Abstract

A single-channel incoherent optical processor that reconstructs an object from its 1-D x-ray projections has been developed. The projection data are recorded directly on x-ray film, which, after development, is used as the input transparency for the processor. The same computing algorithms used by commercial computed tomography units are optically implemented to reconstruct the object from these projections. The totally analog processor employs time-modulated OTF synthesis to achieve the necessary bipolar filtering operations. This technique involves a time-varying pupil plane mask in conjunction with synchronous demodulation of the detector output to obtain the desired response. Results are presented that are similar in quality to reconstructions produced by commercial scanners. The advantage of this system is that it should offer equivalent performance at a reduced cost.

© 1981 Optical Society of America

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References

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  1. For a recent review, see W. Swindell, H. H. Barrett, Phys. Today 30, 32 (1977).
    [CrossRef]
  2. H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
    [CrossRef]
  3. A. F. Gmitro, J. E. Greivenkamp, W. Swindell, H. H. Barrett, M. Y. Chiu, S. K. Gordon, Opt. Eng. 19, 260 (1980).
    [CrossRef]
  4. R. F. Wagner, Photogr. Sci. Eng. 21, 253 (1977).
  5. I. Glaser, H. H. Barrett, Appl. Opt. 18. 2294 (1979).
    [CrossRef] [PubMed]
  6. H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
    [CrossRef]
  7. A. R. Shulman, Optical Data Processing (Wiley, New York, 1970).
  8. J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).
  9. G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
    [CrossRef] [PubMed]
  10. H. H. Barrett, S. K. Gordon, R. S. Hershel, Comput. Biol. Med. 6, 307 (1976).
    [CrossRef] [PubMed]
  11. M. A. Kujoori, B. Hillman, H. H. Barrett, Invest. Radiol. 15, 145 (1980).
    [CrossRef]

1980 (2)

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

M. A. Kujoori, B. Hillman, H. H. Barrett, Invest. Radiol. 15, 145 (1980).
[CrossRef]

1979 (1)

1978 (1)

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

1977 (3)

R. F. Wagner, Photogr. Sci. Eng. 21, 253 (1977).

For a recent review, see W. Swindell, H. H. Barrett, Phys. Today 30, 32 (1977).
[CrossRef]

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

1976 (1)

H. H. Barrett, S. K. Gordon, R. S. Hershel, Comput. Biol. Med. 6, 307 (1976).
[CrossRef] [PubMed]

1971 (1)

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

Barrett, H. H.

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

M. A. Kujoori, B. Hillman, H. H. Barrett, Invest. Radiol. 15, 145 (1980).
[CrossRef]

I. Glaser, H. H. Barrett, Appl. Opt. 18. 2294 (1979).
[CrossRef] [PubMed]

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

For a recent review, see W. Swindell, H. H. Barrett, Phys. Today 30, 32 (1977).
[CrossRef]

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

H. H. Barrett, S. K. Gordon, R. S. Hershel, Comput. Biol. Med. 6, 307 (1976).
[CrossRef] [PubMed]

Chiu, M. Y.

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

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

Gaskill, J. D.

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

Glaser, I.

Gmitro, A. F.

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

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

Gordon, S. K.

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

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

H. H. Barrett, S. K. Gordon, R. S. Hershel, Comput. Biol. Med. 6, 307 (1976).
[CrossRef] [PubMed]

Greivenkamp, J. E.

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

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

Hershel, R. S.

H. H. Barrett, S. K. Gordon, R. S. Hershel, Comput. Biol. Med. 6, 307 (1976).
[CrossRef] [PubMed]

Hillman, B.

M. A. Kujoori, B. Hillman, H. H. Barrett, Invest. Radiol. 15, 145 (1980).
[CrossRef]

Kujoori, M. A.

M. A. Kujoori, B. Hillman, H. H. Barrett, Invest. Radiol. 15, 145 (1980).
[CrossRef]

Lakshminarayanan, A. V.

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

Ramachandran, G. N.

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

Shulman, A. R.

A. R. Shulman, Optical Data Processing (Wiley, New York, 1970).

Swindell, W.

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

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

For a recent review, see W. Swindell, H. H. Barrett, Phys. Today 30, 32 (1977).
[CrossRef]

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

Wagner, R. F.

R. F. Wagner, Photogr. Sci. Eng. 21, 253 (1977).

Appl. Opt. (1)

Comput. Biol. Med. (1)

H. H. Barrett, S. K. Gordon, R. S. Hershel, Comput. Biol. Med. 6, 307 (1976).
[CrossRef] [PubMed]

Invest. Radiol. (1)

M. A. Kujoori, B. Hillman, H. H. Barrett, Invest. Radiol. 15, 145 (1980).
[CrossRef]

Opt. Commun. (1)

H. H. Barrett, J. E. Greivenkamp, S. K. Gordon, A. F. Gmitro, M. Y. Chiu, W. Swindell, Opt. Commun. 28, 287 (1978).
[CrossRef]

Opt. Eng. (1)

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

Photogr. Sci. Eng. (1)

R. F. Wagner, Photogr. Sci. Eng. 21, 253 (1977).

Phys. Today (1)

For a recent review, see W. Swindell, H. H. Barrett, Phys. Today 30, 32 (1977).
[CrossRef]

Proc. IEEE (1)

H. H. Barrett, W. Swindell, Proc. IEEE 65, 89 (1977).
[CrossRef]

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

G. N. Ramachandran, A. V. Lakshminarayanan, Proc. Natl. Acad. Sci. USA 68, 2236 (1971).
[CrossRef] [PubMed]

Other (2)

A. R. Shulman, Optical Data Processing (Wiley, New York, 1970).

J. D. Gaskill, Linear Systems, Fourier Transforms, and Optics (Wiley, New York, 1978).

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Figures (22)

Fig. 1
Fig. 1

Projection fϕ(x′) of an object μ(x,y).

Fig. 2
Fig. 2

Filter function h(x) and transfer function H(ξ) required for the convolution algorithm.

Fig. 3
Fig. 3

Device for recording the 1-D x-ray projections of an object on film.

Fig. 4
Fig. 4

Three-point object and its sinogram.

Fig. 5
Fig. 5

Sinogram produced by the commercial EMI phantom [Fig. 20(a)].

Fig. 6
Fig. 6

Projections of a helical absorber as it is rotated about its axis. All these projections are portions of a sine curve, and this object will be reconstructed as a point.

Fig. 7
Fig. 7

Transmission vs absorber thickness for a film–screen combination (DuPont Cronex IV film with DuPont Quanta II screens).

Fig. 8
Fig. 8

Incoherent optical system for producing an unfiltered reconstruction.

Fig. 9
Fig. 9

Shape of the slit required to reconstruct a point with coordinates (r,θ).

Fig. 10
Fig. 10

Processing mask used to produce an unfiltered reconstruction of the entire object.

Fig. 11
Fig. 11

Contributions of unwanted sine curves that produce a 1/r PSF in the reconstruction.

Fig. 12
Fig. 12

Method used to produce the processing loops.

Fig. 13
Fig. 13

Complete system for producing unfiltered reconstructions.

Fig. 14
Fig. 14

Transfer function obtained with OTF synthesis by placing a Ronchi ruling in the pupil plane of one lens of a two-channel optical system: (A) the Ronchi ruling; (B) negative channel OTF; (C) positive channel OTF, and (D) composite OTF.

Fig. 15
Fig. 15

PSF generated by the Ronchi pupil and OTF synthesis.

Fig. 16
Fig. 16

Effect on the image plane slit on the PSF and OTF of the two-channel system with a Ronchi pupil.

Fig. 17
Fig. 17

Disk spun in front of the imaging lens to implement time-modulated OTF synthesis.

Fig. 18
Fig. 18

Electronic processing applied to the detector output to demodulate it.

Fig. 19
Fig. 19

Complete single-channel incoherent optical system used to reconstruct an object from its corresponding sinogram.

Fig. 20
Fig. 20

Commercial EMI phantom: (A) diagram; (B) optical reconstruction; and (C) reconstruction produced by digitizing the sinogram.

Fig. 21
Fig. 21

Phantom consisting of a hexagonal array of glass tubes: (A) diagram; (B) optical reconstruction.

Fig. 22
Fig. 22

Reconstruction of a slice through the abdomen and front paws of a dog: (A) optical reconstruction; (B) diagram.

Equations (19)

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f ϕ ( x ) = - μ ( x , y ) d y .
I ϕ ( x ) = I 0 exp [ - - μ ( x , y ) d y ] ,
f ϕ ( x ) = - ln [ I ϕ ( x ) / I 0 ] .
μ ^ ( r , θ ) = 1 π 0 π d ϕ - d x f ϕ ( x ) h [ r cos ( θ - ϕ ) - x ] ,
p ( r , θ ) = 1 π 0 π h [ r cos ( θ - ϕ ) ] d ϕ .
p u ( r , θ ) = 1 π 0 π δ [ r cos ( θ - ϕ ) ] d ϕ .
δ [ y ( ϕ ) ] = y ( ϕ ) / ϕ ϕ = ϕ 0 - 1 δ ( ϕ - ϕ 0 ) ,
p u ( r ) = 1 / ( π r ) .
H ( ξ ) = A ( ξ ) ξ ,
μ ^ u ( r , θ ) = 1 π 0 π f ϕ [ r cos ( θ - ϕ ) ] d ϕ .
D ( y ) = m r sin ( π y m L - θ ) ,
h ( x ) = h + ( x ) - h - ( x ) .
H ( ξ , η ) [ p ( x , y ) p * ( x , y ) ] x = ξ λ f , y = η λ f ,
h ( x ) = δ ( x ) - n odd ( 2 π n ) 2 δ ( x - n λ f d ) .
Φ ( r , θ ) 0 π d ϕ - d x s [ r cos ( θ - ϕ ) - x ] × - d x f ϕ ( x m ) h ( x - x ) .
Φ ( r , θ ) 0 π d ϕ - d x f ϕ ( x / m ) - d x s ( x ) × h [ r cos ( θ - ϕ ) - x - x ] ,
Φ ( r , θ ) 0 π d ϕ - d x f ϕ ( x / m ) h [ r cos ( θ - ϕ ) - x ] ,
h ( x ) = s ( x ) * h ( x ) .
D center ( SNR ) 2 η 3 t ,

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