Abstract

The problem of determining the structure of an object from knowledge of its projections along straight lines arises in a variety of optical contexts. The general solution to this problem is given by the inverse Radon transform of the projections. A superposition principle is developed to show the relation between this general inversion and the more familiar inverse Abel transform that is associated with radially symmetric objects.

© 1974 Optical Society of America

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  1. In this paper the object field, in any coordinate system, is always denoted by f, and its projection by Φ, i.e., f(x,y,z) and f(r,ϕ,z) denote the same field.
  2. M. V. Berry and D. F. Gibbs, Proc. R. Soc. (Lond.) A 314, 143 (1970).
    [CrossRef]
  3. P. D. Rowley, J. Opt. Soc. Am. 59, 1496 (1969).
    [CrossRef]
  4. I. J. Good, Phys. Lett. 31A, 155 (1970).
  5. H. G. Junginger and W. van Haeringer, Opt. Commun. 5, 1 (1972).
    [CrossRef]
  6. C. M. Vest, J. Opt. Soc. Am. 63, 486A (1973).
  7. A. M. Cormack, Phys. Med. Biol. 18, 195 (1973).
    [CrossRef] [PubMed]
  8. J. Radon, Berichte Saechsische Akademie der Wissenschaften (Leipzig) 69, 262 (1917).
  9. I. M. Gel’Fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. V (Academic, New York, 1966).
  10. F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations (Wiley–Interscience, New York, 1955).
  11. R. N. Bracewell, Australian J. Phys. 9, 198 (1956).
    [CrossRef]
  12. D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
    [CrossRef]
  13. The value of a three-dimensional field at a single point can be determined in terms of the integrals of f over all planes passing through the point. This is clear from the discussions of the Radon transform in Refs. 7, 9, and 10.
  14. P. F. C. Gilbert, Proc. R. Soc. (Lond.) B 182, 89 (1972).
    [CrossRef]
  15. D. W. Sweeney and C. M. Vest, Appl. Opt. 12, 2649 (1973).
    [CrossRef] [PubMed]

1973 (3)

C. M. Vest, J. Opt. Soc. Am. 63, 486A (1973).

A. M. Cormack, Phys. Med. Biol. 18, 195 (1973).
[CrossRef] [PubMed]

D. W. Sweeney and C. M. Vest, Appl. Opt. 12, 2649 (1973).
[CrossRef] [PubMed]

1972 (2)

P. F. C. Gilbert, Proc. R. Soc. (Lond.) B 182, 89 (1972).
[CrossRef]

H. G. Junginger and W. van Haeringer, Opt. Commun. 5, 1 (1972).
[CrossRef]

1970 (2)

I. J. Good, Phys. Lett. 31A, 155 (1970).

M. V. Berry and D. F. Gibbs, Proc. R. Soc. (Lond.) A 314, 143 (1970).
[CrossRef]

1969 (1)

1962 (1)

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

1956 (1)

R. N. Bracewell, Australian J. Phys. 9, 198 (1956).
[CrossRef]

1917 (1)

J. Radon, Berichte Saechsische Akademie der Wissenschaften (Leipzig) 69, 262 (1917).

Berry, M. V.

M. V. Berry and D. F. Gibbs, Proc. R. Soc. (Lond.) A 314, 143 (1970).
[CrossRef]

Bracewell, R. N.

R. N. Bracewell, Australian J. Phys. 9, 198 (1956).
[CrossRef]

Cormack, A. M.

A. M. Cormack, Phys. Med. Biol. 18, 195 (1973).
[CrossRef] [PubMed]

Gel’Fand, I. M.

I. M. Gel’Fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. V (Academic, New York, 1966).

Gibbs, D. F.

M. V. Berry and D. F. Gibbs, Proc. R. Soc. (Lond.) A 314, 143 (1970).
[CrossRef]

Gilbert, P. F. C.

P. F. C. Gilbert, Proc. R. Soc. (Lond.) B 182, 89 (1972).
[CrossRef]

Good, I. J.

I. J. Good, Phys. Lett. 31A, 155 (1970).

Graev, M. I.

I. M. Gel’Fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. V (Academic, New York, 1966).

John, F.

F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations (Wiley–Interscience, New York, 1955).

Junginger, H. G.

H. G. Junginger and W. van Haeringer, Opt. Commun. 5, 1 (1972).
[CrossRef]

Middleton, D.

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Petersen, D. P.

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

Radon, J.

J. Radon, Berichte Saechsische Akademie der Wissenschaften (Leipzig) 69, 262 (1917).

Rowley, P. D.

Sweeney, D. W.

van Haeringer, W.

H. G. Junginger and W. van Haeringer, Opt. Commun. 5, 1 (1972).
[CrossRef]

Vest, C. M.

Vilenkin, N. Ya.

I. M. Gel’Fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. V (Academic, New York, 1966).

Appl. Opt. (1)

Australian J. Phys. (1)

R. N. Bracewell, Australian J. Phys. 9, 198 (1956).
[CrossRef]

Berichte Saechsische Akademie der Wissenschaften (Leipzig) (1)

J. Radon, Berichte Saechsische Akademie der Wissenschaften (Leipzig) 69, 262 (1917).

Inf. Control (1)

D. P. Petersen and D. Middleton, Inf. Control 5, 279 (1962).
[CrossRef]

J. Opt. Soc. Am. (2)

C. M. Vest, J. Opt. Soc. Am. 63, 486A (1973).

P. D. Rowley, J. Opt. Soc. Am. 59, 1496 (1969).
[CrossRef]

Opt. Commun. (1)

H. G. Junginger and W. van Haeringer, Opt. Commun. 5, 1 (1972).
[CrossRef]

Phys. Lett. (1)

I. J. Good, Phys. Lett. 31A, 155 (1970).

Phys. Med. Biol. (1)

A. M. Cormack, Phys. Med. Biol. 18, 195 (1973).
[CrossRef] [PubMed]

Proc. R. Soc. (Lond.) A (1)

M. V. Berry and D. F. Gibbs, Proc. R. Soc. (Lond.) A 314, 143 (1970).
[CrossRef]

Proc. R. Soc. (Lond.) B (1)

P. F. C. Gilbert, Proc. R. Soc. (Lond.) B 182, 89 (1972).
[CrossRef]

Other (4)

The value of a three-dimensional field at a single point can be determined in terms of the integrals of f over all planes passing through the point. This is clear from the discussions of the Radon transform in Refs. 7, 9, and 10.

In this paper the object field, in any coordinate system, is always denoted by f, and its projection by Φ, i.e., f(x,y,z) and f(r,ϕ,z) denote the same field.

I. M. Gel’Fand, M. I. Graev, and N. Ya. Vilenkin, Generalized Functions, Vol. V (Academic, New York, 1966).

F. John, Plane Waves and Spherical Means Applied to Partial Differential Equations (Wiley–Interscience, New York, 1955).

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Figures (4)

Fig. 1
Fig. 1

Notation for analysis and reconstruction of an asymmetric field, f(r,ϕ).

Fig. 2
Fig. 2

Notation for analysis and reconstruction of a radially symmetric field, f(r).

Fig. 3
Fig. 3

Notation for application of the two-dimensional sampling theorem, Eq. (11).

Fig. 4
Fig. 4

Notation for analysis and reconstruction of a radially symmetric component field, σij(ρij), centered on the sampling point denoted by (i,j).

Equations (24)

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Φ k = S k f ( r , ϕ , z ) d S k ,
f ˆ ( ξ , p ) = f ( x ) δ ( p - ( ξ , x ) ) d x ,
( ξ , x ) = ξ 1 x 1 + + ξ n x n = p
f ( x ) = ( - 1 ) n / 2 ( n - 1 ) ! ( 2 π ) n × Γ [ - f ˆ ( ξ , p ) { p - ( ξ , x ) } - n d p ] ω ( ξ ) ,
ω ( ξ ) = k = 1 n ( - 1 ) k - 1 ξ k d ξ 1 d ξ k - 1 d ξ k + 1 d ξ n
Φ ( ψ , p ) = f ( x , y ) δ [ p - r sin ( ϕ - ψ ) ] d x = f ˆ ( ψ , p ) .
f ( r , ϕ ) = - 1 4 π 2 - π π d ψ - Φ ( ψ , p ) [ p - r sin ( ϕ - ψ ) ] 2 d p .
f ( r , ϕ ) = - 1 2 π 2 - π / 2 π / 2 d ψ - Φ ( ψ , p ) [ p - r sin ( ϕ - ψ ) ] 2 d p ,
f ( r , ϕ ) = 1 2 π 2 - π / 2 π / 2 d ψ - Φ / p r sin ( ϕ - ψ ) - p d p .
Φ ( x ) = x f ( r ) d ( r 2 ) ( r 2 - x 2 ) 1 2 ,
f ( r ) = - 1 π r Φ ( x ) d x ( x 2 - r 2 ) 1 2 ,
f ( x , y ) = n = - m = - f ( n / 2 B , m / 2 B ) × { 2 π B 2 J 1 [ 2 π B { ( x - n / 2 B ) 2 + ( y - m / 2 B ) 2 } 1 2 ] 2 π B { ( x - n / 2 B ) 2 + ( y - m / 2 B ) 2 } 1 2 } ,
f ( x , y ) = i = - j = - 2 π B 2 f i j besc ( B ρ i j ) ;
sinc ( x ) = sin ( π x ) / ( π x ) .
Φ ( p , ψ ) = i = - j = - Φ i j ( μ i j ) ,
Φ i j ( μ i j ) = 2 π B 2 f i j μ i j besc ( B ρ i j ) d ( ρ i j 2 ) ( ρ i j 2 - μ i j 2 ) 1 2 .
σ i j ( ρ i j ) = - 1 π ρ i j Φ i j ( μ i j ) d μ i j ( ρ i j 2 - μ i j 2 ) 1 2 ,
σ i j ( ρ i j ) = - 1 π - H [ 1 - ( ρ i j / μ i j ) 2 ] ( μ i j 2 - ρ i j 2 ) 1 2 Φ i j d μ i j ,
H ( x ) = { 1 if x 0 0 if x < 0
- π π d ψ ρ i j sin ( γ i j - ψ ) - μ i j = - 2 π H [ 1 - ( ρ i j / μ i j ) 2 ] ( δ i j 2 - ρ i j 2 ) 1 2 ,
σ i j ( ρ i j ) = 1 2 π 2 - Φ i j d μ i j - π π d ψ ρ i j sin ( γ i j - ψ ) - μ i j = 1 2 π 2 - π π d ψ - Φ i j d μ i j ρ i j sin ( γ i j - ψ ) - μ i j .
σ i j ( r , ϕ ) = 1 2 π - π π d ψ - ( Φ i j / p ) d p r sin ( ϕ - ψ ) - p .
f ( r , ϕ ) = i = - j = - σ i j ( r , ϕ ) , Φ ( p , ψ ) = i = - j = - Φ i j ( p , ψ ) ,
f ( r , ϕ ) = 1 π 2 - π / 2 π / 2 d ψ - ( Φ / p ) d p r sin ( ϕ - ψ ) - p ,