Abstract

In coded-aperture imaging, a technique useful in nuclear medicine, a coded image of dimensionality one less than that of the object is recorded. It is possible to obtain a crude reconstruction of the object with the use of a form of correlation decoding of the coded image, but to date these methods have been unsatisfactory. We present an iterative reconstruction algorithm that makes use of a priori knowledge of object border and object postivity. Reconstructions obtained in simulations are consistent with the constraints and coded image and are surprisingly accurate in light of the extensive information loss involved in the image-formation process.

© 1984 Optical Society of America

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References

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  1. B. P. Medoff, W. R. Brody, A. Macovski, J. Opt. Soc. Am. 73, 1493 (1983).
    [CrossRef]
  2. H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation Detection and Processing (Academic, New York, 1981).
  3. R. G. Simpson, H. H. Barrett, “Coded-Aperture Imaging,” in Imaging in Diagnostic Medicine, S. Nudelman, Ed. (Plenum, New York, 1980).
  4. L. T. Chang, B. MacDonald, V. Perez-Mendez, IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
    [CrossRef]
  5. M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Gindi, J. Opt. Soc. Am. 69, 1323 (1979).
    [CrossRef]
  6. N. Ohyama, T. Honda, J. Tsujiuschi, Opt. Commun. 27, 339 (1978).
    [CrossRef]
  7. A. Steinbach, “An Analysis of the Depth Resolution Problem in One-Dimensional Coded Aperture Imaging,” Ph.D. Thesis, Stanford U. (1977).
  8. W. J. Wild, Opt. Lett. 8, 247 (1983).
    [CrossRef] [PubMed]
  9. D. G. Leunberger, Optimization by Vector Space Methods (Wiley, New York, 1969).
  10. R. W. Schafer, R. M. Mersereau, M. A. Richards, Proc. IEEE 69, 432 (1981).
    [CrossRef]
  11. W. E. Smith, H. H. Barrett, R. G. Paxman, Opt. Lett. 9, 199 (1983).
    [CrossRef]
  12. M. Tipton, Proc. Soc. Photo-Opt. Instrum. Eng. 152, 113 (1978).
  13. R. G. Simpson, H. H. Barrett, J. G. Kelley, K. T. Stalker, “Some Applications of One-dimensional Coded Apertures,” SPIE/SPSE Technical Symposium, Reston, Va. (1977).
  14. D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

1983 (3)

1981 (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, Proc. IEEE 69, 432 (1981).
[CrossRef]

1979 (1)

1978 (2)

N. Ohyama, T. Honda, J. Tsujiuschi, Opt. Commun. 27, 339 (1978).
[CrossRef]

M. Tipton, Proc. Soc. Photo-Opt. Instrum. Eng. 152, 113 (1978).

1976 (1)

L. T. Chang, B. MacDonald, V. Perez-Mendez, IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[CrossRef]

Arendt, J. W.

Barrett, H. H.

W. E. Smith, H. H. Barrett, R. G. Paxman, Opt. Lett. 9, 199 (1983).
[CrossRef]

M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Gindi, J. Opt. Soc. Am. 69, 1323 (1979).
[CrossRef]

H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation Detection and Processing (Academic, New York, 1981).

R. G. Simpson, H. H. Barrett, “Coded-Aperture Imaging,” in Imaging in Diagnostic Medicine, S. Nudelman, Ed. (Plenum, New York, 1980).

R. G. Simpson, H. H. Barrett, J. G. Kelley, K. T. Stalker, “Some Applications of One-dimensional Coded Apertures,” SPIE/SPSE Technical Symposium, Reston, Va. (1977).

Brody, W. R.

Chang, L. T.

L. T. Chang, B. MacDonald, V. Perez-Mendez, IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[CrossRef]

Chiu, M. Y.

Chou, C.

Devaux, J. Y.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

Fonroget, J.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

Gindi, G. R.

Guilhem, J. B.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

Guiraud, R.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

Honda, T.

N. Ohyama, T. Honda, J. Tsujiuschi, Opt. Commun. 27, 339 (1978).
[CrossRef]

Kelley, J. G.

R. G. Simpson, H. H. Barrett, J. G. Kelley, K. T. Stalker, “Some Applications of One-dimensional Coded Apertures,” SPIE/SPSE Technical Symposium, Reston, Va. (1977).

Lefkoupoulos, D.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

Leunberger, D. G.

D. G. Leunberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

MacDonald, B.

L. T. Chang, B. MacDonald, V. Perez-Mendez, IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[CrossRef]

Macovski, A.

Medoff, B. P.

Mersereau, R. M.

R. W. Schafer, R. M. Mersereau, M. A. Richards, Proc. IEEE 69, 432 (1981).
[CrossRef]

Ohyama, N.

N. Ohyama, T. Honda, J. Tsujiuschi, Opt. Commun. 27, 339 (1978).
[CrossRef]

Paxman, R. G.

Perez-Mendez, V.

L. T. Chang, B. MacDonald, V. Perez-Mendez, IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[CrossRef]

Richards, M. A.

R. W. Schafer, R. M. Mersereau, M. A. Richards, Proc. IEEE 69, 432 (1981).
[CrossRef]

Roucayrol, J. C.

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

Schafer, R. W.

R. W. Schafer, R. M. Mersereau, M. A. Richards, Proc. IEEE 69, 432 (1981).
[CrossRef]

Simpson, R. G.

M. Y. Chiu, H. H. Barrett, R. G. Simpson, C. Chou, J. W. Arendt, G. R. Gindi, J. Opt. Soc. Am. 69, 1323 (1979).
[CrossRef]

R. G. Simpson, H. H. Barrett, “Coded-Aperture Imaging,” in Imaging in Diagnostic Medicine, S. Nudelman, Ed. (Plenum, New York, 1980).

R. G. Simpson, H. H. Barrett, J. G. Kelley, K. T. Stalker, “Some Applications of One-dimensional Coded Apertures,” SPIE/SPSE Technical Symposium, Reston, Va. (1977).

Smith, W. E.

Stalker, K. T.

R. G. Simpson, H. H. Barrett, J. G. Kelley, K. T. Stalker, “Some Applications of One-dimensional Coded Apertures,” SPIE/SPSE Technical Symposium, Reston, Va. (1977).

Steinbach, A.

A. Steinbach, “An Analysis of the Depth Resolution Problem in One-Dimensional Coded Aperture Imaging,” Ph.D. Thesis, Stanford U. (1977).

Swindell, W.

H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation Detection and Processing (Academic, New York, 1981).

Tipton, M.

M. Tipton, Proc. Soc. Photo-Opt. Instrum. Eng. 152, 113 (1978).

Tsujiuschi, J.

N. Ohyama, T. Honda, J. Tsujiuschi, Opt. Commun. 27, 339 (1978).
[CrossRef]

Wild, W. J.

IEEE Trans. Nucl. Sci. (1)

L. T. Chang, B. MacDonald, V. Perez-Mendez, IEEE Trans. Nucl. Sci. NS-23, 568 (1976).
[CrossRef]

J. Opt. Soc. Am. (2)

Opt. Commun. (1)

N. Ohyama, T. Honda, J. Tsujiuschi, Opt. Commun. 27, 339 (1978).
[CrossRef]

Opt. Lett. (2)

Proc. IEEE (1)

R. W. Schafer, R. M. Mersereau, M. A. Richards, Proc. IEEE 69, 432 (1981).
[CrossRef]

Proc. Soc. Photo-Opt. Instrum. Eng. (1)

M. Tipton, Proc. Soc. Photo-Opt. Instrum. Eng. 152, 113 (1978).

Other (6)

R. G. Simpson, H. H. Barrett, J. G. Kelley, K. T. Stalker, “Some Applications of One-dimensional Coded Apertures,” SPIE/SPSE Technical Symposium, Reston, Va. (1977).

D. Lefkoupoulos, J. Fonroget, J. Y. Devaux, J. B. Guilhem, J. C. Roucayrol, R. Guiraud, “Quantitative 3D Imaging with Coded Apertures by Using SVD Decomposition of the Transmission Matrix,” in Proceedings, Third World Congress of Nuclear Medicine and Biology, C. Raynaud, Ed. (Pergamon, Paris, 1982), p. 503.

D. G. Leunberger, Optimization by Vector Space Methods (Wiley, New York, 1969).

H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation Detection and Processing (Academic, New York, 1981).

R. G. Simpson, H. H. Barrett, “Coded-Aperture Imaging,” in Imaging in Diagnostic Medicine, S. Nudelman, Ed. (Plenum, New York, 1980).

A. Steinbach, “An Analysis of the Depth Resolution Problem in One-Dimensional Coded Aperture Imaging,” Ph.D. Thesis, Stanford U. (1977).

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Figures (4)

Fig. 1
Fig. 1

Coded-aperture imaging: (a) for pinhole imaging, points A and B form point images at A′ and B′, (b) in coded-aperture imaging, the pinhole is replaced by a code (here an annulus), and each object point casts a shifted and magnified version of the code.

Fig. 2
Fig. 2

Geometry for the encoding and decoding operations via projected and backprojected rays. (a) Encoding step. Shown is a ray which emanates from the object, passes through the coded aperture, and strikes the detector. (b) Decoding step. Shown is a backprojected ray which is weighted by its points of intersection with the coded-image plane and the decoding template and smeared along a line in the layergram space (rectangle). In each case, the aperture plane is located at z = 0, and only one of many possible rays is shown.

Fig. 3
Fig. 3

Layout for the simulations. The aperture pinholes are indicated by arrows.

Fig. 4
Fig. 4

(a) Original object; (b) layergram; (c) unconstrained estimate; (d) constrained estimate; (e) constrained estimate obtained from noisy coded image; (f) constrained estimate with smoothing at each iteration.

Equations (25)

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h ( r ) = ( z s ) 2 f ( - z s r ) ,
h ( r ) = d 2 r f ( r ) g ( r ) ,
r = z s + z r + s s + z r .
h ( r ) = ( z s ) 2 f ˜ ( r ) * * g ˜ ( r ) ,
f ˜ ( r ) f ( - z s r ) ,
g ˜ ( r ) g ( z z + s r ) .
f ^ ( r ) = h ( r ) d ˜ ( r ) = f ˜ ( r ) * * { g ˜ ( r ) d ˜ ( r ) ] .
f ^ ( r ) = d 2 r h ( r ) d ( r )
h ( r ) = d 2 r d z f ( r , z ) g ( r ) P [ f ( r , z ) ] ,
f ^ ( r , z ) = d 2 r h ( r ) d ( r ) B [ h ( r ) ] ,
f ^ ( r , z ) = B P [ f ( r , z ) ] ,
f ^ ( r , z ) = [ f ¯ ( r , z 0 ) * * g ˜ ( r , z 0 ) d z 0 ] d ˜ ( r , z ) ,
f ^ ( r , z ) = f ( r , z 0 ) * * p ( r ; z 0 , z ) d z 0 ,
p ( r ; z 0 , z ) g ˜ ( r , z 0 ) d ˜ ( r , z )
f ^ j ( r ) = i = 1 N f i ( r ) * * p i j ( r ) ,
F ^ j ( ξ , η ) = i = 1 N F i ( ξ , η ) P i j ( ξ , η ) ,
p i j ( r ) = g ˜ i ( r ) d ˜ j ( r )
P i j ( ξ , η ) = G ˜ i ( ξ , η ) D ˜ j * ( ξ , η ) ,
f ^ ( x , z ) = B P [ f ( x , z ) ] .
C [ u ( x ) ] = { u ( x ) at x where constraint holds , zero otherwise .
C = C P C L .
f ( x , z ) = C [ f ( x , z ) ] .
f ( x , z ) = C [ f ( x , z ) ] + λ { f ^ ( x , z ) - B P C [ f ( x , z ) ] } T [ f ( x , z ) ] .
f ^ k + 1 ( x , z ) = T [ f ^ k ( x , z ) ] = C [ f ^ k ( x , z ) ] + λ { f ^ ( x , z ) - B P C [ f ^ k ( x , z ) ] } ,
f ^ 0 ( x , z ) = f ^ ( x , z ) .

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