Abstract

It is shown that the Abel inversion, onion-peeling, and filtered backprojection methods can be intercompared without assumptions about the object being deconvolved. If the projection data are taken at equally spaced radial positions, the deconvolved field is given by weighted sums of the projections divided by the data spacing. The weighting factors are independent of the data spacing. All the methods are remarkably similar and have Abelian behavior: the field at a radial location is primarily determined by the weighted differences of a few projections around the radial position. Onion-peeling and an Abel inversion using two-point interpolation are similar. When the Shepp–Logan filtered backprojection method is reduced to one dimension, it is essentially identical to an Abel inversion using three-point interpolation. The weighting factors directly determine the relative noise performance: the three-point Abel inversion is the best, while onion peeling is the worst with approximately twice the noise. Based on ease of calculation, robustness, and noise, the three-point Abel inversion is recommended.

© 1992 Optical Society of America

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References

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  1. N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
    [CrossRef]
  2. R. J. Hall, P. A. Bonzyk, “Sooting flame thermometry using emission/absorption tomography,” Appl. Opt. 29, 4590–4598 (1990).
    [CrossRef] [PubMed]
  3. H. Uchiyama, M. Nakajima, S. Yuta, “Measurement of flame temperature distribution by IR emission computed tomography,” Appl. Opt. 24, 4111–4115 (1985).
    [CrossRef] [PubMed]
  4. G. N. Ramachandran, A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
    [CrossRef] [PubMed]
  5. L. A. Shepp, B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1974).
    [CrossRef]
  6. A. M. Cormack, “Computed tomography: some history and recent developments,” Computed Tomography, Vol. 27 of Proceedings of Symposia in Applied Mathematics, L. A. Shepp, ed. (American Mathematical Society, Providence, R.I., 1982), pp. 35–42.
  7. H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
    [CrossRef]
  8. R. N. Bracewell, A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
    [CrossRef]
  9. S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, New York, 1979), p. 9–79.
    [CrossRef]
  10. I. S. Gradshteyn, I. M. Rydzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Eqs. (2.261) and (2.264.1).

1990 (1)

1985 (1)

1977 (1)

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

1974 (1)

L. A. Shepp, B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1974).
[CrossRef]

1971 (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef] [PubMed]

1967 (1)

R. N. Bracewell, A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

1826 (1)

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
[CrossRef]

Abel, N. H.

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
[CrossRef]

Barret, H. H.

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Bonzyk, P. A.

Bracewell, R. N.

R. N. Bracewell, A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

Cormack, A. M.

A. M. Cormack, “Computed tomography: some history and recent developments,” Computed Tomography, Vol. 27 of Proceedings of Symposia in Applied Mathematics, L. A. Shepp, ed. (American Mathematical Society, Providence, R.I., 1982), pp. 35–42.

Gradshteyn, I. S.

I. S. Gradshteyn, I. M. Rydzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Eqs. (2.261) and (2.264.1).

Hall, R. J.

Lakshminarayanan, A. V.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef] [PubMed]

Logan, B. F.

L. A. Shepp, B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1974).
[CrossRef]

Nakajima, M.

Ramachandran, G. N.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef] [PubMed]

Riddle, A. C.

R. N. Bracewell, A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

Rowland, S. W.

S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, New York, 1979), p. 9–79.
[CrossRef]

Rydzhik, I. M.

I. S. Gradshteyn, I. M. Rydzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Eqs. (2.261) and (2.264.1).

Shepp, L. A.

L. A. Shepp, B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1974).
[CrossRef]

Swindel, W.

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Uchiyama, H.

Yuta, S.

Appl. Opt. (2)

Astrophys. J. (1)

R. N. Bracewell, A. C. Riddle, “Inversion of fan-beam scans in radio astronomy,” Astrophys. J. 150, 427–434 (1967).
[CrossRef]

IEEE Trans. Nucl. Sci. (1)

L. A. Shepp, B. F. Logan, “Reconstructing interior head tissue from x-ray transmissions,” IEEE Trans. Nucl. Sci. NS-21, 228–236 (1974).
[CrossRef]

J. Reine Angew. Math. (1)

N. H. Abel, “Auflosung einer mechanischen Aufgabe,” J. Reine Angew. Math. 1, 153–157 (1826).
[CrossRef]

Proc. IEEE (1)

H. H. Barret, W. Swindel, “Analog reconstruction methods for transaxial tomography,” Proc. IEEE 65, 89–107 (1977).
[CrossRef]

Proc. Natl. Acad. Sci. USA (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-dimensional reconstruction from radiographs and electron micrographs: application of convolutions instead of Fourier transforms,” Proc. Natl. Acad. Sci. USA 68, 2236–2240 (1971).
[CrossRef] [PubMed]

Other (3)

A. M. Cormack, “Computed tomography: some history and recent developments,” Computed Tomography, Vol. 27 of Proceedings of Symposia in Applied Mathematics, L. A. Shepp, ed. (American Mathematical Society, Providence, R.I., 1982), pp. 35–42.

S. W. Rowland, “Computer implementation of image reconstruction formulas,” in Image Reconstruction from Projections, G. T. Herman, ed. (Springer-Verlag, New York, 1979), p. 9–79.
[CrossRef]

I. S. Gradshteyn, I. M. Rydzhik, Tables of Integrals, Series, and Products (Academic, New York, 1965), Eqs. (2.261) and (2.264.1).

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Figures (7)

Fig. 1
Fig. 1

Tomographic arrangement using parallel beams.

Fig. 2
Fig. 2

Elements of the Abel deconvolution operator.

Fig. 3
Fig. 3

Comparison of the deconvolution operators. The D4j coefficients are given.

Fig. 4
Fig. 4

Elements of the onion-peeling deconvolution operator.

Fig. 5
Fig. 5

Comparison of noise coefficients [see Eq. (20)].

Fig. 6
Fig. 6

Reconstruction errors of a Gaussian field distribution. The error is the percentage of deviation of the reconstruction from the actual field distribution. The Gaussian e−1 point is at 8 Δr.

Fig. 7
Fig. 7

Reconstruction of a ring distribution. The ring lies between 7.25 Δr and 14.75 Δr and has a value of 1.

Tables (1)

Tables Icon

Table I Performance of the Deconvolution Methods for Cylindrical Objects

Equations (31)

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F ( r i ) = 1 Δ r j = 0 D i j P ( r j ) ,
P ( r ) = 2 0 F [ ( r 2 + y 2 ) 1 / 2 ] d y
= 2 r r F ( r ) ( r 2 r 2 ) 1 / 2 d r .
F ( r ) = 1 π r P ( r ) ( r 2 r 2 ) 1 / 2 d r ,
D i j ( asymptotic ) = { 0 j < i 1 π j ( j 2 i 2 ) 3 / 2 j > i ,
D i j ( three-point Abel ) = { 0 j < i 1 I i , j + 1 ( 0 ) I i , j + 1 ( 1 ) j = i 1 I i , j + 1 ( 0 ) I i , j + 1 ( 1 ) + 2 I i j ( 1 ) j = i , I i , j + 1 ( 0 ) I i , j + 1 ( 1 ) + 2 I i j ( 1 ) I i , j 1 ( 0 ) I i , j 1 ( 1 ) j i + 1 I i , j + 1 ( 0 ) I i , j + 1 ( 1 ) + 2 I i j ( 1 ) 2 I i , j 1 ( 1 ) i = 0 , j = 1
I i j ( 0 ) = { 0 j = i = 0 or j < i 1 2 π ln { [ ( 2 j + 1 ) 2 4 i 2 ] 1 / 2 + 2 j + 1 2 j } j = i 0 , 1 2 π ln { [ ( 2 j + 1 ) 2 4 i 2 ] 1 / 2 + 2 j + 1 [ ( 2 j 1 ) 2 4 i 2 ] 1 / 2 + 2 j 1 } j > i
I i j ( 1 ) = { 0 j < i 1 2 π [ ( 2 j + 1 ) 2 4 i 2 ] 1 / 2 + 2 j I i j ( 0 ) j = i 1 2 π { [ ( 2 j + 1 ) 2 4 i 2 ] 1 / 2 [ ( 2 j 1 ) 2 4 i 2 ] 1 / 2 } + 2 j I i j ( 0 ) j > i .
D i j ( 2 -pt . Abel ) = { 0 j < i J i j j = i J i j J i , j 1 j > i ,
J i j = { 0 j < i 2 π j = i = 0 1 π In { [ ( j + 1 ) 2 i 2 ] 1 / 2 + j + 1 [ ( j 1 ) 2 i 2 ] 1 / 2 + j } j i .
P ( r i ) = Δ r j = 1 W i j F ( r j ) ,
W i j = { 0 j < i [ ( 2 j + 1 ) 2 4 i 2 ] 1 / 2 j = i [ ( 2 j + 1 ) 2 4 i 2 ] 1 / 2 [ ( 2 j 1 ) 2 4 i 2 ] 1 / 2 j > i .
D i j ( onion-peeling ) = ( W 1 ) i j .
F ( r , θ ) = Δ r 2 0 π d ϕ d r P ( r , ϕ ) × C 2 { [ r r cos ( θ ϕ ) ] / Δ r }
C 2 ( n ) = { 1 / 4 n = 0 1 / π 2 n 2 n = odd 0 n = even 0 linear interpolation n integer , ,
C 2 ( n ) = { 2 / π 2 n = 0 2 / π 2 ( 4 n 2 1 ) n = integer 0 linear interpolation n integer .
F ( r ) = Δ r 2 0 P ( r ) C 1 ( r / Δ r , r / Δ r ) d r ,
C 1 ( x , x ) = 2 0 C 2 ( x x cos ϕ ) d ϕ .
D i j ( FBP ) = { C 1 ( i , j ) / 2 j = 0 C 1 ( i , j ) j > i .
C 1 ( i , j ) = 2 k = j i j + i 1 ϕ k ϕ k + 1 { C 2 ( k ) + [ C 2 ( k + 1 ) C 2 ( k ) ] ( j i cos ϕ k ) } d ϕ = 2 k = j i j + i 1 [ ( 1 j + k ) C 2 ( k ) + ( j k ) C 2 ( k + 1 ) ] ( ϕ k + 1 ϕ k ) i [ C 2 ( k + 1 ) C 2 ( k ) ] ( sin ϕ k + 1 sin ϕ k ) ,
δ F ( r i ) = N Δ r ( j = 0 D i j 2 ) 1 / 2 ,
F ( r i ) = Δ r j = 0 D i j P ( r j ) = j = 0 D i j j P ( r j ) .
F ( r i ) / δ F ( r i ) D i ( j = 0 D i j 2 ) 1 / 2 [ P ( r i ) Δ r N ] ,
F ( r i ) = 1 π j = i 0 , j = i Δ r / 2 , j > i Δ r / 2 P ( r j + δ ) [ ( r j + δ ) 2 r i 2 ] 1 / 2 d δ .
P ( r j + δ ) = [ P ( r j + 1 ) P ( r j 1 ) ] / 2 Δ r + [ P ( r j + 1 ) + P ( r j 1 ) 2 P ( r j ) ] δ / Δ r 2 .
F ( r i ) = 1 Δ r j = i [ I i j ( 1 ) I i j ( 0 ) ] P ( r j 1 ) 2 I i j ( 1 ) P ( r j ) + [ I i j ( 1 ) + I i j ( 0 ) ] P ( r j + 1 ) ,
I i j ( n ) = 1 2 π 0 , j = i 1 , j > i 1 δ n [ ( 2 j + δ ) 2 4 i 2 ] 1 / 2 d δ .
F ( r i ) = 1 π j = i 0 Δ r P ( r j + δ ) [ ( r j + δ ) 2 r i 2 ] 1 / 2 d δ .
P ( r j + δ ) = [ P ( r j + 1 ) P ( r j ) ] / Δ r .
F ( r i ) = 1 Δ r j = i J i j [ P ( r j + 1 ) P ( r j ) ] ,
J i j = 1 π 0 1 1 [ ( j + δ ) 2 i 2 ] 1 / 2 d δ .

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