Abstract

A method for optimizing the aperture size in emission imaging is presented that takes into account limitations due to the Poisson nature of the detected radiation stream as well as the conspicuity limitation imposed by a spatially varying background. System assessment is based on the calculated performance of two model observers: the best linear observer, also called the Hotelling observer, and the nonprewhitening matched-filter observer. The tasks are the detection of a Gaussian signal and the discrimination of a single from a double Gaussian signal. When the background is specified, detection is optimized by enlarging the aperture; an inhomogeneous background results in an optimum aperture size matched naturally to the signal. The discrimination task has a finite optimum aperture for a flat background; a nonuniform background drives the optimum toward still-finer resolution.

© 1990 Optical Society of America

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References

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  1. H. H. Barrett, W. Swindell, Radiological Imaging: The Theory of Image Formation, Detection and Processing (Academic, New York, 1981).
  2. H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Vols. I, II, and III.
  3. A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971).
  4. B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
    [CrossRef] [PubMed]
  5. R. F. Wagner, D. G. Brown, C. E. Metz, “On the multiplex advantage of coded source/aperture photon imaging,” in Digital Radiography, W. Brody, ed., Proc. Soc. Photo-Opt. Instrum. Eng.314, 72–76 (1981).
    [CrossRef]
  6. J. L. Harris, “Resolving power and decision theory,” J. Opt. Soc. Am. 54, 606–611 (1964).
    [CrossRef]
  7. H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
    [CrossRef] [PubMed]
  8. R. F. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
    [CrossRef]
  9. H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
    [CrossRef]
  10. K. Fukunaga, Introduction to Statistical Pattern Recognition (Academic, New York, 1972).
  11. H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrumen. Eng.535, 65–69 (1985).
    [CrossRef]
  12. H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Application of Optical Instrumentation in Medicine XIV, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).
  13. W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3, 717–725 (1986).
    [CrossRef]
  14. R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
    [CrossRef] [PubMed]
  15. J. P. Rolland, Factors Influencing Lesion Detectability in Medical Imaging, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990).
  16. Z. H. Gu, S. Lee, “Optical implementation of the Hotelling trace criterion for image classification,” Opt. Eng. 23, 727–731 (1984).
    [CrossRef]
  17. H. H. Barrett, J. P. Rolland, R. F. Wagner, K. J. Myers, “Detection and discrimination of known signals in inhomogeneous random backgrounds,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 176–182 (1989).
    [CrossRef]
  18. R. Shaw, “Evaluating the efficiency of imaging processes,” Rep. Prog. Phys. 41, 1103–1155 (1978).
    [CrossRef]
  19. J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974).
  20. R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
    [CrossRef]
  21. A. E. Burgess, R. F. Wagner, R. J. Jennings, H. B. Barlow, “Efficiency of human visual discrimination,” Science 214, 93–94 (1981).
    [CrossRef] [PubMed]
  22. K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
    [CrossRef] [PubMed]
  23. K. J. Myers, R. F. Wagner, D. G. Brown, H. H. Barrett, “Efficient utilization of aperture and detector by optimal coding,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 164–175 (1989).
    [CrossRef]
  24. K. M. Hanson, “Method of evaluating image-recovery algorithms based on task performance,” J. Opt. Soc. Am. A 7, 1294–1304 (1990).
    [CrossRef]

1990 (2)

1987 (1)

1986 (1)

1985 (2)

1984 (1)

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

1981 (1)

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

1978 (2)

R. Shaw, “Evaluating the efficiency of imaging processes,” Rep. Prog. Phys. 41, 1103–1155 (1978).
[CrossRef]

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[CrossRef] [PubMed]

1964 (1)

1931 (1)

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Atkins, F. B.

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[CrossRef] [PubMed]

Barlow, H. B.

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

Barrett, H. H.

H. H. Barrett, “Objective assessment of image quality: effects of quantum noise and object variability,” J. Opt. Soc. Am. A 7, 1266–1278 (1990).
[CrossRef] [PubMed]

R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
[CrossRef] [PubMed]

W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3, 717–725 (1986).
[CrossRef]

K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
[CrossRef] [PubMed]

K. J. Myers, R. F. Wagner, D. G. Brown, H. H. Barrett, “Efficient utilization of aperture and detector by optimal coding,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 164–175 (1989).
[CrossRef]

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

H. H. Barrett, J. P. Rolland, R. F. Wagner, K. J. Myers, “Detection and discrimination of known signals in inhomogeneous random backgrounds,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 176–182 (1989).
[CrossRef]

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrumen. Eng.535, 65–69 (1985).
[CrossRef]

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Application of Optical Instrumentation in Medicine XIV, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

Beck, R. N.

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[CrossRef] [PubMed]

Borgstrom, M. C.

Brown, D. G.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

R. F. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[CrossRef]

R. F. Wagner, D. G. Brown, C. E. Metz, “On the multiplex advantage of coded source/aperture photon imaging,” in Digital Radiography, W. Brody, ed., Proc. Soc. Photo-Opt. Instrum. Eng.314, 72–76 (1981).
[CrossRef]

K. J. Myers, R. F. Wagner, D. G. Brown, H. H. Barrett, “Efficient utilization of aperture and detector by optimal coding,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 164–175 (1989).
[CrossRef]

Burgess, A. E.

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

R. F. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[CrossRef]

Dainty, J. C.

J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974).

Fiete, R. D.

R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
[CrossRef] [PubMed]

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrumen. Eng.535, 65–69 (1985).
[CrossRef]

Fukunaga, K.

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

Gu, Z. H.

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

Hanson, K. M.

Harris, J. L.

Hotelling, H.

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

Jennings, R. J.

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

Lee, S.

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

Metz, C. E.

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[CrossRef] [PubMed]

R. F. Wagner, D. G. Brown, C. E. Metz, “On the multiplex advantage of coded source/aperture photon imaging,” in Digital Radiography, W. Brody, ed., Proc. Soc. Photo-Opt. Instrum. Eng.314, 72–76 (1981).
[CrossRef]

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 Application of Optical Instrumentation in Medicine XIII, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrumen. Eng.535, 65–69 (1985).
[CrossRef]

Myers, K. J.

R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
[CrossRef] [PubMed]

K. J. Myers, H. H. Barrett, M. C. Borgstrom, D. D. Patton, G. W. Seeley, “Effect of noise correlation on detectability of disk signals in medical imaging,” J. Opt. Soc. Am. A 2, 1752–1759 (1985).
[CrossRef] [PubMed]

K. J. Myers, R. F. Wagner, D. G. Brown, H. H. Barrett, “Efficient utilization of aperture and detector by optimal coding,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 164–175 (1989).
[CrossRef]

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrumen. Eng.535, 65–69 (1985).
[CrossRef]

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Application of Optical Instrumentation in Medicine XIV, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

R. F. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[CrossRef]

H. H. Barrett, J. P. Rolland, R. F. Wagner, K. J. Myers, “Detection and discrimination of known signals in inhomogeneous random backgrounds,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 176–182 (1989).
[CrossRef]

Patton, D. D.

Rolland, J. P.

H. H. Barrett, J. P. Rolland, R. F. Wagner, K. J. Myers, “Detection and discrimination of known signals in inhomogeneous random backgrounds,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 176–182 (1989).
[CrossRef]

J. P. Rolland, Factors Influencing Lesion Detectability in Medical Imaging, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990).

Seeley, G. W.

Shaw, R.

R. Shaw, “Evaluating the efficiency of imaging processes,” Rep. Prog. Phys. 41, 1103–1155 (1978).
[CrossRef]

J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974).

Smith, W. E.

R. D. Fiete, H. H. Barrett, W. E. Smith, K. J. Myers, “The Hotelling trace criterion and its correlation with human observer performance,” J. Opt. Soc. Am. A 4, 945–953 (1987).
[CrossRef] [PubMed]

W. E. Smith, H. H. Barrett, “The Hotelling trace criterion as a figure of merit for the optimization of imaging systems,” J. Opt. Soc. Am. A 3, 717–725 (1986).
[CrossRef]

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrumen. Eng.535, 65–69 (1985).
[CrossRef]

Starr, S. J.

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[CrossRef] [PubMed]

Swindell, W.

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

Tapiovaara, M. J.

R. F. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[CrossRef]

Tsui, B. M. W.

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[CrossRef] [PubMed]

Van Trees, H. L.

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Vols. I, II, and III.

Wagner, R. F.

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

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

H. H. Barrett, J. P. Rolland, R. F. Wagner, K. J. Myers, “Detection and discrimination of known signals in inhomogeneous random backgrounds,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 176–182 (1989).
[CrossRef]

R. F. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[CrossRef]

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Application of Optical Instrumentation in Medicine XIV, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

R. F. Wagner, D. G. Brown, C. E. Metz, “On the multiplex advantage of coded source/aperture photon imaging,” in Digital Radiography, W. Brody, ed., Proc. Soc. Photo-Opt. Instrum. Eng.314, 72–76 (1981).
[CrossRef]

K. J. Myers, R. F. Wagner, D. G. Brown, H. H. Barrett, “Efficient utilization of aperture and detector by optimal coding,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 164–175 (1989).
[CrossRef]

Whalen, A. D.

A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971).

Ann. Math. Stat. (1)

H. Hotelling, “The generalization of Student’s ratio,” Ann. Math. Stat. 2, 360 (1931).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (5)

Opt. Eng. (1)

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

Phys. Med. Biol. (2)

B. M. W. Tsui, C. E. Metz, F. B. Atkins, S. J. Starr, R. N. Beck, “A comparison of optimum spatial resolution in nuclear imaging based on statistical theory and on observer performance,” Phys. Med. Biol. 23, 654–676 (1978).
[CrossRef] [PubMed]

R. F. Wagner, D. G. Brown, “Unified SNR analysis of medical imaging systems,” Phys. Med. Biol. 30, 489–518 (1985).
[CrossRef]

Rep. Prog. Phys. (1)

R. Shaw, “Evaluating the efficiency of imaging processes,” Rep. Prog. Phys. 41, 1103–1155 (1978).
[CrossRef]

Science (1)

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

Other (12)

K. J. Myers, R. F. Wagner, D. G. Brown, H. H. Barrett, “Efficient utilization of aperture and detector by optimal coding,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 164–175 (1989).
[CrossRef]

J. C. Dainty, R. Shaw, Image Science (Academic, London, 1974).

H. H. Barrett, J. P. Rolland, R. F. Wagner, K. J. Myers, “Detection and discrimination of known signals in inhomogeneous random backgrounds,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 176–182 (1989).
[CrossRef]

J. P. Rolland, Factors Influencing Lesion Detectability in Medical Imaging, Ph.D. dissertation (University of Arizona, Tucson, Ariz., 1990).

R. F. Wagner, K. J. Myers, D. G. Brown, M. J. Tapiovaara, A. E. Burgess, “Higher-order tasks: human vs. machine performance,” in Medical Imaging III: Image Formation, R. H. Schneider, S. J. Dwyer, R. G. Jost, eds., Proc. Soc. Photo-Opt. Instrum. Eng.1090, 183–194 (1989).
[CrossRef]

R. F. Wagner, D. G. Brown, C. E. Metz, “On the multiplex advantage of coded source/aperture photon imaging,” in Digital Radiography, W. Brody, ed., Proc. Soc. Photo-Opt. Instrum. Eng.314, 72–76 (1981).
[CrossRef]

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

H. L. Van Trees, Detection, Estimation, and Modulation Theory (Wiley, New York, 1968), Vols. I, II, and III.

A. D. Whalen, Detection of Signals in Noise (Academic, New York, 1971).

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

H. H. Barrett, W. E. Smith, K. J. Myers, T. D. Milster, R. D. Fiete, “Quantifying the performance of imaging systems,” in Application of Optical Instrumentation in Medicine XIII, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrumen. Eng.535, 65–69 (1985).
[CrossRef]

H. H. Barrett, K. J. Myers, R. F. Wagner, “Beyond signal-detection theory,” in Application of Optical Instrumentation in Medicine XIV, R. H. Schneider, S. J. Dwyer, eds., Proc. Soc. Photo-Opt. Instrum. Eng.626, 231–239 (1986).

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Figures (14)

Fig. 1
Fig. 1

Illustrative images of a signal on various backgrounds. The left-hand column shows Gaussian-pinhole (σp = 0.4σs) images of a Gaussian signal (5% contrast, σs = 10 units, centered in the object array) superimposed upon lumpy backgrounds with a correlation length of 30 units for Wf(0) = 0, 400, 4000, 40,000 from top to bottom. The second column shows the same objects imaged by a Gaussian aperture with σp = 0.8σs. The third and fourth columns show the same objects imaged by an aperture size σp that is equal to 1.6σs and 3.4σs, respectively. The parameters used here are not the same as on the graphs below but rather are chosen to demonstrate detection limitations due to both background inhomogeneity and to Poisson noise.

Fig. 2
Fig. 2

Family of Hotelling SNR2 curves as a function of the ratio of the Gaussian aperture width to the signal width for a detection task. The plots show the effect of increasing the value of the power spectral density of the background at zero spatial frequency, Wf(0). The signal width is 10 units, and the width of the autocorrelation function of the background is 30 units. The signal contrast is 5%.

Fig. 3
Fig. 3

Plot of the NPWMF SNR2 as a function of the Gaussian aperture width to the signal width for the same detection task as in Fig. 2.

Fig. 4
Fig. 4

Plot of the Hotelling SNR2 for the detection task with a square aperture of length L on a side. The abscissa is the ratio of the square aperture length to the signal width. The signal width is 10 units, and the width of the autocorrelation function of the background is 30 units. The signal contrast is 5%.

Fig. 5
Fig. 5

Plot of the NPWMF SNR2 as a function of the ratio of the side length of the square aperture to the signal width. The signal and background parameters are the same as for Fig. 4.

Fig. 6
Fig. 6

NPWMF SNR2 for a high-contrast detection task. The signal strength is 10 times the background, the signal width is 5 units, and the background autocorrelation length is 15 units.

Fig. 7
Fig. 7

Hotelling SNR2 as a function of exposure time for a detection task. The signal width is 10 units, and the background autocorrelation length is 30 units. The Gaussian aperture width is equal to the signal width, and the signal contrast is 5%.

Fig. 8
Fig. 8

NPWMF SNR2 versus exposure time for the same detection task as in Fig. 7.

Fig. 9
Fig. 9

Schematic of the Hotelling template along a radial axis in the space domain for the detection of a Gaussian object on a lumpy background imaged through a Gaussian aperture.

Fig. 10
Fig. 10

Schematic of the Fourier filter corresponding to the Hotelling template for the detection of a Gaussian object on a lumpy background imaged through a Gaussian aperture.

Fig. 11
Fig. 11

Plot of the Hotelling SNR2 for the Rayleigh discrimination task for Gaussian apertures of varying size σp. The source width is 10 units, and the pair separation is 20 units. The contrast and background parameters are the same as in Fig. 2.

Fig. 12
Fig. 12

Plot of the Hotelling SNR2 for the Rayleigh discrimination task for square apertures of varying length L on a side. The source width is 10 units, and the pair separation is 30 units. The contrast and background parameters are the same as in Fig. 2.

Fig. 13
Fig. 13

Plot of the NPWMF observer SNR2 for the Rayleigh resolution task with a Gaussian aperture, as described in Fig. 11.

Fig. 14
Fig. 14

Plot of the NPWMF observer SNR2 for the Rayleigh resolution task with a square aperture as described in Fig. 12.

Equations (73)

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g = Hf + n ,
f ( r ) = κ h ( r ) * * f ( r ) ,
κ = T 4 π ( l 1 + l 2 ) 2 .
s ( r ) = s ( 0 ) exp ( | r | 2 / 2 σ s 2 ) .
W f ( ρ ) = W f ( 0 ) exp ( 4 π 2 | ρ | 2 σ B 2 ) ,
λ = a t g ,
SNR λ 2 = d a 2 = Δ λ ¯ 2 σ λ 2 ,
S 1 = k = 1 2 P k [ g k g 0 ] [ g k g 0 ] t ,
g 0 = P 1 g 1 + P 2 g 2 .
S 1 = P 1 P 2 [ g 2 g 1 ] [ g 2 g 1 ] t
S 2 = P 1 C 1 + P 2 C 2 ,
C k = [ g g k ] [ g g k ] t .
λ Hot = [ g 2 g 1 ] t S 2 1 g = a Hot t g .
J = tr [ S 2 1 S 1 ] ,
J = P 1 P 2 SNR Hot 2 .
f = g n | f .
k = g k = f f | k = g n | f f | k ,
S 1 = ¼ ( 2 1 ) ( 2 1 ) t = ¼ ( Δ ) ( Δ ) t .
Δ = H ( f ¯ 2 f ¯ 1 ) = H Δ s ,
S 1 = ¼ ( H Δ s ) ( H Δ s ) t .
C k = ( g k ) ( g k ) t n | f f | k .
S 2 ( i , j ) = C ( i , j ) = R g ( i , j ) + κ A ap B ¯ δ i j ,
λ Hot = [ g 2 g 1 ] t S 2 1 g = Δ t S 2 1 g = ( H Δ s ) t C 1 g .
SNR Hot 2 = 4 J = 4 tr [ S 2 1 S 1 ] = tr [ C 1 ( H Δ s ) ( H Δ s ) t ] = Δ s t H t C 1 H Δ s .
SNR Hot 2 = d 2 ρ | Δ s ¯ ( ρ ) | 2 | H ( ρ ) | 2 [ κ A ap B ¯ + | H ( ρ ) | 2 W f ( ρ ) ] ,
NEQ ( ρ ) = | H ( ρ ) | 2 [ κ A ap B ¯ + | H ( ρ ) | 2 W f ( ρ ) ] .
λ npw = ( H Δ s ) t g .
SNR npw 2 = [ d 2 ρ | Δ s ( ρ ) | 2 | H ( ρ ) | 2 ] 2 d 2 ρ | Δ s ( ρ ) | 2 | H ( ρ ) | 2 κ A ap B ¯ + d 2 ρ | Δ s ( ρ ) | 2 | H ( ρ ) | 4 W f ( ρ ) .
SNR npw 2 = tr [ S 1 ] 2 tr [ S 1 S 2 ] .
SNR npw 2 α T 4 β T 3 + γ T 4 .
g = Hf + n .
C k = ( g k ) ( g k ) t n | f f | k ,
C = ( g f + f ) ( g f + f ) t n | f f .
M 1 = ( g f ) ( g f ) t n | f f ,
M 2 = ( f ) ( f ) t n | f f ,
M 3 = ( f ) ( g f ) t n | f f ,
M 4 = ( g f ) ( f ) t n | f f .
M 1 ( i , j ) = [ g ( i ) f ( i ) ] [ g ( j ) f ( j ) ] n | f f .
M 1 ( i , j ) = g ( i ) f ( i ) n | f g ( j ) f ( j ) n | f f = 0 i j .
M 1 ( i , i ) = [ g ( i ) f ( i ) ] 2 n | f f = f ( i ) f = ( i ) = κ A ap B ¯ .
M 2 ( i , j ) = ( i ) f ( i ) ] [ ( j ) f ( j ) ] n | f f = [ ( i ) f ( i ) ] [ ( j ) f ( j ) ] f = R g ( i , j ) ,
M 3 ( i , j ) = [ ( i ) f ( i ) ] [ g ( j ) f ( j ) ] n | f f = [ ( i ) f ( i ) ] [ g ( j ) f ( j ) ] n | f f = 0 ,
M 4 ( i , j ) = [ g ( i ) f ( i ) ] [ ( j ) f ( j ) ] n | f f = [ g ( i ) f ( i ) ] n | f [ ( j ) f ( j ) ] f = 0 .
C = ( i , j ) = R g ( i , j ) + κ A ap B ¯ δ i j ,
SNR Hot 2 = Δ s t H t C 1 H Δ s ,
C ( i , j ) = R g ( i , j ) + κ A ap B ¯ δ i j .
F Δ s = Δ s ,
Δ s = ( F Δ s ) Δ s t F .
FHF = H ,
H ( i , j ) = H ( i ) δ i j .
FCF = C ,
C ( i , j ) = [ W g ( i ) + κ A ap B ¯ ] δ i j = [ | H ( i ) | 2 W f ( i ) + κ A ap B ¯ ] δ i j
SNR Hot 2 = Δ s t H t C 1 H Δ s = Δ s t ( F F ) H t ( F F ) C 1 ( F F ) H ( F F ) Δ s = ( Δ s t F ) ( F H t F ) ( F C 1 F ) ( FHF ) ( F Δ s ) = Δ s H C 1 H Δ s .
SNR Hot 2 = i j k l Δ s * ( i ) H * ( i , j ) C 1 ( j , k ) H ( k , l ) Δ s ( l ) = i Δ s * ( i ) H * ( i ) C 1 ( i ) H ( i ) Δ s ( i ) = i | Δ s ( i ) | 2 | H ( i ) | 2 [ C ( i ) ] = i | Δ s ( i ) | 2 | H ( i ) | 2 [ κ A ap B + | H ( i ) | 2 W f ( i ) ] ,
SNR Hot 2 = d 2 ( ρ ) | Δ s ( ρ ) | 2 | H ( ρ ) | 2 [ κ A ap B ¯ + | H ( ρ ) | 2 W f ( ρ ) ] .
λ npw = Δ s t H t g .
SNR npw 2 = ( λ ¯ 2 λ ¯ 1 ) 2 ½ ( σ 1 2 + σ 2 2 ) .
SNR npw 2 = [ Δ λ ¯ ] 2 σ 2 .
λ ¯ k = λ n | f f | k .
λ ¯ k = Δ s t H t g n | f f | k = Δ s t H t k .
Δ λ ¯ = Δ s t H t Δ .
Δ λ ¯ = Δ s t H t H Δ s .
σ 2 = ( λ λ ¯ ) 2 n | f f .
σ 2 = ( λ λ ¯ f + λ ¯ f λ ¯ ) 2 n | f f = ( λ λ ¯ f ) 2 n | f f + ( λ ¯ λ ¯ ) 2 n | f f .
( λ λ ¯ f ) 2 n | f f = ( Δ s t H t g Δ s t H t f ) 2 n | f f = ( Δ s t H t ( g f ) ( g f ) t H Δ s n | f f = Δ s t H t ( g f ) ( g f ) t n | f f H Δ s = Δ s t H t M 1 H Δ s ,
M 1 = κ A ap B ¯ I
( λ ¯ f λ ¯ ) 2 n | f f = ( Δ s t H t f Δ s t H t ) 2 n | f f = Δ s t H t ( f ) ( f ) t n | f f H Δ s = Δ s t H t R g H Δ s ,
SNR npw 2 = ( Δ s t H t H Δ s ) 2 Δ s t H t ( κ A ap B ¯ I + R g ) H Δ s .
SNR npw 2 = ( Δ s H H Δ s ) 2 Δ s H ( κ A ap B ¯ I + W g ) H Δ s .
SNR npw 2 = [ i | Δ s ( i ) | 2 | H ( i ) | 2 ] 2 i | Δ s ( i ) | 2 | H ( i ) | 2 [ κ A ap B ¯ + W g ( i ) ] = [ i | Δ s ( i ) | 2 | H ( i ) | 2 ] 2 i | Δ s ( i ) | 2 | H ( i ) | 2 [ κ A ap B ¯ + | H ( i ) | 2 W f ( i ) ] .
SNR npw 2 = I 1 2 κ A ap B ¯ I 1 + I 2 ,
I 1 = d 2 ρ | Δ s ( ρ ) | 2 | H ( ρ ) | 2
I 2 = d 2 ρ | Δ s ( ρ ) | 2 | H ( ρ ) | 4 W f ( ρ ) .

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