Abstract

A method of image reconstruction from projections is described which processes the data in polar rather than rectangular coordinates and which does not require back projection. It is based on the decomposition of the object and its shadow (set of projections) into circular harmonics or radial modulators of angular Fourier components. The radial modulators of the object may be reconstructed from those of the shadow using a space-variant system which becomes space-invariant under a coordinate transformation. Experiments using digital and optical implementations are described.

© 1981 Optical Society of America

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  1. R. A. Brooks, G. DiChiro, Phys. Med. Biol. 21, 689 (1976).
    [CrossRef] [PubMed]
  2. A. C. Kak, Proc. IEEE 67, 1245 (1979).
    [CrossRef]
  3. G. T. Herman, Ed., Image Reconstruction from Projections, Implementation and Applications (Springer, New York, 1979).
    [CrossRef]
  4. A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
    [CrossRef]
  5. B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
    [CrossRef] [PubMed]
  6. E. W. Hansen, J. Opt. Soc. Am. 71, 304 (1981).
    [CrossRef]
  7. A. M. Cormack, J. Appl. Phys. 34, 2722 (1963).
    [CrossRef]
  8. M. Ein-Gal, “The Shadow Transform: An Approach to Cross Sectional Imaging,” Ph.D. Dissertation, Stanford U. (1974);available from University Microfilms, Ann Arbor, Mich.
  9. R. M. Perry, “Reconstructing a Function by Circular Harmonic Analysis of its Line Integrals,” in Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Stanford University Institute for Electronics & Medicine and Optical Society of America, Washington, D.C., 1975).
  10. E. W. Hansen, J. W. Goodman, Opt. Commun. 28, 268 (1978).
    [CrossRef]
  11. I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Chap. 8.940.
  12. M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), Sec. 22.3.
  13. E. W. Hansen, Image Reconstruction from Projections Using Circular Harmonic Expansion, Ph.D. Dissertation, Stanford U. (1980);available from University Microfilms, Ann Arbor, Mich.
  14. A. V. Oppenheim, R. W. Schaefer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.5.
  15. Numerical interpolation is treated in most numerical analysis texts;for example, G. Dahlquist, A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974).
  16. L. R. Rabiner, R. W. Schafer, Bell Syst. Tech. J. 53, 333 (1974).
  17. R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 14ff.
  18. N. Freeman, Introduction to Statistical Inference (Addison-Wesley, Reading, Mass., 1963), pp. 179–181.
  19. J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.
  20. A. W. Lohmann, D. P. Paris, Appl. Opt. 6, 1739 (1967).
    [CrossRef] [PubMed]
  21. S. Lowenthal, P. Chavel, Appl. Opt. 13, 718 (1974).
    [CrossRef] [PubMed]

1981

1980

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

1979

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

1978

E. W. Hansen, J. W. Goodman, Opt. Commun. 28, 268 (1978).
[CrossRef]

1976

R. A. Brooks, G. DiChiro, Phys. Med. Biol. 21, 689 (1976).
[CrossRef] [PubMed]

1974

L. R. Rabiner, R. W. Schafer, Bell Syst. Tech. J. 53, 333 (1974).

S. Lowenthal, P. Chavel, Appl. Opt. 13, 718 (1974).
[CrossRef] [PubMed]

1967

1963

A. M. Cormack, J. Appl. Phys. 34, 2722 (1963).
[CrossRef]

Atkins, D. E.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Barrett, H. H.

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Bjorck, A.

Numerical interpolation is treated in most numerical analysis texts;for example, G. Dahlquist, A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974).

Bracewell, R. N.

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 14ff.

Brooks, R. A.

R. A. Brooks, G. DiChiro, Phys. Med. Biol. 21, 689 (1976).
[CrossRef] [PubMed]

Chavel, P.

Chiu, M. Y.

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Chu, A.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Cormack, A. M.

A. M. Cormack, J. Appl. Phys. 34, 2722 (1963).
[CrossRef]

Dahlquist, G.

Numerical interpolation is treated in most numerical analysis texts;for example, G. Dahlquist, A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974).

DiChiro, G.

R. A. Brooks, G. DiChiro, Phys. Med. Biol. 21, 689 (1976).
[CrossRef] [PubMed]

Ein-Gal, M.

M. Ein-Gal, “The Shadow Transform: An Approach to Cross Sectional Imaging,” Ph.D. Dissertation, Stanford U. (1974);available from University Microfilms, Ann Arbor, Mich.

Freeman, N.

N. Freeman, Introduction to Statistical Inference (Addison-Wesley, Reading, Mass., 1963), pp. 179–181.

Gilbert, B. K.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Gmitro, A. F.

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Goodman, J. W.

E. W. Hansen, J. W. Goodman, Opt. Commun. 28, 268 (1978).
[CrossRef]

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.

Gordon, S. K.

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Gradshtein, I. S.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Chap. 8.940.

Grievenkamp, J. E.

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Hansen, E. W.

E. W. Hansen, J. Opt. Soc. Am. 71, 304 (1981).
[CrossRef]

E. W. Hansen, J. W. Goodman, Opt. Commun. 28, 268 (1978).
[CrossRef]

E. W. Hansen, Image Reconstruction from Projections Using Circular Harmonic Expansion, Ph.D. Dissertation, Stanford U. (1980);available from University Microfilms, Ann Arbor, Mich.

Kak, A. C.

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

Lohmann, A. W.

Lowenthal, S.

Oppenheim, A. V.

A. V. Oppenheim, R. W. Schaefer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.5.

Paris, D. P.

Perry, R. M.

R. M. Perry, “Reconstructing a Function by Circular Harmonic Analysis of its Line Integrals,” in Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Stanford University Institute for Electronics & Medicine and Optical Society of America, Washington, D.C., 1975).

Rabiner, L. R.

L. R. Rabiner, R. W. Schafer, Bell Syst. Tech. J. 53, 333 (1974).

Ritman, E. L.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Ryzhik, I. M.

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Chap. 8.940.

Schaefer, R. W.

A. V. Oppenheim, R. W. Schaefer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.5.

Schafer, R. W.

L. R. Rabiner, R. W. Schafer, Bell Syst. Tech. J. 53, 333 (1974).

Swartzlander, E. E.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

Swindell, W.

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Appl. Opt.

Bell Syst. Tech. J.

L. R. Rabiner, R. W. Schafer, Bell Syst. Tech. J. 53, 333 (1974).

Comput. Biomed. Res.

B. K. Gilbert, A. Chu, D. E. Atkins, E. E. Swartzlander, E. L. Ritman, Comput. Biomed. Res. 12, 17 (1979).
[CrossRef] [PubMed]

J. Appl. Phys.

A. M. Cormack, J. Appl. Phys. 34, 2722 (1963).
[CrossRef]

J. Opt. Soc. Am.

Opt. Commun.

E. W. Hansen, J. W. Goodman, Opt. Commun. 28, 268 (1978).
[CrossRef]

Opt. Eng.

A. F. Gmitro, J. E. Grievenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
[CrossRef]

Phys. Med. Biol.

R. A. Brooks, G. DiChiro, Phys. Med. Biol. 21, 689 (1976).
[CrossRef] [PubMed]

Proc. IEEE

A. C. Kak, Proc. IEEE 67, 1245 (1979).
[CrossRef]

Other

G. T. Herman, Ed., Image Reconstruction from Projections, Implementation and Applications (Springer, New York, 1979).
[CrossRef]

M. Ein-Gal, “The Shadow Transform: An Approach to Cross Sectional Imaging,” Ph.D. Dissertation, Stanford U. (1974);available from University Microfilms, Ann Arbor, Mich.

R. M. Perry, “Reconstructing a Function by Circular Harmonic Analysis of its Line Integrals,” in Image Processing for 2-D and 3-D Reconstruction from Projections: Theory and Practice in Medicine and the Physical Sciences (Stanford University Institute for Electronics & Medicine and Optical Society of America, Washington, D.C., 1975).

I. S. Gradshtein, I. M. Ryzhik, Table of Integrals, Series and Products (Academic, New York, 1965), Chap. 8.940.

M. Abramowitz, I. A. Stegun, Eds., Handbook of Mathematical Functions (Dover, New York, 1970), Sec. 22.3.

E. W. Hansen, Image Reconstruction from Projections Using Circular Harmonic Expansion, Ph.D. Dissertation, Stanford U. (1980);available from University Microfilms, Ann Arbor, Mich.

A. V. Oppenheim, R. W. Schaefer, Digital Signal Processing (Prentice-Hall, Englewood Cliffs, N.J., 1975), Sec. 5.5.

Numerical interpolation is treated in most numerical analysis texts;for example, G. Dahlquist, A. Bjorck, Numerical Methods (Prentice-Hall, Englewood Cliffs, N.J., 1974).

R. N. Bracewell, The Fourier Transform and Its Applications (McGraw-Hill, New York, 1978), pp. 14ff.

N. Freeman, Introduction to Statistical Inference (Addison-Wesley, Reading, Mass., 1963), pp. 179–181.

J. W. Goodman, Introduction to Fourier Optics (McGraw-Hill, New York, 1968), Chap. 7.

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Figures (8)

Fig. 1
Fig. 1

Projection geometry.

Fig. 2
Fig. 2

Two-dimensional noncausal point spread functions for the inverse CHT: left to right, from top: N = 0, 1, 2, 4, 8, 16, 32, 64.

Fig. 3
Fig. 3

Off-axis disk phantom used in experiments.

Fig. 4
Fig. 4

Digital reconstruction of phantom a: left to right from top: N = 0, 1, 2, 4, 8, 16, 32, 64.

Fig. 5
Fig. 5

Digital reconstruction: from top, phantoms f, d, e.

Fig. 6
Fig. 6

Image quality vs highest order modulator: top, phantoms a, b, c; bottom, phantoms d, e, f.

Fig. 7
Fig. 7

Experimental optical system.

Fig. 8
Fig. 8

Results of optical reconstruction of individual radial modulators f ˇ n ( t ): from top, n = 0, 1, 2, 3, 4. Digital simulation is on the left, output of optical system on the right.

Equations (24)

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g ( R , ϕ ) = L ( R , ϕ ) f ( r , θ ) d = 0 2 π 0 f ( r , θ ) δ [ r cos ( ϕ θ ) R ] rdrd θ ,
g n ( R ) = 0 f n ( r ) c n ( R / r ) dr ,
g n ( R ) = 1 2 π 0 2 π g ( R , ϕ ) exp ( jn ϕ ) d ϕ ,
g ( R , ϕ ) = n = g n ( R ) exp ( jn ϕ ) ,
f n ( r ) = 1 2 π 0 2 π f ( r , θ ) exp ( jn θ ) d θ ,
f ( r , θ ) = n = f n ( r ) exp ( jn θ ) .
c n ( R / r ) = 0 2 π r δ ( r cos ϕ R ) exp ( jn ϕ ) d ϕ = 2 T n ( R / r ) u ( 1 R / r ) 1 ( R / r ) 2 ,
u ( x ) = { 1 x > 0 , ½ x = 0 , 0 x < 0 .
f n ( r ) = 0 g n ( R ) h n ( r / R ) dR R ,
h n ( r ) = { 1 π T | n | ( 1 / r ) 1 r 2 , 1 r 0 , 0 , r 1 .
h n ( r ) = { 1 π 1 1 r 2 [ r 1 + 1 r 2 ] | n | 1 > r 0 , 1 π r U | n | 1 ( 1 / r ) r 1 ,
U n ( x ) = sin ( ( n + 1 ) cos 1 x ) sin ( cos 1 x )
f ˇ n ( t ) = exp ( τ ) g ˇ n ( τ ) h ˇ n ( t τ ) d τ ,
n = N N h ˇ n ( t ) exp ( jn θ )
f ˇ n ( t ) = u ( t ) 0 exp ( τ ) g n ( τ ) h ˇ n ( t τ ) d τ .
f ˇ n ( t ) = u ( t ) exp ( σ t ) 0 exp [ ( 1 σ ) τ ] g ˇ n ( τ ) × [ exp [ σ ( t τ ) ] h ˇ n ( t τ ) ] d τ .
f ˇ n ( t ) = u ( t ) exp ( σ t ) 1 [ G n ( ν ; σ ) H n ( ν ; σ ) ] ,
G n ( ν ; σ ) = { exp [ ( 1 σ ) t ] g ˇ n ( t ) } ,
H n ( ν ; σ ) = [ exp ( σ t ) h ˇ n ( t ) ] ,
N R / N t ln R min ,
2 ¯ = 1 N i = 1 N [ f i ( a f ̂ i + b ) ] 2 ,
Q = 1 2 ¯ s f 2 = ( 1 N f i f ̂ i μ f μ f ̂ ) 2 s f 2 s f ̂ 2 ,
f ˇ n ( t ) = u ( t ) 1 { H n ( ν ) [ e t g n ( t ) ] } ,
g ( R , ϕ ) = g 0 ( R ) + 2 n = 1 Re [ g n ( R ) exp ( jn ϕ ) ] ,

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