Abstract

A scalar figure of merit is presented that characterizes how well two or more object classes are separated for classification purposes. This scalar, the Hotelling trace criterion (HTC), can also be used as a measure of the performance of an imaging system in maintaining object-class separability. As an example, the HTC is applied to finding the optimum placement of pinholes in a simple coded-aperture system. Such a system could be used for tomographic imaging in nuclear medicine.

© 1986 Optical Society of America

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References

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  1. D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).
  2. C. E. Metz, “Basic principles of ROC analysis,” Semin. Nucl. Med. 8, 283–298 (1978).
    [CrossRef] [PubMed]
  3. J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–121 (1979).
    [CrossRef] [PubMed]
  4. W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
    [CrossRef]
  5. R. F. Wagner, “Decision theory and the detail signal-to-noise ratio of Otto Schade,” Photogr. Sci. Eng. 22, 41–46 (1978).
  6. A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
    [CrossRef] [PubMed]
  7. W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt. Soc. Am. A 2, 491–500 (1985).
    [CrossRef] [PubMed]
  8. W. E. Smith, “Simulated annealing and estimation theory in coded-aperture imaging,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1985).
  9. R. G. Paxman, “Coordinated design of restoration algorithm and coded aperture,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1984).
  10. Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
    [CrossRef]
  11. Y. Bizais, R. W. Rowe, I. G. Zubal, G. W. Bennett, A. B. Brill, “Coded aperture tomography revisited,” presented at Information Processing in Medical Imaging VIIIth IPMI Conference, Brussels, Belgium, August 29–September 2, 1983.
  12. M. D. Tipton, J. Dowdy, E. M. Stokely, “Background suppression of multiple pinhole-coded aperture scintigrams,” presented at the Fourth International Conference of the American Association of Physicists in Medicine on Medical Physics, Ottawa, July 1976.
  13. L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
    [PubMed]
  14. R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101 (1968).
    [CrossRef]
  15. R. G. Simpson, “Annular coded-aperture system for nuclear medicine,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1978).
  16. H. H. Barrett, “Fresnel zone plate imaging in nuclear medicine,”J. Nucl. Med. 13, 382 (1972).
    [PubMed]
  17. W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, “Application of a Fresnel zone plate to gamma-ray imaging,”J. Nucl. Med. 13, 612 (1972).
    [PubMed]
  18. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).
  19. H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).
  20. Z. H. Gu, S. H. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
    [CrossRef]
  21. H. H. Barrett, K. J. Myers, M. O. Stempski, R. F. Wagner, “Beyond signal detection theory,” presented at SPIE Medicine XIV, Newport Beach, Calif., February 1986.
  22. H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection, and Processing (Academic, New York, 1981), Vols. I and II.
  23. W. E. Smith, R. G. Paxman, H. H. Barrett, “Application of simulated annealing to coded-aperture design and tomographic reconstruction,”IEEE Trans. Nucl. Sci. NS-32, 758–761 (1985).
    [CrossRef]

1985 (3)

W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt. Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).

W. E. Smith, R. G. Paxman, H. H. Barrett, “Application of simulated annealing to coded-aperture design and tomographic reconstruction,”IEEE Trans. Nucl. Sci. NS-32, 758–761 (1985).
[CrossRef]

1984 (1)

Z. H. Gu, S. H. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

1983 (1)

Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
[CrossRef]

1981 (1)

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

1979 (1)

J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–121 (1979).
[CrossRef] [PubMed]

1978 (2)

C. E. Metz, “Basic principles of ROC analysis,” Semin. Nucl. Med. 8, 283–298 (1978).
[CrossRef] [PubMed]

R. F. Wagner, “Decision theory and the detail signal-to-noise ratio of Otto Schade,” Photogr. Sci. Eng. 22, 41–46 (1978).

1974 (1)

L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
[PubMed]

1972 (2)

H. H. Barrett, “Fresnel zone plate imaging in nuclear medicine,”J. Nucl. Med. 13, 382 (1972).
[PubMed]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, “Application of a Fresnel zone plate to gamma-ray imaging,”J. Nucl. Med. 13, 612 (1972).
[PubMed]

1968 (1)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101 (1968).
[CrossRef]

1958 (1)

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Barlow, H. B.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Barrett, H. H.

W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt. Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

W. E. Smith, R. G. Paxman, H. H. Barrett, “Application of simulated annealing to coded-aperture design and tomographic reconstruction,”IEEE Trans. Nucl. Sci. NS-32, 758–761 (1985).
[CrossRef]

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).

H. H. Barrett, “Fresnel zone plate imaging in nuclear medicine,”J. Nucl. Med. 13, 382 (1972).
[PubMed]

H. H. Barrett, K. J. Myers, M. O. Stempski, R. F. Wagner, “Beyond signal detection theory,” presented at SPIE Medicine XIV, Newport Beach, Calif., February 1986.

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

Beierwaltes, W. H.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, “Application of a Fresnel zone plate to gamma-ray imaging,”J. Nucl. Med. 13, 612 (1972).
[PubMed]

Bennett, G. W.

Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
[CrossRef]

Y. Bizais, R. W. Rowe, I. G. Zubal, G. W. Bennett, A. B. Brill, “Coded aperture tomography revisited,” presented at Information Processing in Medical Imaging VIIIth IPMI Conference, Brussels, Belgium, August 29–September 2, 1983.

Birdsall, T. G.

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Bizais, Y.

Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
[CrossRef]

Y. Bizais, R. W. Rowe, I. G. Zubal, G. W. Bennett, A. B. Brill, “Coded aperture tomography revisited,” presented at Information Processing in Medical Imaging VIIIth IPMI Conference, Brussels, Belgium, August 29–September 2, 1983.

Brill, A. B.

Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
[CrossRef]

Y. Bizais, R. W. Rowe, I. G. Zubal, G. W. Bennett, A. B. Brill, “Coded aperture tomography revisited,” presented at Information Processing in Medical Imaging VIIIth IPMI Conference, Brussels, Belgium, August 29–September 2, 1983.

Burgess, A. E.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Chang, L. T.

L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
[PubMed]

Dicke, R. H.

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101 (1968).
[CrossRef]

Dowdy, J.

M. D. Tipton, J. Dowdy, E. M. Stokely, “Background suppression of multiple pinhole-coded aperture scintigrams,” presented at the Fourth International Conference of the American Association of Physicists in Medicine on Medical Physics, Ottawa, July 1976.

Fiete, R. D.

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).

Fukunaga, K.

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

Green, D. M.

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

Gu, Z. H.

Z. H. Gu, S. H. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

Han, K. S.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, “Application of a Fresnel zone plate to gamma-ray imaging,”J. Nucl. Med. 13, 612 (1972).
[PubMed]

Jennings, R. J.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Jones, L. W.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, “Application of a Fresnel zone plate to gamma-ray imaging,”J. Nucl. Med. 13, 612 (1972).
[PubMed]

Kaplan, S. N.

L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
[PubMed]

Lee, S. H.

Z. H. Gu, S. H. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

Macdonald, B.

L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
[PubMed]

Metz, C. E.

C. E. Metz, “Basic principles of ROC analysis,” Semin. Nucl. Med. 8, 283–298 (1978).
[CrossRef] [PubMed]

Milster, T. D.

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).

Myers, K. J.

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).

H. H. Barrett, K. J. Myers, M. O. Stempski, R. F. Wagner, “Beyond signal detection theory,” presented at SPIE Medicine XIV, Newport Beach, Calif., February 1986.

Paxman, R. G.

W. E. Smith, R. G. Paxman, H. H. Barrett, “Application of simulated annealing to coded-aperture design and tomographic reconstruction,”IEEE Trans. Nucl. Sci. NS-32, 758–761 (1985).
[CrossRef]

W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt. Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

R. G. Paxman, “Coordinated design of restoration algorithm and coded aperture,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1984).

Perez-Mendez, V.

L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
[PubMed]

Rogers, W. L.

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, “Application of a Fresnel zone plate to gamma-ray imaging,”J. Nucl. Med. 13, 612 (1972).
[PubMed]

Rowe, R. W.

Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
[CrossRef]

Y. Bizais, R. W. Rowe, I. G. Zubal, G. W. Bennett, A. B. Brill, “Coded aperture tomography revisited,” presented at Information Processing in Medical Imaging VIIIth IPMI Conference, Brussels, Belgium, August 29–September 2, 1983.

Shiraishi, L.

L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
[PubMed]

Simpson, R. G.

R. G. Simpson, “Annular coded-aperture system for nuclear medicine,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1978).

Smith, W. E.

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).

W. E. Smith, R. G. Paxman, H. H. Barrett, “Image reconstruction from coded data: I. Reconstruction algorithms and experimental results,” J. Opt. Soc. Am. A 2, 491–500 (1985).
[CrossRef] [PubMed]

W. E. Smith, R. G. Paxman, H. H. Barrett, “Application of simulated annealing to coded-aperture design and tomographic reconstruction,”IEEE Trans. Nucl. Sci. NS-32, 758–761 (1985).
[CrossRef]

W. E. Smith, “Simulated annealing and estimation theory in coded-aperture imaging,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1985).

Stempski, M. O.

H. H. Barrett, K. J. Myers, M. O. Stempski, R. F. Wagner, “Beyond signal detection theory,” presented at SPIE Medicine XIV, Newport Beach, Calif., February 1986.

Stokely, E. M.

M. D. Tipton, J. Dowdy, E. M. Stokely, “Background suppression of multiple pinhole-coded aperture scintigrams,” presented at the Fourth International Conference of the American Association of Physicists in Medicine on Medical Physics, Ottawa, July 1976.

Swets, J. A.

J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–121 (1979).
[CrossRef] [PubMed]

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

Swindell, W.

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

Tanner, W. P.

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Tipton, M. D.

M. D. Tipton, J. Dowdy, E. M. Stokely, “Background suppression of multiple pinhole-coded aperture scintigrams,” presented at the Fourth International Conference of the American Association of Physicists in Medicine on Medical Physics, Ottawa, July 1976.

Wagner, R. F.

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

R. F. Wagner, “Decision theory and the detail signal-to-noise ratio of Otto Schade,” Photogr. Sci. Eng. 22, 41–46 (1978).

H. H. Barrett, K. J. Myers, M. O. Stempski, R. F. Wagner, “Beyond signal detection theory,” presented at SPIE Medicine XIV, Newport Beach, Calif., February 1986.

Zubal, I. G.

Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
[CrossRef]

Y. Bizais, R. W. Rowe, I. G. Zubal, G. W. Bennett, A. B. Brill, “Coded aperture tomography revisited,” presented at Information Processing in Medical Imaging VIIIth IPMI Conference, Brussels, Belgium, August 29–September 2, 1983.

Astrophys. J. (1)

R. H. Dicke, “Scatter-hole cameras for x-rays and gamma rays,” Astrophys. J. 153, L101 (1968).
[CrossRef]

IEEE Trans. Nucl. Sci. (2)

Y. Bizais, I. G. Zubal, R. W. Rowe, G. W. Bennett, A. B. Brill, “Dual seven pinhole tomography,”IEEE Trans. Nucl. Sci. NS-30, 703 (1983).
[CrossRef]

W. E. Smith, R. G. Paxman, H. H. Barrett, “Application of simulated annealing to coded-aperture design and tomographic reconstruction,”IEEE Trans. Nucl. Sci. NS-32, 758–761 (1985).
[CrossRef]

Invest. Radiol. (1)

J. A. Swets, “ROC analysis applied to the evaluation of medical imaging techniques,” Invest. Radiol. 14, 109–121 (1979).
[CrossRef] [PubMed]

J. Acoust. Soc. Am. (1)

W. P. Tanner, T. G. Birdsall, “Definitions of d′ and η as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

J. Nucl. Med. (3)

H. H. Barrett, “Fresnel zone plate imaging in nuclear medicine,”J. Nucl. Med. 13, 382 (1972).
[PubMed]

W. L. Rogers, K. S. Han, L. W. Jones, W. H. Beierwaltes, “Application of a Fresnel zone plate to gamma-ray imaging,”J. Nucl. Med. 13, 612 (1972).
[PubMed]

L. T. Chang, S. N. Kaplan, B. Macdonald, V. Perez-Mendez, L. Shiraishi, “A method of tomographic imaging using a multiple pinhole coded aperture,”J. Nucl. Med. 15, 1063 (1974).
[PubMed]

J. Opt. Soc. Am. A (1)

Opt. Eng. (1)

Z. H. Gu, S. H. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
[CrossRef]

Photogr. Sci. Eng. (1)

R. F. Wagner, “Decision theory and the detail signal-to-noise ratio of Otto Schade,” Photogr. Sci. Eng. 22, 41–46 (1978).

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

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Proc. Soc. Photo-Opt. Instrum. Eng. 535,Applications of Optical Instrumentation in Medicine XIII, 65–69 (1985).

Science (1)

A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual signal discrimination,” Science 214, 93–94 (1981).
[CrossRef] [PubMed]

Semin. Nucl. Med. (1)

C. E. Metz, “Basic principles of ROC analysis,” Semin. Nucl. Med. 8, 283–298 (1978).
[CrossRef] [PubMed]

Other (9)

D. M. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1966).

W. E. Smith, “Simulated annealing and estimation theory in coded-aperture imaging,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1985).

R. G. Paxman, “Coordinated design of restoration algorithm and coded aperture,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1984).

K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).

Y. Bizais, R. W. Rowe, I. G. Zubal, G. W. Bennett, A. B. Brill, “Coded aperture tomography revisited,” presented at Information Processing in Medical Imaging VIIIth IPMI Conference, Brussels, Belgium, August 29–September 2, 1983.

M. D. Tipton, J. Dowdy, E. M. Stokely, “Background suppression of multiple pinhole-coded aperture scintigrams,” presented at the Fourth International Conference of the American Association of Physicists in Medicine on Medical Physics, Ottawa, July 1976.

R. G. Simpson, “Annular coded-aperture system for nuclear medicine,” Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1978).

H. H. Barrett, K. J. Myers, M. O. Stempski, R. F. Wagner, “Beyond signal detection theory,” presented at SPIE Medicine XIV, Newport Beach, Calif., February 1986.

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

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Figures (5)

Fig. 1
Fig. 1

The coded-aperture geometry incorporating three views. If the object is a square of side 1 unit, each aperture is at a distance of 1.0 unit from the object center. All three apertures have identical pinhole patterns for a given pinhole permutation, with a separation of 1.0 unit between extreme pinhole candidate positions. The eight pinhole candidate positions are uniformly spaced. Each detector is 2.0 units long, a distance of 1.83 units from the object center.

Fig. 2
Fig. 2

A, The first 36 objects of the tumor class, which contains a total of 225 15 × 15 pixel objects. B, The first 36 objects of the tumorless class, which again contains a total of 225 15 × 15 pixel objects.

Fig. 3
Fig. 3

Plot of Jg versus aperture configuration for the first 127 permutations. The horizontal axis starts at 1, representing the pinhole code 0 0 0 0 0 0 0 1, and increases to 127, representing the pinhole code 0 1 1 1 1 1 1 1. All possible pinhole permutations occurring in between are in order of their binary-equivalent value. Tic marks represent cases where only one pinhole is open, i.e., 1, 2, 4, 8, 16, 32, and 64.

Fig. 4
Fig. 4

Plot of Jg versus aperture configuration for the remaining permutations. The horizontal axis starts at 128, representing the pinhole code 1 0 0 0 0 0 0 0, and increases to 255, representing the pinhole code 1 1 1 1 1 1 1 1. Again, all possible pinhole permutations occurring in between are in order of their binary-equivalent value. A tic mark represents the only single-pinhole case, namely at 128.

Fig. 5
Fig. 5

A, Four objects from the original object classes. The first and third objects are samples from the tumor class, and the second and fourth objects are samples from the tumorless class. B, Reconstructions of the original objects using the Monte Carlo algorithm, acting on data obtained with the worst aperture, for the case αβ = 111000.0. C, Reconstructions of the original objects using the Monte Carlo algorithm, acting on data obtained with the best aperture, again for the case αβ = 111000.0.

Tables (2)

Tables Icon

Table 1 The Apertures Corresponding to the Maximum and Minimum Jg Values

Tables Icon

Table 2 Jg Values for the Optimum Apertures at the Three Noise Levels

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

f ˜ k f k f ˜ k p ( f ˜ k ) ,
C k = ( f ˜ k - f ˜ k ) ( f ˜ k - f ˜ k ) T k f k ( f ˜ k - f ˜ k ) ( f ˜ k - f ˜ k ) T p ( f ˜ k ) ,
S 1 k = 1 K P k ( f ˜ k - f ˜ 0 ) ( f ˜ k - f ˜ 0 ) T ,
f ˜ 0 k = 1 K P k f ˜ k .
S 2 k = 1 K P k C k .
J Tr [ S 2 - 1 S 1 ] ,
g ˜ k = H f ˜ k + n ˜ k ,
g ˜ k = f ˜ k n ˜ k ( H f ˜ k + n ˜ k ) p ( n ˜ k f ˜ k ) p ( f ˜ k ) = H f ˜ k ,
g ˜ 0 = H f ˜ 0 .
S 1 g k = 1 K P k ( g ˜ k - g ˜ 0 ) ( g ˜ k - g ˜ 0 ) T = H S 1 H T ,
C g k f ˜ k n ˜ k ( g ˜ k - g ˜ k ) ( g ˜ k - g ˜ k ) T p ( n ˜ k f ˜ k ) p ( f ˜ k ) = f ˜ k n ˜ k [ H ( f ˜ k - f ˜ k ) + n ˜ k ] [ H ( f ˜ k - f ˜ k ) + n ˜ k ] T × p ( n ˜ k f ˜ k ) p ( f ˜ k ) = H C k H T + C n k ,
C n k f ˜ k n ˜ k n ˜ k n ˜ k T p ( n ˜ k f ˜ k ) p ( f ˜ k )
S 2 g = k = 1 K P k C g k = k = 1 K P k [ H C k H T + C n k ] = H S 2 H T + C n ,
C n k = 1 K P k C n k .
J g Tr [ S 2 g - 1 S 1 g ] ,
J g = Tr [ ( H S 2 H T ) - 1 H S 1 H T ] = Tr [ H T - 1 S 2 - 1 S 1 H T ] = Tr [ S 2 - 1 S 1 H T H T - 1 ] = J ;
f ˜ = 37 , 000 × A ρ V N f ˜ ,
H = ω T η H ,
ω = solid angle of pinhole 4 π π ( d 2 ) 2 cos θ 4 π R 2 = ( d 4 R ) 2 cos θ ,
p ( N ˜ ) = N ˜ N ˜ N ˜ ! exp ( - N ˜ ) ,
σ N 2 = N ˜ 2 - N ˜ 2 = N ˜ .
SNR N ˜ σ N ˜ = N ˜ .
g ˜ f ˜ k g ˜ k g ˜ k p ( g ˜ k f ˜ k ) = n ˜ k ( H f ˜ k + n ˜ k ) p ( n ˜ k f ˜ k ) = H f ˜ k ,
n ˜ k ( g k - g ˜ f ˜ k ) ( g ˜ k - g ˜ f ˜ k ) T p ( n ˜ k f ˜ k ) = diag ( g ˜ f ˜ k ) = diag ( H f ˜ k ) ,
n ˜ k = g ˜ k - H f ˜ k = g ˜ k - g ˜ f ˜ k ,
n ˜ k n ˜ k n ˜ k T p ( n ˜ k f ˜ k ) = diag ( H f ˜ k ) ,
C n k = f ˜ k diag ( H f ˜ k ) p ( f ˜ k ) = diag ( H f ˜ k ) .
C n = k = 1 K P k C n k = k = 1 K P k diag ( H f ˜ k ) = diag ( H f ˜ 0 ) ,
α 37 , 000 × A ρ V N
β ω T η .
S 1 = α 2 S 1
S 2 = α 2 S 2 ,
J g = Tr { [ α 2 β 2 H S 2 H T + ( α β ) diag ( H f ˜ 0 ) ] - 1 α 2 β 2 H S 1 H T } = Tr { [ H S 2 H T + ( 1 α β ) diag ( H f ˜ 0 ) ] - 1 H S 1 H T } .

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