Abstract

The properties of many three-dimensional radiographic imaging systems are examined within a common analytic framework. It is found, by performing an important coordinate transformation, that the projection data of these systems can be transformed to a form amenable to analysis by the central-slice theorem. Therefore, a clear relationship between the measured data set and the three-dimensional Fourier transform of the object can be established. For the Fourier aperture system, each measurement in the detector plane gives directly one point in the three-dimensional Fourier transform of the object. The limited view angle of these systems manifests itself in the incomplete collection of the Fourier transform of the object. This “missing cones” region in the Fourier space produces a point-spread function that has long-range conical ridges radiating from the central core. It is shown that degradations in linear reconstructions of extended objects are not as disastrous as might have been expected.

© 1979 Optical Society of America

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References

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  1. C. Chou and H. H. Barrett, “Gamma-ray imaging in Fourier space,” Opt. Lett. 3, 187–189 (1978).
    [Crossref] [PubMed]
  2. L. Mertz, “Applicability of the rotation collimator to nuclear medicine,” Opt. Commun. 12, 216–219 (1974).
    [Crossref]
  3. L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci.,  NS-23, 568–572 (1976).
    [Crossref]
  4. W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).
  5. R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
    [PubMed]
  6. J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comp. Graph. Img. Proc. 6, 513–537 (1977).
    [Crossref]
  7. K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Three-dimensional object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
    [Crossref]
  8. B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
    [Crossref]
  9. A. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttgart) 51, 105–117 (1978).
  10. R. N. Bracewell and S. J. Wernecke, “Image reconstruction over a finite field of view,” J. Opt. Soc. Am. 65, 1342–1346 (1975).
    [Crossref]
  11. T. Ionoye, “Image reconstructing with limited angle projection data,” IEEE Trans. Nucl. Sci.,  NS-26, 2666–2669 (1979).
  12. A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
    [Crossref]
  13. A good review of this topic can be found in B. R. Frieden, “Image Enhancement and Restoration,” Chap. 5 of Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, Heidelberg, 1975).
  14. J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
    [Crossref] [PubMed]
  15. R. G. Pridham, “Imaging of volume object by shadow casting,” Ph.D. thesis, University of Rhode Island, 1977, Appendix M (unpublished).

1979 (2)

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Three-dimensional object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

T. Ionoye, “Image reconstructing with limited angle projection data,” IEEE Trans. Nucl. Sci.,  NS-26, 2666–2669 (1979).

1978 (5)

J. R. Fienup, “Reconstruction of an object from the modulus of its Fourier transform,” Opt. Lett. 3, 27–29 (1978).
[Crossref] [PubMed]

A. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttgart) 51, 105–117 (1978).

C. Chou and H. H. Barrett, “Gamma-ray imaging in Fourier space,” Opt. Lett. 3, 187–189 (1978).
[Crossref] [PubMed]

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

1977 (1)

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comp. Graph. Img. Proc. 6, 513–537 (1977).
[Crossref]

1976 (1)

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci.,  NS-23, 568–572 (1976).
[Crossref]

1975 (2)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[Crossref]

R. N. Bracewell and S. J. Wernecke, “Image reconstruction over a finite field of view,” J. Opt. Soc. Am. 65, 1342–1346 (1975).
[Crossref]

1974 (1)

L. Mertz, “Applicability of the rotation collimator to nuclear medicine,” Opt. Commun. 12, 216–219 (1974).
[Crossref]

1967 (1)

Barrett, H. H.

Bracewell, R. N.

Chang, L. T.

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci.,  NS-23, 568–572 (1976).
[Crossref]

Chou, C.

Colsher, J. G.

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comp. Graph. Img. Proc. 6, 513–537 (1977).
[Crossref]

Fienup, J. R.

Frieden, B. R.

B. R. Frieden, “Optical transfer of the three-dimensional object,” J. Opt. Soc. Am. 57, 56–66 (1967).
[Crossref]

A good review of this topic can be found in B. R. Frieden, “Image Enhancement and Restoration,” Chap. 5 of Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, Heidelberg, 1975).

Ionoye, T.

T. Ionoye, “Image reconstructing with limited angle projection data,” IEEE Trans. Nucl. Sci.,  NS-26, 2666–2669 (1979).

Keyes, J. W.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).

Kirch, D.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

Koral, K. F.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).

LeFree, M.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

Leonard, P. F.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).

Lohmann, A.

A. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttgart) 51, 105–117 (1978).

Macdonald, B.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Three-dimensional object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci.,  NS-23, 568–572 (1976).
[Crossref]

Mayans, R.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).

Mertz, L.

L. Mertz, “Applicability of the rotation collimator to nuclear medicine,” Opt. Commun. 12, 216–219 (1974).
[Crossref]

Papoulis, A.

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[Crossref]

Perez-Mendez, V.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Three-dimensional object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci.,  NS-23, 568–572 (1976).
[Crossref]

Pridham, R. G.

R. G. Pridham, “Imaging of volume object by shadow casting,” Ph.D. thesis, University of Rhode Island, 1977, Appendix M (unpublished).

Rogers, W. L.

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).

Steele, P.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

Tam, K. C.

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Three-dimensional object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

Vogel, R. A.

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

Wernecke, S. J.

Comp. Graph. Img. Proc. (1)

J. G. Colsher, “Iterative three-dimensional image reconstruction from tomographic projections,” Comp. Graph. Img. Proc. 6, 513–537 (1977).
[Crossref]

IEEE Trans. Circuits Syst. (1)

A. Papoulis, “A new algorithm in spectral analysis and band-limited extrapolation,” IEEE Trans. Circuits Syst. 22, 735–742 (1975).
[Crossref]

IEEE Trans. Nucl. Sci. (3)

T. Ionoye, “Image reconstructing with limited angle projection data,” IEEE Trans. Nucl. Sci.,  NS-26, 2666–2669 (1979).

K. C. Tam, V. Perez-Mendez, and B. Macdonald, “Three-dimensional object reconstruction in emission and transmission tomography with limited angular input,” IEEE Trans. Nucl. Sci. NS-26, 2797–2805 (1979).
[Crossref]

L. T. Chang, B. Macdonald, and V. Perez-Mendez, “Axial tomography and three-dimensional image reconstruction,” IEEE Trans. Nucl. Sci.,  NS-23, 568–572 (1976).
[Crossref]

J. Nucl. Med. (2)

W. L. Rogers, K. F. Koral, R. Mayans, P. F. Leonard, and J. W. Keyes, “Coded aperture imaging of the heart,” J. Nucl. Med. 19, 730–730 (1978).

R. A. Vogel, D. Kirch, M. LeFree, and P. Steele, “A new method of multiplanar emission tomography using a seven pinhole collimator and an Anger scintillation camera,” J. Nucl. Med. 19, 648–654 (1978).
[PubMed]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

L. Mertz, “Applicability of the rotation collimator to nuclear medicine,” Opt. Commun. 12, 216–219 (1974).
[Crossref]

Opt. Lett. (2)

Optik (Stuttgart) (1)

A. Lohmann, “Three-dimensional properties of wave fields,” Optik (Stuttgart) 51, 105–117 (1978).

Other (2)

R. G. Pridham, “Imaging of volume object by shadow casting,” Ph.D. thesis, University of Rhode Island, 1977, Appendix M (unpublished).

A good review of this topic can be found in B. R. Frieden, “Image Enhancement and Restoration,” Chap. 5 of Picture Processing and Digital Filtering, edited by T. S. Huang (Springer-Verlag, Heidelberg, 1975).

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Figures (16)

FIG. 1
FIG. 1

Central-slice theorem in three dimensions. (a) Two-dimensional projection obtained by integrating along a set of parallel lines yields information on a plane in the three-dimensional Fourier space. (b) One-dimensional projection obtained by integrating over a set of parallel planes yields information on a line in the three-dimensional Fourier space.

FIG. 2
FIG. 2

Two types of radiographic imaging systems. In transmission mode, the radiation source is outside the patient’s body, while for emission mode, the source is within the human body.

FIG. 3
FIG. 3

Basic principle of the Fourier aperture. The detector looking through the moiré grids toward the planar object is equivalent to projecting fringes on the object and then integrating the whole pattern.

FIG. 4
FIG. 4

Geometry for three-dimensional imaging through the Fourier aperture. The aperture plane is chosen at z = 0.

FIG. 5
FIG. 5

Relation between object and Fourier space. (a) Detector P collects cone beam radiation through a moiré grid of frequency ρg. (b) The data obtained by detector P give a direct measure of a point Q in the distorted Fourier space. The whole array of detectors PP gives a line QQ parallel to the ζ ˜ axis.

FIG. 6
FIG. 6

Missing cones in Fourier space. The cone half-angle is ψ.

FIG. 7
FIG. 7

(a) Point on the detector plane looking toward the source through the moiré grids can be thought of as projecting three-dimensional divergent fringes over the object. (b) After coordinate transformation, the divergent fringes are converted into parallel fringes.

FIG. 8
FIG. 8

Multiple-pinhole imaging systems used by Chang and Vogel. In Chang’s work, each pinhole is opened one at a time and the projection images are recorded sequentially. Vogel adds one pinhole at the center. By using an Anger camera of large field of view, he is able to obtain seven nonoverlapped projections from seven pinholes simultaneously.

FIG. 9
FIG. 9

Geometry for three-dimensional imaging through a pinhole. The z = 0 plane is located at the center of the object.

FIG. 10
FIG. 10

Off-axis rotating slit system. The aperture is a narrow slit displaced a distance l from the rotation axis. Both the slit and the one-dimensional detector array rotate synchronously.

FIG. 11
FIG. 11

Conventional tomographic imaging system. The detector (film) and the source move synchronously such that only a particular plane (fulcrum plane) is in focus while others are blurred.

FIG. 12
FIG. 12

Three-dimensional multiplexing properties for some coded aperture systems.

FIG. 13
FIG. 13

(a) Point-spread function for the missing-cone system. No apodization. Cone half-angle = 60°. (b) Same as (a) except with Gaussian apodization.

FIG. 14
FIG. 14

Three-dimensional reconstruction of various objects: (a) solid sphere, (b) shell, (c) disc. The cone half-angles are 60°, 45°, and 30° for each image. All pictures are 64×64 pixels. Only alternate longitudinal (z direction) sections are shown.

FIG. 15
FIG. 15

Geometry for the lens imaging system. P,P are front and rear focal planes, F,F are focal points, and C,C are two focal lengths away from the principal planes. The point Q at (r,z) of the object space is conjugate to Q at ( r ˜ , z ˜ ) of the image space.

FIG. 16
FIG. 16

Geometry for the two-zone-plate coded aperture system.

Equations (62)

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d Ω 4 π = 1 4 π d x d y ( s + z ) 2 cos 3 θ .
h ( r ) d 2 r = d 2 r 4 π d 2 r d z f ( r , z ) g ( r ) 1 ( s + z ) 2 ,
r = a r + b r ,
a = z / ( s + z ) , b = s / ( s + z ) .
g c ( r ) = e j 2 π ρ g · r + e j 2 π ρ g · r ,
g s ( r ) = e j 2 π ρ g · r e j 2 π ρ g · r ,
g ( r ) = e j 2 π ρ g · r .
h ( r ) = 1 4 π d 2 r d z f ( r , z ) ( s + z ) 2 e j 2 π ρ g · ( a r + b r ) .
z ˜ = s s + z z , x ˜ = s s + z x , y ˜ = s x + z y ,
z = s s z ˜ z ˜ , x = s s z ˜ x ˜ , y = s s z ˜ y ˜ ,
f ˜ ( x ˜ , y ˜ , z ˜ ) d x ˜ d y ˜ d z ˜ [ s / ( s + z ) ] 2 f ( x , y , z ) d x d y d z .
a = z ˜ / s , b x = x ˜ , b y = y ˜ .
h ( r ) = 1 4 π s 2 d x ˜ d y ˜ d z ˜ f ˜ ( x ˜ , y ˜ , z ˜ ) × exp [ j 2 π ( ρ g · r s z ˜ + ρ g · r ˜ ) ] = 1 4 π s 2 F ˜ ( ξ g , η g , ρ g · r s ) ,
ζ ˜ max = ρ g D / 2 s = ρ g tan Ө ,
x = ( l + m s ) + m z ,
s x ˜ / ( s z ˜ ) = ( l + m s ) + m s z ˜ / ( s z ˜ )
x ˜ = ( l + m s ) ( l / s ) z ˜ .
h ( r ) = 1 4 π d 2 r d z f ( r , z ) g ( a r + b r ) ( s 1 + s 2 + z ) 2 ,
a = ( s 1 + z ) / ( s 1 + s 2 + z ) , b = s 2 / ( s 1 + s 2 + z ) .
g ( r ) = δ ( r r 0 ) .
z ˜ = s 1 s 1 + z z , r ˜ = s 1 s 1 + z r ,
f ˜ ( x ˜ , y ˜ , z ˜ ) d x ˜ d y ˜ d z ˜ [ s 1 / ( s 1 + z ) ] 2 f ( x , y , z ) d x d y d z .
h δ ( r ) = 1 4 π d 2 r ˜ d z ˜ f ˜ ( r ˜ , z ˜ ) ( s 1 + z s 1 ) 2 × δ ( a r + b r r 0 ) ( s 1 + s 2 + z ) 2 = 1 4 π s 1 2 d 2 r ˜ d z ˜ f ˜ ( r ˜ , z ˜ ) × δ ( r + b a r 1 a r 0 ) .
H δ ( ρ ) = 1 4 π s 1 2 d 2 r ˜ d z ˜ d 2 r e j 2 π ρ · r × f ˜ ( r ˜ , z ˜ ) δ ( r + b a r 1 a r 0 ) = 1 4 π s 1 2 d 2 r ˜ d z ˜ f ˜ ( r ˜ , z ˜ ) × exp [ j 2 π ρ · ( b a r + 1 a r 0 ) ]
( b / a ) r = ( s 2 / s 1 ) r ˜ ,
1 / a = ( 1 + s 2 / s 1 ) ( s 2 / s 1 2 ) z ˜ .
f ˜ ( x ˜ , y ˜ , z ˜ ) = d ξ ˜ d η ˜ d ζ ˜ F ˜ ( ξ ˜ , η ˜ , ζ ˜ ) e j 2 π ( ξ ˜ x ˜ + η ˜ y ˜ + ζ ˜ z ˜ ) .
H δ ( ρ ) = 1 4 π s 1 2 d x ˜ d y ˜ d z ˜ exp [ j 2 π ( 1 + s 2 / s 1 ) ρ · r 0 ] exp [ j 2 π ρ · ( s 2 s 1 r ˜ s 2 z ˜ s 1 2 r 0 ) ] × d ξ ˜ d η ˜ d ζ ˜ F ˜ ( ξ ˜ , η ˜ , ζ ˜ ) e j 2 π ( ξ ˜ x ˜ + η ˜ y ˜ + ζ ˜ z ˜ ) = exp [ j 2 π ( 1 + s 2 / s 1 ) ρ · r 0 ] 4 π s 1 2 d ξ ˜ d η ˜ d ζ ˜ F ˜ ( ξ ˜ , η ˜ , ζ ˜ ) { d x ˜ exp [ j 2 π ( ξ ˜ s 2 s 1 ξ ) x ˜ ] } × d y ˜ exp [ j 2 π ( η ˜ s 2 s 1 η ) y ˜ ] d z ˜ exp [ j 2 π ( s 2 s 1 2 ρ · r 0 ζ ˜ ) z ˜ ] .
d x e j 2 π ξ x = δ ( ξ ) ,
H δ ( ρ ) = 1 4 π s 1 2 exp [ j 2 π ( 1 + s 2 / s 1 ) ρ · r 0 ] × F ˜ ( s 2 s 1 ξ , s 2 s 1 η , s 2 s 1 2 ρ · r 0 ) .
H ( ρ ) = 1 4 π s 1 2 d 2 r 0 g ( r 0 ) exp [ j 2 π ( 1 + s 2 / s 1 ) ρ · r 0 ] × F ˜ ( s 2 s 1 ξ , s 2 s 1 η , s 2 s 1 2 ρ · r 0 ) .
ψ = π / 2 l / s ,
H ( ρ ) = 1 4 π s 1 2 d 2 r 0 g ( r 0 ) × F ˜ ( s 2 s 1 ξ , s 2 s 1 η , s 2 s 1 2 ρ · r 0 ) .
F ˜ ( ξ ˜ , η ˜ , ζ ˜ ) = M ˜ ( ξ ˜ , η ˜ ) e j 2 π ζ ˜ z ˜ 0 ,
H ( ρ ) = 1 4 π s 1 2 M ˜ ( s 2 s 1 ξ , s 2 s 1 η ) T ( ρ ; z ˜ 0 ) ,
T ( ρ ; z ˜ 0 ) = d 2 r 0 g ( r 0 ) exp [ j 2 π ( s 2 / s 1 2 ) z ˜ 0 ρ · r 0 ] = G [ ( s 2 / s 1 2 ) z ˜ 0 ρ ] ,
h ( r ) = [ 1 4 π s 2 2 ( s 1 2 s 2 z ˜ 0 ) ] μ ˜ ( s 1 s 2 r ) * * g ( s 1 2 s 2 z ˜ 0 r ) .
p ( r , θ r , z ) = 0 ρ 0 d ρ ρ / tan ψ ρ / tan ψ d ζ 0 2 π d θ ρ ρ × exp j 2 π [ ρ r cos ( θ ρ θ r ) + ζ z ] ,
p ( r , z ) = 0 ρ 0 d ρ ρ ρ / tan ψ ρ / tan ψ d ζ e j 2 π ζ z × 0 2 π d θ ρ e j 2 π ρ r cos θ ρ = 2 z 0 ρ 0 d ρ ρ J 0 ( 2 π ρ r ) sin ( 2 π ρ z tan ψ ) .
= ρ / ρ 0 , α = 2 π r ρ 0 , β = 2 π z ρ 0 / tan ψ ,
p ( r , z ) = 2 ρ 0 2 z 0 1 J 0 ( α ) sin ( β ) d .
J 0 ( z ) ~ ( 2 / π z ) 1 / 2 cos ( z π / 4 ) | z | .
A ( ξ , η , ζ ) = e γ ρ 2 / ρ 0 2
( z + f ) ( z ˜ + f ) = f 2 ,
z ˜ = f f + z z .
m = f / ( f + z ) .
r ˜ = m r = f f + z r .
h ( r ) = 1 4 π d 2 r d z f ( r , z ) × g 1 ( r ) g 2 ( r ) 1 ( s 2 + z ) 2 .
r = a 1 r + b 1 r , r = a 2 r + b 2 r ,
a 1 = ( z s 1 ) / ( z + s 2 ) , b 1 = ( s 1 + s 2 ) / ( z + s 2 ) ;
a 2 = ( z + s 1 ) / ( z + s 2 ) , b 2 = ( s 2 s 1 ) / ( z + s 2 ) .
g 2 ( r ) = g 1 ( s 2 + s 1 s 2 s 1 r ) = g 1 ( b 1 b 2 r ) ,
h ( r ) = 1 4 π d 2 r d z f ( r , z ) ( s 2 + z ) 2 × g 1 ( a 1 r + b 1 r ) g 1 ( a 2 b 1 b 2 r + b 1 r ) .
g 1 ( r ) = cos α r 2 = 1 2 ( e j α r 2 + e j α r 2 ) .
h ( r ) = 1 16 π d 2 r d z f ( r , z ) ( s 2 + z ) 2 × [ exp ( j α | a 1 r + b 1 r | 2 ) × exp ( j α | a 2 b 1 b 2 r + b 1 r | 2 ) + c . c . + exp ( j α | a 1 r + b 1 r | 2 ) × exp ( j α | a 2 b 1 b 2 r + b 1 r | ) + c . c . ] ,
h ( r ) = 1 16 π d 2 r d z f ( r , z ) ( s 2 + z ) × exp [ j α ( a 2 2 b 1 2 b 2 2 a 1 2 ) r 2 ] × exp [ j 2 α ( a 2 b 1 2 b 2 a 1 b 1 ) r · r ] .
r ˜ = ( a 2 b 1 2 b 2 a 1 b 1 ) r = ( 2 s 1 ( s 1 + s 2 ) ( s 2 s 1 ) ( z + s 2 ) ) r ,
z ˜ = ( s 2 s 1 ) ( s 1 + s 2 ) 2 s 1 ( a 2 2 b 1 2 b 2 2 a 1 2 ) = ( 2 s 1 ( s 1 + s 2 ) ( s 2 s 1 ) ( z + s 2 ) ) ( z + s 1 2 s 2 ) .
f ˜ ( r ˜ , z ˜ ) d 2 r ˜ d z ˜ [ s 2 / ( s 2 + z ) ] 2 f ( r , z ) d 2 r d z .
h ( r ) = 1 16 π s 2 2 d 2 r ˜ d z ˜ f ˜ ( r ˜ , z ˜ ) × exp [ j 2 α ( s 1 s 2 2 s 1 2 ) r 2 z ˜ ] exp ( j 2 α r · r ˜ ) = 1 16 π s 2 2 F ˜ [ α x π , α y π , α r 2 π ( s 1 s 2 2 s 1 2 ) ] ,
α = π / r 1 2 .
h ( r ) = 1 16 π s 2 2 F ˜ [ x r 1 2 , y r 1 2 , ( s 1 s 2 2 s 1 2 ) r 2 r 1 2 ] .