Abstract

An algorithm for computer tomography has been developed that is applicable to reconstruction from data having incomplete projections because an opaque object blocks some of the probing radiation as it passes through the object field. The algorithm is based on iteration between the object domain and the projection (Radon transform) domain. Reconstructions are computed during each iteration by the well-known convolution method. Although it is demonstrated that this algorithm does not converge, an empirically justified criterion for terminating the iteration when the most accurate estimate has been computed is presented. The algorithm has been studied by using it to reconstruct several different object fields with several different opaque regions. It also has been used to reconstruct aerodynamic density fields from interferometric data recorded in wind tunnel tests.

© 1984 Optical Society of America

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References

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  1. S. R. Deans, The Radon Transform and Some of its Applications (Wiley, New York, 1983).
  2. G. T. Herman, “Computerized Tomography,” Proc. IEEE 71, 291 (1983).
    [CrossRef]
  3. T.-F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J 13, 841 (1975).
    [CrossRef]
  4. S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
    [CrossRef] [PubMed]
  5. R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections. III: Projection Completion Methods (Theory),” Optik 50, 189 (1978); “Image Reconstruction from Projections: IV: Projection Completion Methods (Computational Examples),” Optik 50, 269 (1978).
  6. T. Sato, S. J. Norton, M. Linzer, O. Ikeda, M. Hirama, “Tomographic Image Reconstruction from Limited Projections Using Iterative Revisions in Image and Transform Space,” Appl. Opt. 20, 395 (1981).
    [CrossRef] [PubMed]
  7. J. R. Fienup, “Phase Retrieval Algorithms: a Comparison,” Appl. Opt. 21, 2758 (1982).
    [CrossRef] [PubMed]
  8. W. Braga, C. M. Vest, “Computer Tomography by Iteration Between Image and Projection Spaces,” J. Opt. Soc. Am. 71, 1642A (1981).
  9. B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative Convolution Backprojection Algorithms for Image Reconstruction from Limited Data,” J. Opt. Soc. Am. 73, 1493 (1983).
    [CrossRef]
  10. A. Klug, R. A. Crowther, “Three-Dimensional Image Reconstruction from the Viewpoint of Information Theory,” Nature London 238, 435 (1972).
    [CrossRef]
  11. G. N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs, Part III,” Ind. J. Pure Appl. Phys. 9, 997 (1971).
  12. I. Prikryl, C. M. Vest, “Computer Tomography of Flows External to Test Models,” Report INTFL-8202, Department of Mechanical Engineering and Applied Mechanics, U. Michigan, Ann Arbor 48109 (1982).
  13. D. Ludwig, “The Radon Transform on Euclidean Space,” Commun. Pure Appl. Math. 19, 49 (1966).
    [CrossRef]

1983 (2)

1982 (1)

1981 (3)

1978 (1)

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections. III: Projection Completion Methods (Theory),” Optik 50, 189 (1978); “Image Reconstruction from Projections: IV: Projection Completion Methods (Computational Examples),” Optik 50, 269 (1978).

1975 (1)

T.-F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J 13, 841 (1975).
[CrossRef]

1972 (1)

A. Klug, R. A. Crowther, “Three-Dimensional Image Reconstruction from the Viewpoint of Information Theory,” Nature London 238, 435 (1972).
[CrossRef]

1971 (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs, Part III,” Ind. J. Pure Appl. Phys. 9, 997 (1971).

1966 (1)

D. Ludwig, “The Radon Transform on Euclidean Space,” Commun. Pure Appl. Math. 19, 49 (1966).
[CrossRef]

Bates, R. H. T.

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections. III: Projection Completion Methods (Theory),” Optik 50, 189 (1978); “Image Reconstruction from Projections: IV: Projection Completion Methods (Computational Examples),” Optik 50, 269 (1978).

Braga, W.

W. Braga, C. M. Vest, “Computer Tomography by Iteration Between Image and Projection Spaces,” J. Opt. Soc. Am. 71, 1642A (1981).

Brody, W. R.

Cha, S.

Crowther, R. A.

A. Klug, R. A. Crowther, “Three-Dimensional Image Reconstruction from the Viewpoint of Information Theory,” Nature London 238, 435 (1972).
[CrossRef]

Deans, S. R.

S. R. Deans, The Radon Transform and Some of its Applications (Wiley, New York, 1983).

Fienup, J. R.

Herman, G. T.

G. T. Herman, “Computerized Tomography,” Proc. IEEE 71, 291 (1983).
[CrossRef]

Hirama, M.

Ikeda, O.

Klug, A.

A. Klug, R. A. Crowther, “Three-Dimensional Image Reconstruction from the Viewpoint of Information Theory,” Nature London 238, 435 (1972).
[CrossRef]

Lakshminarayanan, A. V.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs, Part III,” Ind. J. Pure Appl. Phys. 9, 997 (1971).

Lewitt, R. M.

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections. III: Projection Completion Methods (Theory),” Optik 50, 189 (1978); “Image Reconstruction from Projections: IV: Projection Completion Methods (Computational Examples),” Optik 50, 269 (1978).

Linzer, M.

Ludwig, D.

D. Ludwig, “The Radon Transform on Euclidean Space,” Commun. Pure Appl. Math. 19, 49 (1966).
[CrossRef]

Macovski, A.

Medoff, B. P.

Nassi, M.

Norton, S. J.

Prikryl, I.

I. Prikryl, C. M. Vest, “Computer Tomography of Flows External to Test Models,” Report INTFL-8202, Department of Mechanical Engineering and Applied Mechanics, U. Michigan, Ann Arbor 48109 (1982).

Ragsdale, W. C.

T.-F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J 13, 841 (1975).
[CrossRef]

Ramachandran, G. N.

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs, Part III,” Ind. J. Pure Appl. Phys. 9, 997 (1971).

Sato, T.

Spring, W. C.

T.-F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J 13, 841 (1975).
[CrossRef]

Vest, C. M.

S. Cha, C. M. Vest, “Tomographic Reconstruction of Strongly Refracting Fields and its Application to Interferometric Measurement of Boundary Layers,” Appl. Opt. 20, 2787 (1981).
[CrossRef] [PubMed]

W. Braga, C. M. Vest, “Computer Tomography by Iteration Between Image and Projection Spaces,” J. Opt. Soc. Am. 71, 1642A (1981).

I. Prikryl, C. M. Vest, “Computer Tomography of Flows External to Test Models,” Report INTFL-8202, Department of Mechanical Engineering and Applied Mechanics, U. Michigan, Ann Arbor 48109 (1982).

Zien, T.-F.

T.-F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J 13, 841 (1975).
[CrossRef]

AIAA J (1)

T.-F. Zien, W. C. Ragsdale, W. C. Spring, “Quantitative Determination of Three-Dimensional Density Field by Holographic Interferometry,” AIAA J 13, 841 (1975).
[CrossRef]

Appl. Opt. (3)

Commun. Pure Appl. Math. (1)

D. Ludwig, “The Radon Transform on Euclidean Space,” Commun. Pure Appl. Math. 19, 49 (1966).
[CrossRef]

Ind. J. Pure Appl. Phys. (1)

G. N. Ramachandran, A. V. Lakshminarayanan, “Three-Dimensional Reconstruction from Radiographs and Electron Micrographs, Part III,” Ind. J. Pure Appl. Phys. 9, 997 (1971).

J. Opt. Soc. Am. (2)

W. Braga, C. M. Vest, “Computer Tomography by Iteration Between Image and Projection Spaces,” J. Opt. Soc. Am. 71, 1642A (1981).

B. P. Medoff, W. R. Brody, M. Nassi, A. Macovski, “Iterative Convolution Backprojection Algorithms for Image Reconstruction from Limited Data,” J. Opt. Soc. Am. 73, 1493 (1983).
[CrossRef]

Nature London (1)

A. Klug, R. A. Crowther, “Three-Dimensional Image Reconstruction from the Viewpoint of Information Theory,” Nature London 238, 435 (1972).
[CrossRef]

Optik (1)

R. M. Lewitt, R. H. T. Bates, “Image Reconstruction from Projections. III: Projection Completion Methods (Theory),” Optik 50, 189 (1978); “Image Reconstruction from Projections: IV: Projection Completion Methods (Computational Examples),” Optik 50, 269 (1978).

Proc. IEEE (1)

G. T. Herman, “Computerized Tomography,” Proc. IEEE 71, 291 (1983).
[CrossRef]

Other (2)

S. R. Deans, The Radon Transform and Some of its Applications (Wiley, New York, 1983).

I. Prikryl, C. M. Vest, “Computer Tomography of Flows External to Test Models,” Report INTFL-8202, Department of Mechanical Engineering and Applied Mechanics, U. Michigan, Ann Arbor 48109 (1982).

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Figures (8)

Fig. 1
Fig. 1

Object field and its projections.

Fig. 2
Fig. 2

Representation of the iterative convolution algorithm.

Fig. 3
Fig. 3

One of the object functions used in the empirical study of the iterative convolution algorithm. This function is given by Eq. (14): (a) the function; (b) the function with the segment r < rc = 0.6 r0 missing.

Fig. 4
Fig. 4

Dependence of reconstruction error on number of iterations for two different schemes for initializing the missing projection data.

Fig. 5
Fig. 5

Dependence of reconstruction error on number of iterations and number of projections. The number of data points, M = 27, is fixed.

Fig. 6
Fig. 6

Dependence of reconstruction error on amount of data. The number of data points M is fixed for each curve, and the number of projections N is matched to M using Eq. (7).

Fig. 7
Fig. 7

Reconstructed distribution of axisymmetric density change in the plane 12.7 mm behind the tip of a cylindrical projectile in a Mach 0.6 airflow at zero angle of attack.

Fig. 8
Fig. 8

Reconstructed asymmetric distribution of density difference in a Mach 0.6 airflow about a projectile at 5.5° angle of attack.

Tables (1)

Tables Icon

Table I Comparison of Errors Among Reconstruction Obtained by Iterative Convolution and Series Expansion Methods.a

Equations (20)

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Δ Y = 2 r 0 / ( M - 1 ) ,
Δ θ = π / N .
Δ Y 1 / ( 2 ρ 0 ) ;
Δ θ π / ( 2 π ρ 0 r 0 - 3 ) .
M - 1 4 r 0 ;
N + 3 2 π r 0 ρ 0 .
( N + 3 ) / ( M - 1 ) = π / 2.
f ( r , ϕ ) = k g [ μ ( r , ϕ ) Δ Y , θ k ] Δ θ ,
g ( m Δ Y , θ k ) = { ¼ [ f ^ ( m Δ Y , θ k ) ] - 1 π 2 p odd f ^ [ ( m + p ) Δ Y , θ k ] / p 2 } / Δ Y
μ ( Δ Y ) = r sin ( ϕ - θ k ) .
μ = ( β cos θ k - α sin θ k ) / ( Δ Y )
θ k = k ( Δ θ )
k = 0 , 1 , 2 , , N - 1 and β = r sin ( ϕ ) α = r cos ( ϕ ) .
g [ μ Δ Y , θ k ] = ( m + 1 - μ ) g ( m Δ Y , θ k ) + ( μ - m ) g [ ( m + 1 ) Δ Y , θ k ] .
f ( x , y ) = exp { - 6 [ ( x - 0.6 ) 2 + y 2 ] 1 - ( x 2 + y 2 ) } + 0.5 exp { - 6 [ ( x + 0.6 ) 2 + y 2 ] 1 - ( x 2 + y 2 ) } + exp { - 6 [ x 2 + ( y - 0.6 ) 2 ] 1 - ( x 2 + y 2 ) } + 0.5 exp { - 6 [ x 2 + ( y + 0.6 ) 2 ] 1 - ( x 2 + y 2 ) } .
E n = - - f ( x , y ) - f n ( x , y ) 2 d x d y = - - F ( ξ , η ) - F n ( ξ , η ) 2 d x d y = 0 π - F ( ρ , θ ) - F n ( ρ , θ ) 2 ρ d ρ d θ = a n 0 π - F ( ρ , θ ) - F n ( ρ , θ ) 2 d ρ d θ = a n 0 π - f ^ ( Y , θ ) - f ^ n ( Y , θ ) 2 d Y d θ a n 0 π - f ^ ( Y , θ ) - h ^ n ( Y , θ ) 2 d Y d θ = a n 0 π - F ( ρ , θ ) - H n ( ρ , θ ) 2 d ρ d θ = a n b n 0 π - F ( ρ , θ ) - H n ( ρ , θ ) 2 ρ d ρ d θ = a n b n - - F ( ξ , η ) - H n ( ξ , η ) 2 d ξ d η = a n b n - - f ( x , y ) - h n ( x , y ) 2 d x d y a n b n - - f ( x , y ) - f n + 1 ( x , y ) 2 d x d y = a n b n E n + 1 ,
a n = 0 π - F ( ρ , θ ) - F n ( ρ , θ ) 2 ρ d ρ d θ 0 π - F ( ρ , θ ) - F n ( ρ , θ ) 2 d ρ d θ ,
b n = 0 π - F ( ρ , θ ) - H n ( ρ , θ ) 2 ρ d ρ d θ 0 π - F ( ρ , θ ) - H n ( ρ , θ ) 2 d ρ d θ .
- - f ( x , y ) 2 d x d y = 0 π - F - 1 [ ρ 1 / 2 f ^ ˜ ( ρ , θ ) ] 2 d ρ d θ .
E n = - - f ( x , y ) - f n ( x , y ) 2 d x d y = 0 π - Φ - 1 { ρ 1 / 2 [ f ^ ˜ ( ρ , θ ) - f ^ ˜ n ( ρ , θ ) ] } 2 d ρ d θ 0 π - Φ - 1 { ρ 1 / 2 [ f ^ ˜ ( ρ , θ ) - h ^ ˜ n ( ρ , θ ) ] } 2 d ρ d θ = - - f ( x , y ) - h n + 1 ( x , y ) 2 d x d y - - f ( x , y ) - f n + 1 ( x , y ) 2 d x d y = E n + 1 .

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