Abstract

Recent work in coded aperture imaging has shown that the uniformly redundant array (URA) can image distant planar radioactive sources with no artifacts. This paper investigates the performance of two URA apertures when used in a close-up tomographic imaging system. It is shown that a URA based on m sequences is superior to one based on quadratic residues. The m-sequence array not only produces less noticeable defocus artifacts in tomographic imaging but is also more resilient to some described detrimental effects of close-up imaging. It is shown that, in spite of these close-up effects, the URA system retains tomographic depth resolution even as the source is moved close to the detector. The URAs based on m sequences provide better images than those obtained using random arrays. This compliments previous studies that have shown random arrays to have better tomographical properties than Fresnel zone plates and nonredundant arrays.

© 1979 Optical Society of America

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References

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  1. L. Mertz, N. O. Young, in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961).
  2. H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
    [PubMed]
  3. K. F. Koral, W. L. Rogers, G. F. Knoll, J. Nuc. Med., 16, 402 (1975).
  4. L. Chang, B. Macdonald, V. Perez-Mendez, “Comparisons of Coded Aperture Imaging Using Various Aperture and Decoding Methods,” presented at SPIE Symposium on Applications of Optics in Medicine and Biology, San Diego (1976).
  5. E. E. Fenimore, T. M. Cannon, Appl. Opt. 17, 337 (1978).
    [CrossRef] [PubMed]
  6. D. Calabro, J. K. Wolf, Inf. Control 11, 537 (1968).
    [CrossRef]
  7. F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).
  8. F. J. MacWilliams, N. J. Sloan, Proc. IEEE 64, 1715 (1976).
    [CrossRef]
  9. R. H. Dicke, Astrophys. J. 153, L101 (1968).
    [CrossRef]
  10. H. H. Barrett, F. A. Horrigan, Appl. Opt. 12, 2606 (1973).
    [CrossRef]
  11. E. E. Fenimore, Appl. Opt. 17, 3562 (1978).
    [CrossRef] [PubMed]
  12. E. E. Fenimore, T. M. Cannon, E. L. Miller, “A Comparison of Fresnel Zone Plates and Uniformly Redundant Arrays,” presented at SPIE Twenty-Second Technical Symposium, San Diego (1978).

1978 (2)

1976 (2)

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

F. J. MacWilliams, N. J. Sloan, Proc. IEEE 64, 1715 (1976).
[CrossRef]

1975 (1)

K. F. Koral, W. L. Rogers, G. F. Knoll, J. Nuc. Med., 16, 402 (1975).

1973 (1)

H. H. Barrett, F. A. Horrigan, Appl. Opt. 12, 2606 (1973).
[CrossRef]

1972 (1)

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

1968 (2)

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

D. Calabro, J. K. Wolf, Inf. Control 11, 537 (1968).
[CrossRef]

Barrett, H. H.

H. H. Barrett, F. A. Horrigan, Appl. Opt. 12, 2606 (1973).
[CrossRef]

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

Calabro, D.

D. Calabro, J. K. Wolf, Inf. Control 11, 537 (1968).
[CrossRef]

Cannon, T. M.

E. E. Fenimore, T. M. Cannon, Appl. Opt. 17, 337 (1978).
[CrossRef] [PubMed]

E. E. Fenimore, T. M. Cannon, E. L. Miller, “A Comparison of Fresnel Zone Plates and Uniformly Redundant Arrays,” presented at SPIE Twenty-Second Technical Symposium, San Diego (1978).

Chang, L.

L. Chang, B. Macdonald, V. Perez-Mendez, “Comparisons of Coded Aperture Imaging Using Various Aperture and Decoding Methods,” presented at SPIE Symposium on Applications of Optics in Medicine and Biology, San Diego (1976).

Dicke, R. H.

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Fenimore, E. E.

E. E. Fenimore, T. M. Cannon, Appl. Opt. 17, 337 (1978).
[CrossRef] [PubMed]

E. E. Fenimore, Appl. Opt. 17, 3562 (1978).
[CrossRef] [PubMed]

E. E. Fenimore, T. M. Cannon, E. L. Miller, “A Comparison of Fresnel Zone Plates and Uniformly Redundant Arrays,” presented at SPIE Twenty-Second Technical Symposium, San Diego (1978).

Gunson, F.

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

Horrigan, F. A.

H. H. Barrett, F. A. Horrigan, Appl. Opt. 12, 2606 (1973).
[CrossRef]

Knoll, G. F.

K. F. Koral, W. L. Rogers, G. F. Knoll, J. Nuc. Med., 16, 402 (1975).

Koral, K. F.

K. F. Koral, W. L. Rogers, G. F. Knoll, J. Nuc. Med., 16, 402 (1975).

Macdonald, B.

L. Chang, B. Macdonald, V. Perez-Mendez, “Comparisons of Coded Aperture Imaging Using Various Aperture and Decoding Methods,” presented at SPIE Symposium on Applications of Optics in Medicine and Biology, San Diego (1976).

MacWilliams, F. J.

F. J. MacWilliams, N. J. Sloan, Proc. IEEE 64, 1715 (1976).
[CrossRef]

Mertz, L.

L. Mertz, N. O. Young, in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961).

Miller, E. L.

E. E. Fenimore, T. M. Cannon, E. L. Miller, “A Comparison of Fresnel Zone Plates and Uniformly Redundant Arrays,” presented at SPIE Twenty-Second Technical Symposium, San Diego (1978).

Perez-Mendez, V.

L. Chang, B. Macdonald, V. Perez-Mendez, “Comparisons of Coded Aperture Imaging Using Various Aperture and Decoding Methods,” presented at SPIE Symposium on Applications of Optics in Medicine and Biology, San Diego (1976).

Polychronopulos, B.

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

Rogers, W. L.

K. F. Koral, W. L. Rogers, G. F. Knoll, J. Nuc. Med., 16, 402 (1975).

Sloan, N. J.

F. J. MacWilliams, N. J. Sloan, Proc. IEEE 64, 1715 (1976).
[CrossRef]

Wolf, J. K.

D. Calabro, J. K. Wolf, Inf. Control 11, 537 (1968).
[CrossRef]

Young, N. O.

L. Mertz, N. O. Young, in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961).

Appl. Opt. (3)

Astrophys. J. (1)

R. H. Dicke, Astrophys. J. 153, L101 (1968).
[CrossRef]

Inf. Control (1)

D. Calabro, J. K. Wolf, Inf. Control 11, 537 (1968).
[CrossRef]

J. Nuc. Med. (1)

K. F. Koral, W. L. Rogers, G. F. Knoll, J. Nuc. Med., 16, 402 (1975).

J. Nucl. Med. (1)

H. H. Barrett, J. Nucl. Med. 13, 382 (1972).
[PubMed]

Mon. Not. R. Astron. Soc. (1)

F. Gunson, B. Polychronopulos, Mon. Not. R. Astron. Soc. 177, 485 (1976).

Proc. IEEE (1)

F. J. MacWilliams, N. J. Sloan, Proc. IEEE 64, 1715 (1976).
[CrossRef]

Other (3)

E. E. Fenimore, T. M. Cannon, E. L. Miller, “A Comparison of Fresnel Zone Plates and Uniformly Redundant Arrays,” presented at SPIE Twenty-Second Technical Symposium, San Diego (1978).

L. Mertz, N. O. Young, in Proceedings of the International Conference on Optical Instruments and Techniques, K. J. Habell, Ed. (Chapman and Hall, London, 1961).

L. Chang, B. Macdonald, V. Perez-Mendez, “Comparisons of Coded Aperture Imaging Using Various Aperture and Decoding Methods,” presented at SPIE Symposium on Applications of Optics in Medicine and Biology, San Diego (1976).

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Figures (11)

Fig. 1
Fig. 1

Uniformly redundant array based on quadratic residues.

Fig. 2
Fig. 2

A uniformly redundant array based on m sequences.

Fig. 3
Fig. 3

A random array.

Fig. 4
Fig. 4

The method used by coded aperture systems for obtaining depth information is shown here. Sources distant to the detector cast smaller aperture shadows than closer sources. By correlating the recorded image with decoding patterns of different sizes, images of the source distribution at different depths can be retrieved.

Fig. 5
Fig. 5

The effects of close-up imaging are depicted above. The fact that a and b are equal implies that there is no geometrical distortion. However, since Φ is smaller than θ and r′ is longer than r (obliquity and inverse square effects), fewer photons strike the detector at its periphery than at its center (for an on-axis source).

Fig. 6
Fig. 6

Simulated effects of the obliquity and inverse square effects are shown. The attenuated peripheral exposure is the result of imaging an on-axis point source only 7.6 cm from the detector.

Fig. 7
Fig. 7

Representative results of a computer simulation of two URAs and a random array are shown above. From left to right, the results of imaging a point source are shown for a quadratic residue URA, m-sequence URA, and random array, respectively. The top row is a decoded plane 10 cm from the detector; the center row is a plane containing the source 15 cm from the detector; the bottom row is a plane 20 cm from the detector.

Fig. 8
Fig. 8

A URA coded aperture picture of a uranium disk and a plutonium point source 12 m from the detector. Imaging time was 30 min at 120 keV.

Fig. 9
Fig. 9

A pinhole picture of the same scene as Fig. 8. Imaging time was also 30 min at 120 keV.

Fig. 10
Fig. 10

A laboratory experiment produced these images which are similar to those of the computer simulation of Fig. 7. The random array was not included in the experiment. A major difference between these results and Fig. 7 is the blurring caused by the gamma camera imaging system.

Fig. 11
Fig. 11

The above mosaic represents the results of a depth resolution experiment. Two point sources were imaged 15 cm and 20 cm, respectively, from the aperture. The recorded data were decoded at 2.5-cm (1-in.) intervals between 13 cm and 25 cm from the aperture. The resulting six planes are shown above, ordered top to bottom, left to right.

Equations (8)

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P ( x , y ) = z [ O z ( x , y ) * A z ( x , y ) ] ,
m = ( z + f ) / z ,
O ˜ z ( x , y ) = P ( x , y ) * D z ( x , y ) .
O ˜ k ( x , y ) = O k ( x , y ) * A k ( x , y ) * D k ( x , y ) .
O ˜ k ( x , y ) = O k ( x , y ) * δ ( x , y ) = O k ( x , y ) ,
O ˜ k ( x , y ) = z [ O z ( x , y ) * A z ( x , y ) ] * D k ( x , y ) .
E L ( x , y ) = O L ( x , y ) * A L ( x , y ) * D k ( x , y ) .
Φ tan - 1 [ r · a r 2 + ( a · s ) ] .

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