Abstract

A new integral transform, derived from the three-dimensional Radon transform, is introduced. The basis functions for this transform, which may be physically interpreted as sheets of dipoles, are shown to be orthonormal and complete. The inverse transform is derived, and an expression for the Fourier transform of the basis functions is found. It is shown that all spherically symmetric functions retain the same functional form under this transform and that it can be used to reduce certain differential equations, such as the Helmholtz equation, to a spherically symmetric form, even if the original problem has no symmetry at all.

© 1982 Optical Society of America

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  1. J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss. Kl. 69, 262–78 (1917).
  2. M. Y. Chiu, H. H. Barrett, and R. G. Simpson, “Three-dimensional reconstruction from planar projections,” J. Opt. Soc. Am. 70, 755–762 (1980).
    [Crossref]
  3. H. H. Barrett and W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).
  4. E. Tanaka and T. A. Iinuma, “Image processing for coded aperture imaging and an attempt at rotating slit imaging,” presented at Fourth International Conference on Information Processing in Scintigraphy, Orsay, France, July 1975.
  5. W. I. Keyes, “The fan-beam gamma camera,” Phys. Med. Biol. 20, 489 (1975).
    [Crossref] [PubMed]
  6. G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).
  7. Y. Das and W. M. Boerner, “On radar target shape estimation using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
    [Crossref]
  8. E. M. Kennaugh and D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
    [Crossref]
  9. L. A. Shepp, “Computerized tomography and nuclear magnetic resonance,” J. Comput. Assist. Tomog. 4, 94–107 (1980).
    [Crossref]
  10. H. H. Barrett, “Three-dimensional image reconstruction from planar projections, with application to optical data processing,” presented at SPIE Advanced Institute on Transformations in Optical Signal Processing, Seattle, Washington, February 1981.
  11. R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 274.
  12. C. M. Vest and D. G. Steel, “Reconstruction of spherically symmetric objects from slit-imaged emission: application to spatially resolved spectroscopy,” Opt. Lett. 3, 54–56 (1978).
    [Crossref] [PubMed]
  13. J. E. Greivenkamp, W. Swindell, A. F. Gmitro, and H. H. Barrett, “Incoherent optical processor for x-ray transaxial tomography,” Appl. Opt. 20, 264–273 (1981).
    [Crossref] [PubMed]
  14. A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

1981 (2)

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

J. E. Greivenkamp, W. Swindell, A. F. Gmitro, and H. H. Barrett, “Incoherent optical processor for x-ray transaxial tomography,” Appl. Opt. 20, 264–273 (1981).
[Crossref] [PubMed]

1980 (2)

M. Y. Chiu, H. H. Barrett, and R. G. Simpson, “Three-dimensional reconstruction from planar projections,” J. Opt. Soc. Am. 70, 755–762 (1980).
[Crossref]

L. A. Shepp, “Computerized tomography and nuclear magnetic resonance,” J. Comput. Assist. Tomog. 4, 94–107 (1980).
[Crossref]

1978 (2)

Y. Das and W. M. Boerner, “On radar target shape estimation using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[Crossref]

C. M. Vest and D. G. Steel, “Reconstruction of spherically symmetric objects from slit-imaged emission: application to spatially resolved spectroscopy,” Opt. Lett. 3, 54–56 (1978).
[Crossref] [PubMed]

1975 (1)

W. I. Keyes, “The fan-beam gamma camera,” Phys. Med. Biol. 20, 489 (1975).
[Crossref] [PubMed]

1965 (1)

E. M. Kennaugh and D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[Crossref]

1917 (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss. Kl. 69, 262–78 (1917).

Arendt, J.

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Barrett, H. H.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

J. E. Greivenkamp, W. Swindell, A. F. Gmitro, and H. H. Barrett, “Incoherent optical processor for x-ray transaxial tomography,” Appl. Opt. 20, 264–273 (1981).
[Crossref] [PubMed]

M. Y. Chiu, H. H. Barrett, and R. G. Simpson, “Three-dimensional reconstruction from planar projections,” J. Opt. Soc. Am. 70, 755–762 (1980).
[Crossref]

H. H. Barrett, “Three-dimensional image reconstruction from planar projections, with application to optical data processing,” presented at SPIE Advanced Institute on Transformations in Optical Signal Processing, Seattle, Washington, February 1981.

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

H. H. Barrett and W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).

Boerner, W. M.

Y. Das and W. M. Boerner, “On radar target shape estimation using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[Crossref]

Bracewell, R.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 274.

Chiu, M. Y.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

M. Y. Chiu, H. H. Barrett, and R. G. Simpson, “Three-dimensional reconstruction from planar projections,” J. Opt. Soc. Am. 70, 755–762 (1980).
[Crossref]

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Das, Y.

Y. Das and W. M. Boerner, “On radar target shape estimation using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[Crossref]

Ervin, A.

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Giles, C. L.

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Gindi, G. R.

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Gmitro, A. F.

J. E. Greivenkamp, W. Swindell, A. F. Gmitro, and H. H. Barrett, “Incoherent optical processor for x-ray transaxial tomography,” Appl. Opt. 20, 264–273 (1981).
[Crossref] [PubMed]

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

Gordon, S. K.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

Greivenkamp, J. E.

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

J. E. Greivenkamp, W. Swindell, A. F. Gmitro, and H. H. Barrett, “Incoherent optical processor for x-ray transaxial tomography,” Appl. Opt. 20, 264–273 (1981).
[Crossref] [PubMed]

Iinuma, T. A.

E. Tanaka and T. A. Iinuma, “Image processing for coded aperture imaging and an attempt at rotating slit imaging,” presented at Fourth International Conference on Information Processing in Scintigraphy, Orsay, France, July 1975.

Kennaugh, E. M.

E. M. Kennaugh and D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[Crossref]

Keyes, W. I.

W. I. Keyes, “The fan-beam gamma camera,” Phys. Med. Biol. 20, 489 (1975).
[Crossref] [PubMed]

Kujoory, M. A.

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Miller, E. L.

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Moffatt, D. L.

E. M. Kennaugh and D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[Crossref]

Radon, J.

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss. Kl. 69, 262–78 (1917).

Shepp, L. A.

L. A. Shepp, “Computerized tomography and nuclear magnetic resonance,” J. Comput. Assist. Tomog. 4, 94–107 (1980).
[Crossref]

Simpson, R. G.

M. Y. Chiu, H. H. Barrett, and R. G. Simpson, “Three-dimensional reconstruction from planar projections,” J. Opt. Soc. Am. 70, 755–762 (1980).
[Crossref]

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

Steel, D. G.

Swindell, W.

J. E. Greivenkamp, W. Swindell, A. F. Gmitro, and H. H. Barrett, “Incoherent optical processor for x-ray transaxial tomography,” Appl. Opt. 20, 264–273 (1981).
[Crossref] [PubMed]

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

H. H. Barrett and W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).

Tanaka, E.

E. Tanaka and T. A. Iinuma, “Image processing for coded aperture imaging and an attempt at rotating slit imaging,” presented at Fourth International Conference on Information Processing in Scintigraphy, Orsay, France, July 1975.

Vest, C. M.

Appl. Opt. (1)

Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss. Kl. (1)

J. Radon, “Über die Bestimmung von Funktionen durch ihre Integralwerte längs gewisser Mannigfaltigkeiten,” Ber. Verh. Saechs. Akad. Wiss. Leipzig, Math. Naturwiss. Kl. 69, 262–78 (1917).

IEEE Trans. Antennas Propag. (1)

Y. Das and W. M. Boerner, “On radar target shape estimation using algorithms for reconstruction from projections,” IEEE Trans. Antennas Propag. AP-26, 274–279 (1978).
[Crossref]

J. Comput. Assist. Tomog. (1)

L. A. Shepp, “Computerized tomography and nuclear magnetic resonance,” J. Comput. Assist. Tomog. 4, 94–107 (1980).
[Crossref]

J. Opt. Soc. Am. (1)

Opt. Eng. (1)

A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, and S. K. Gordon, “Optical computers for reconstructing objects from their x-ray projections,” Opt. Eng. 19, 260 (1981).

Opt. Lett. (1)

Phys. Med. Biol. (1)

W. I. Keyes, “The fan-beam gamma camera,” Phys. Med. Biol. 20, 489 (1975).
[Crossref] [PubMed]

Proc. IEEE (1)

E. M. Kennaugh and D. L. Moffatt, “Transient and impulse response approximations,” Proc. IEEE 53, 893–901 (1965).
[Crossref]

Other (5)

G. R. Gindi, J. Arendt, H. H. Barrett, M. Y. Chiu, A. Ervin, C. L. Giles, M. A. Kujoory, E. L. Miller, and R. G. Simpson, “Imaging with rotating slit apertures and collimators,” Med. Phys. (to be published).

H. H. Barrett and W. Swindell, Radiological Imaging: Theory of Image Formation, Detection and Processing (Academic, New York, 1981).

E. Tanaka and T. A. Iinuma, “Image processing for coded aperture imaging and an attempt at rotating slit imaging,” presented at Fourth International Conference on Information Processing in Scintigraphy, Orsay, France, July 1975.

H. H. Barrett, “Three-dimensional image reconstruction from planar projections, with application to optical data processing,” presented at SPIE Advanced Institute on Transformations in Optical Signal Processing, Seattle, Washington, February 1981.

R. Bracewell, The Fourier Transform and its Applications (McGraw-Hill, New York, 1965), p. 274.

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Figures (4)

Fig. 1
Fig. 1

Top: Plane of integration in the Radon transform is specified by its normal unit vector n ˆ and distance from the origin p. Bottom: The vector n ˆ is specified in spherical coordinates by azimuth ϕn and colatitude θn.

Fig. 2
Fig. 2

Schematic representation of the dipole-sheet function ψ(p, r). The function vanishes except in the neighborhood of plane p = r · n ˆ , drawn here with n ˆ in the xy plane. The function is uniform in planes perpendicular to n ˆ and has the form of the derivative of a 1D delta function in the direction n ˆ . It may be regarded as a sheet of value +∞ just to one side of the plane p = r · n ˆ and a sheet of value −∞ just to the other side, but its actual mathematical behavior is governed by the definition of a derivative of a delta function.

Fig. 3
Fig. 3

The surface p = r · n ˆ describes a sphere passing through the origin in p space (where p ≡ p n ˆ ). The function ψ(p, r) is zero except in the neighborhood of this spherical surface.

Fig. 4
Fig. 4

(a) Geometry for illustrating the orthogonality of the ψ(p, r) for the case where p and p′ are parallel. Then, if pp′ is finite, there is no overlap between ψ(p, r) and ψ(p′, r). (b) The case in which p and p′ are not parallel, drawn with both vectors in the plane of the figure. Now ψ(p, r) and ψ(p′, r) do overlap, but their product has equal positive and negative components.

Equations (61)

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p = r · n ˆ .
f R ( p , n ˆ ) = d 3 r f ( r ) δ ( p - r · n ˆ ) .
f R ( p , n ˆ ) = f R ( - p , - n ˆ )
p = p n ˆ .
F 1 [ f R ( p , n ˆ ) ] = - d p exp ( - 2 π i p ν ) f R ( p , n ˆ ) = - d p exp ( - 2 π i p ν ) d 3 r f ( r ) δ ( p - r · n ˆ ) .
F 1 [ f R ( p , n ˆ ) ] = d 3 r exp ( - 2 π i r · n ˆ ν ) f ( r ) .
F ( σ ) = F 3 [ f ( r ) ] = d 3 r exp ( - 2 π i σ · r ) f ( r ) ,
F 1 [ f R ( p , n ˆ ) ] = F 3 [ f ( r ) ] σ = n ˆ ν = F ( n ˆ ν ) .
f ( r ) = - 1 8 π 2 2 4 π d Ω n f R ( p , n ˆ ) p = r · n ˆ ,
- 1 8 π 2 2 - 1 1 d ( cos θ n ) 0 2 π d ϕ n δ ( r cos θ n ) = - 1 4 π 2 ( 1 r ) = δ ( r ) ,
2 g ( r · n ˆ ) = d 2 g ( z ) d z 2 | z = r · n ˆ = g ( r · n ˆ ) ,
f ( r ) = - 1 8 π 2 4 π d Ω n d 2 f R ( p , n ˆ ) d p 2 | p = r · n = - 1 8 π 2 4 π d Ω n f R ( r · n ˆ , n ˆ ) ,
f ( r ) = - 1 8 π 2 - d p 4 π d Ω n f R ( p , n ˆ ) δ ( p - r · n ˆ ) = - 1 8 π 2 - d p 4 π d Ω n f R ( p , n ˆ ) δ ( p - r · n ˆ ) ,
f ( r ) = - 1 4 π 2 d 3 p f R ( p ) δ ( p - r · n ˆ ) / p 2 ,
f D ( p ) = i 2 π d 3 r f ( r ) δ ( p - r · n ˆ ) / p .
f ( r ) = - i 2 π d 3 p f D ( p ) δ ( p - r · n ˆ ) / p .
f D ( p ) = d 3 r ψ ( p , r ) f ( r ) = D [ f ( r ) ] ,
f ( r ) = d 3 p ψ * ( p , r ) f D ( p ) = D - 1 [ f D ( p ) ] ,
ψ ( p , r ) i 2 π δ ( p - r · n ˆ ) p
f D ( p ) = K p p f R ( p ) ,
f ( r ) = ½ K 2 4 π d Ω n { 2 p 2 [ f R ( p ) ] } p = r · n ˆ = ½ K 2 4 π d Ω n { p [ 1 K p f D ( p ) ] } p = r · n ˆ .
f ( r ) = ½ K - d p 4 π d Ω n { p [ p f D ( p ) ] } δ ( p - r · n ˆ ) .
f ( r ) = - ½ K - d p 4 π d Ω n p f D ( p ) δ ( p - r · n ˆ ) .
f ( r ) = - K d 3 p f D ( p ) δ ( p - r · n ˆ ) / p ,
I ( p , p ) = d 3 r ψ * ( p , r ) ψ ( p , r ) = 1 4 π 2 p p d 3 r δ ( p - r · n ˆ ) δ ( p - r · n ˆ ) ,
I ( p , p ) = 1 4 π 2 p p - d x - d y - d z δ ( p - z ) × δ ( p - α x - β y - γ z ) .
I ( p , p ) = γ 4 π 2 p p - d x - d y δ ( p - α x - β y - γ p ) .
d 3 p I ( p , p ) = 0 p 2 d p 4 π d Ω n d 3 r 1 4 π 2 p p × δ ( p - r · n ˆ ) δ ( p - r · n ˆ ) .
4 π d Ω n δ ( p - r · n ˆ ) = 2 π - 1 1 d ( cos θ n ) δ ( p - r cos θ n ) = 2 π r - r r d u δ ( p - u ) = 2 π r - d u rect ( u 2 r ) δ ( p - u ) = - 2 π r δ ( p - r ) ,
d 3 p I ( p , p ) = - 1 4 π 2 p 0 p d p × d 3 r 2 π r δ ( p - r ) δ ( p - r · n ˆ ) = - 1 2 π p 0 r 2 d r 4 π d Ω δ ( p - r · n ˆ ) .
4 π d Ω δ ( p - r · n ˆ ) = - 2 π r δ ( p - r ) .
d 3 p I ( p , p ) = 1 p 0 r d r δ ( p - r ) = 1.
d 3 p I ( p , p ) = 1.
d 3 r ψ * ( p , r ) ψ ( p , r ) = δ ( p - p ) .
J ( r , r ) = d 3 p ψ * ( p , r ) ψ ( p , r ) .
J ( r , r ) = 1 4 π 2 0 p 2 d p 4 π d Ω n 1 p 2 δ ( p - r · n ˆ ) × δ ( p - r · n ˆ ) = 1 8 π 2 - d p 4 π d Ω n δ ( p - r · n ˆ ) δ ( p - r · n ˆ ) = - 1 8 π 2 4 π d Ω n δ [ ( r - r ) · n ˆ ] .
J ( r , r ) = - 1 8 π 2 2 4 π d Ω n δ [ ( r - r ) · n ˆ ] .
4 π d Ω n δ [ ( r - r ) · n ˆ ] = 2 π - 1 1 d ( cos θ n ) × δ [ r - r cos θ n ] = 2 π r - r .
J ( r , r ) = - 1 4 π 2 1 r - r = δ ( r - r ) ,
- 1 8 π 2 4 π d Ω n δ ( r · n ˆ ) = δ ( r ) .
F 3 [ ψ ( p , r ) ] = d 3 r exp ( - 2 π i σ · r ) i 2 π p δ ( p - r · n ˆ ) .
δ ( p - r · n ˆ ) = - d ν exp [ 2 π i ν ( p - r · n ˆ ) ] 2 π i ν .
F 3 [ ψ ( p , r ) ] = d 3 r - d ν exp ( - 2 π i σ · r ) × exp [ 2 π i ν ( p - r · n ˆ ) ] ( - ν / p ) = d ν δ ( σ + n ˆ ν ) ( - ν / p ) exp ( 2 π i ν p ) .
F 3 [ ψ ( p , r ) ] = δ ( ξ ) δ ( η ) ( ζ / p ) exp ( - 2 π i ζ p ) .
D [ f ( r ) ] = f D ( p ) = i 2 π p 0 r 2 d r 4 π d Ω δ ( p - r · n ˆ ) f ( r ) .
f D ( p ) = i 2 π p 0 r 2 d r ( - 2 π r ) δ ( p - r ) f ( r ) = - i f ( p ) .
f ( r ) = { 1 if r < a 0 if r > a .
f D ( p ) = 2 π - 1 1 d ( cos θ ) 0 a r 2 d r i 2 π p δ ( p - r · n ˆ ) .
f D ( p ) = i p 0 a r 2 d r 1 r - r r d u δ ( p - u ) , = i p 0 a r d r - rect ( u 2 r ) δ ( p - u ) d u , = i p 0 a r d r [ δ ( p + r ) - δ ( p - r ) ] , = { - i if p < a 0 if p > a ,
f D ( p ) = i 2 π p d d p f R ( p ) .
f ( p ) = - 1 2 π p d d p f R ( p ) .
2 ψ ( p , r ) = i 2 π p 2 δ ( p - r · n ˆ ) = i 2 π p δ ( p - r · n ˆ ) ,
2 ψ ( p , r ) = i 2 π p 2 p 2 δ ( p - r · n ) = 1 p 2 p 2 [ p ψ ( p , r ) ] .
( 2 + k 2 ) f ( r ) = g ( r ) .
V ( u 2 v - v 2 u ) d 3 r = S ( u v - v u ) · d a ,
u = f ( r ) ,             v = ψ ( p , r ) .
2 u = 2 f ( r ) = g ( r ) - k 2 f ( r ) ,
2 v = 2 ψ ( p , r ) = 1 p 2 p 2 [ p ψ ( p , r ) ] .
V { f ( r ) 1 p 2 p 2 [ p ψ ( p , r ) ] - g ( r ) ψ ( p , r ) + k 2 f ( r ) ψ ( p , r ) } d 3 r = S [ f ( r ) ψ ( p , r ) - ψ ( p , r ) f ( r ) ] · da .
( 1 p 2 p 2 p + k 2 ) V f ( r ) ψ ( p , r ) d 3 r = V g ( r ) ψ ( p , r ) d 3 r
( 1 p 2 p 2 p + k 2 ) f D ( p ) = g D ( p ) .