented as a line integral across the pupil. Integration path Γ is confined to overlap region .

Fig. 4
Fig. 4

Determination of F for a diffraction-limited, circular pupil. F=AA′/BB′.

Fig. 5
Fig. 5

Bandpass volume ∑ in Ω-space for F(Ω) and I(Ω). Region ∑ is generated by rotation of the curve ω3=ω(2k)−1(2α0ω) about the ω3-axis.

Equations (113)

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r = ( x , y , z ) r G = ( x G , y G , z G ) r = ( x , y , z ) d r = d x d y d z
d r = d x d y d z .
d N ( r ) = a σ ( r ) d σ ( r )
d N ( r ) = a V ( r ) d r ,
s ( r ; r ) = s ( r - r G ) .
i ( r ) = s ( r 1 ; r ) + s ( r 2 ; r ) .
i ( r ) = 2 s ( r 1 ; r ) .
i ( r ) = i = 1 N o ( r i ) s ( r i ; r ) .
i ( r ) = V o V ( r ) s ( r ; r ) d r ,
i ( r ) = σ o σ ( r ) s ( r ; r ) d σ ( r ) .
and            o V a V o σ a σ ,
or            o σ ( r ) = 0 o V ( r ) = 0 }             for            z f .
i ( r ) = { V o V ( r ) s ( r - r G ) d r            or σ o σ ( r ) s ( r - r G ) d σ ( r ) ,
r G = f r ( z - f ) - 1 .
i ( r ) = - o σ ( r ) s ( r - m r ) d x d y ,
m = f ( z - f ) - 1 .
I ( Ω ) = ( 2 π ) - 3 2 Image i ( r ) exp ( - j Ω · r ) d r ,
Ω = ( ω 1 , ω 2 , ω 3 )
j = ( - 1 ) 1 2 .
ω = ( ω 1 , ω 2 ) .
- i ( x , y , z ) d x d y = P i
I ( 0 , 0 , ω 3 ) = ( 2 π ) - 1 2 P i δ ( ω 3 ) ,
i ( r ) = ( 2 π ) - 3 2 Σ I ( Ω ) exp ( j Ω · r ) d Ω ,
d Ω = d ω 1 d ω 2 d ω 3 ,
I ( Ω ) = ( 2 π ) - 3 2 σ d σ ( r ) o ( r ) All space d r s ( r - r G ) × exp ( - j Ω · r ) .
ϱ = r - r G ( α , β , γ ) ,             d ϱ d α d β d γ ,
I ( Ω ) = 0 ( Ω ) · F ( Ω ) ,
0 ( Ω ) = σ d σ ( r ) o σ ( r ) exp [ - j f ( z - f ) - 1 Ω · r ]
F ( Ω ) = ( 2 π ) - 3 2 All space d ϱ s ( ϱ ) exp ( - j Ω · ϱ ) .
0 ( Ω ) = V d r o V ( r ) exp [ - j f ( z - f ) - 1 Ω · r ] .
I t ( ω ; z ) = 0 t ( ω ; z ) τ ( ω ; z - m z ) ,
I t ( ω ; z ) = ( 2 π ) - 1 - i ( r ) × exp [ - j ( ω 1 x + ω 2 y ) ] d x d y ,
0 t ( ω ; z ) = - o ( r ) × exp [ - j m ( ω 1 x + ω 2 y ) ] d x d y ,
τ ( ω ; z - m z ) = ( 2 π ) - 1 - s ( α , β , z - m z ) × exp [ - j ( ω 1 α + ω 2 β ) ] d α d β ,
F ( 0 , 0 , ω 3 ) = ( 2 π ) - 1 2 P δ ( ω 3 ) ,
F ( Ω ) = ( 2 π ) - 1 2 - d γ τ ( ω ; γ ) exp ( - j ω 3 γ ) .
τ ( ω ; γ ) = ( 2 π ) - 1 2 - d ω 3 F ( Ω ) exp ( j ω 3 γ ) ,
F ( Ω ) = F ( Ω ) exp [ - j Λ ( Ω ) ] ,
s ( ϱ ) = ( 2 π ) - 3 2 Σ F ( Ω ) exp ( j Ω · ϱ ) d Ω .
u ( α , β , γ ) = ( λ r G ) - 1 - d p d q U ( p , q ) × exp [ j k ( 2 r G 2 ) - 1 ( p 2 + q 2 ) γ + j k r G - 1 ( p α + q β ) ] ,
s = u · u *
τ ( ω ; γ ) = ( 2 π ) - 1 - d p d q U ( p , q ) × U * [ p - r G ω 1 / k , q - r G ω 2 / k ] × exp [ j γ ( p ω 1 / r G + q ω 2 / r G - ω 2 / 2 k ) ] .
F ( Ω ) = ( 2 π ) - 3 2 - d p d q U ( p , q ) × U * [ p - r G ω 1 / k , q - r G ω 2 / k ] × - d γ exp [ j γ ( p ω 1 / r G + q ω 2 / r G - ω 2 / 2 k - ω 3 ) ] .
2 π r G ω 1 - 1 δ [ p + ω 1 - 1 ( ω 2 q - r G ω 2 / 2 k - r G ω 3 ) ] .
F ( Ω ) = ( 2 π ) - 1 2 r G ω 1 - 1 - d q U [ p ( q ) , q ] × U * [ p ( q ) - r G ω 1 k - 1 , q - r G ω 2 k - 1 ] ,
p ( q ) = ω 1 - 1 ( r G ω 2 / 2 k + r G ω 3 - q ω 2 ) .
p = p - r G ω 1 k - 1
q = q - r G ω 2 k - 1 .
b ( Ω ) = r G ω ( 2 k ) - 1 + r G ( ω 3 / ω ) .
b ( ω 1 , ω 2 , 0 ) = 1 2 ( 0 0 ) ,
d q = ( ω 1 / ω ) d l .
F ( Ω ) = ( 2 π ) - 1 2 r G ω - 1 A A d l ( U · U * ) Γ ,
Ω is in passband Σ .
F ( Ω ) = ω K ( ω ) F ( Ω ) ,
K ( ω ) = ( 2 π ) 1 2 r G - 1 / - U ( η , θ ) 2 d η .
θ = tan - 1 ( ω 1 / ω 2 ) .
F ( Ω ) = A A d l ( U · U * ) Γ / - U ( η , θ ) 2 d η .
F ( Ω ) F 0 ( Ω ) ,
U ( η , θ ) = { 1 for η r 0 0 for η > r 0 ,
F 0 ( Ω ) = A A / B B ,
B B = 2 r 0 .
A A = 2 [ r 0 2 - ( r G ω / 2 k + r G ω 3 / ω ) 2 ] 1 2 ,
F 0 ( Ω ) = { [ 1 - ( r G / r 0 ) 2 ( ω / 2 k + ω 3 / ω ) 2 ] 1 2             for            Ω in Σ 0             for            Ω not in Σ .
F ( Ω ) = 0             when            ω 2 α 0 ,
α 0 = k r 0 / r G .
ω 2 α 0 ,             ω 3 ( ω / 2 k ) ( 2 α 0 - ω ) ;
ω 3 α 0 2 / 2 k ,
ω α 0 - ( α 0 2 - 2 k ω 3 ) 1 2
ω α 0 + ( α 0 2 - 2 k ω 3 ) 1 2 .
ω 3 = ± ( ω / 2 k ) ( 2 α 0 - ω ) .
ω 1 > 2 π R             or            ω 2 > 2 π R             or            ω 3 > 2 π R 3 ,
R = α 0 / π             and            R 3 = α 0 2 / ( 4 π k ) .
R 3 / R = α 0 / 4 k = ( 8 f # ) - 1 ,
f # = r G / 2 r 0 .
R 3 R .
F ( Ω ) = { ( 2 π ) - 3 2 l , m , n s ( l π / 2 α 0 , m π / 2 α 0 , 2 n π k / α 0 2 ) × exp [ - j π ( l ω 1 / 2 α 0 + m ω 2 / 2 α 0 + 2 n k ω 3 / α 0 2 ) ] for            ω 1 2 α 0 , ω 2 2 α 0 , ω 3 α 0 2 / 2 k ; or 0 for Ω not in Σ ;
s ( α , β , γ ) = I , m , n s ( l π / 2 α 0 , m π / 2 α 0 , 2 n π k / α 0 2 ) × sinc ( 2 α 0 α - l π ) sinc ( 2 α 0 β - m π ) × sinc ( α 0 2 γ / 2 k - n π ) .
s ( α , β , 0 ) = l , m s ( l π / 2 α 0 , m π / 2 α 0 , 0 ) sinc ( 2 α 0 α - l π ) × sinc ( 2 α 0 β - m π )
s ( 0 , 0 , γ ) = n s ( 0 , 0 , 2 n π k / α 0 2 ) sinc ( α 0 2 γ / 2 k - n π ) .
F ( Ω ) = ( 2 π ) - 1 2 n τ ( ω ; 2 k π n / α 0 2 ) × exp ( - 2 j k π n ω 3 / α 0 2 )
τ ( ω ; γ ) = n τ ( ω ; 2 k π n / α 0 2 ) sinc ( α 0 2 γ / 2 k - n π ) .
( α , β , γ ) ( x , y , z ) s i
F I
o V ( r ) o ( x , y ) δ ( z - z 0 )
1 / ( λ 3 f # 4 )             degrees of freedom / volume
i ( r ) f 4 ( z - f ) - 4 o V [ f r ( z - f ) - 1 ] .
o V ( r ) = ( 2 π ) - 3 f 4 ( z - f ) - 4 - d Ω 0 ( Ω ) × exp [ j f ( z - f ) - 1 Ω · r ] .
i ( r ) = g ( r ) 2 .
W ( Ω ) = ( 2 π ) - 3 2 All space d ϱ u ( ϱ ) exp ( - j Ω · ϱ ) .
U ( r G ω 1 / k , r G ω 2 / k ) δ [ ω 2 / 2 k - ω 3 ] .
ω α 0 , ω 3 α 0 2 / 2 k .
- d ω 1 d ω 2 U ( r G ω 1 / k , r G ω 2 / k ) 0 ( ω 1 , ω 2 , ω 2 / 2 k ) × exp [ j ( ω 1 x + ω 2 y ) + j z ω 2 / 2 k ] .
U ( p , q ) = ( E / 4 π ) 1 2 r - 1 .
u = u ( ν , γ ) = ( π E ) 1 2 ( λ r G r ) - 1 0 r 0 d μ · μ J 0 ( k μ ν / r G ) × exp ( j k μ 2 γ / 2 r G 2 ) ,
ν = ( α 2 + β 2 ) 1 2             and            μ = ( p 2 + q 2 ) 1 2 .
u ( ν , 0 ) 2 = s ( ν , 0 ) = ( π E / 4 λ 2 ) ( r 0 / r ) 4 ( r - f ϕ ) 2 f ϕ - 2 × [ 2 J 1 ( α 0 ν ) / α 0 ν ] 2 ,
f ϕ = f sec ( ϕ ) ,
u ( 0 , γ ) 2 = s ( 0 , γ ) = ( π E / 4 λ 2 ) ( r 0 / r ) 4 ( r - f ϕ ) 2 f ϕ - 2 × [ sin ( α 0 2 γ / 4 k ) / ( α 0 2 γ / 4 k ) ] 2 .
s ( 0 , 0 ) = ( π E / 4 λ 2 ) ( r 0 / r ) 4 ( r - f ϕ ) 2 f ϕ - 2 ,
R ν = ( 1.64 λ / r 0 ) f ϕ r ( r - f ϕ ) - 1 ,
Z γ = ( 2 λ / r 0 2 ) ( f ϕ r ) 2 ( r - f ϕ ) - 2 .
( 2 π ) - 1 - d ω exp ( j ω x ) - d x f ( x ) exp ( - j ω x )
( θ ) = - U ( η , θ ) 2 d η
s ( ϱ ) = ( 2 π ) - 2 r G Σ F ( Ω ) ω - 1 ( θ ) exp ( j Ω · ϱ ) d Ω .
α = ν sin Φ β = ν cos Φ }
ω 1 = ω sin θ ω 2 = ω cos θ . }
s ( ν , Φ , γ ) = ( 2 π ) - 2 r G 0 2 π 0 2 α 0 - g ( ω ) g ( ω ) d ω 3 d ω d θ F ( ω , θ , ω 3 ) ( θ ) × exp [ j ω ν cos ( Φ - θ ) + j ω 3 γ ] ,
g ( ω ) = ( ω / 2 k ) ( 2 α 0 - ω )             and            ω 2 α 0 .
τ ( ω ; γ ) = ( 2 π ) - 1 2 - g ( ω ) g ( ω ) d ω 3 F ( Ω ) exp ( j ω 3 γ ) .
g ( ω ) α 0 2 / 2 k
τ ( 0 , 0 ; γ ) = ( 2 π ) - 1 P ,
x = r sin ψ y = r cos ψ , }
i ( r ) = ( 2 π ) - 2 r G × 0 2 π 0 2 α 0 - g ( ω ) g ( ω ) d ω 3 d ω d θ ( θ ) 0 ( ω , θ , ω 3 ) F ( ω , θ , ω 3 ) × exp [ j ω r cos ( θ - ψ ) + j ω 3 z ] .