Abstract

The efficiencies of the human observer and the channelized-Hotelling observer relative to the ideal observer for signal-detection tasks are discussed. Both signal-known-exactly (SKE) tasks and signal-known-statistically (SKS) tasks are considered. Signal location is uncertain for the SKS tasks, and lumpy backgrounds are used for background uncertainty in both cases. Markov chain Monte Carlo methods are employed to determine ideal-observer performance on the detection tasks. Psychophysical studies are conducted to compute human-observer performance on the same tasks. Efficiency is computed as the squared ratio of the detectabilities of the observer of interest to the ideal observer. Human efficiencies are approximately 2.1% and 24%, respectively, for the SKE and SKS tasks. The results imply that human observers are not affected as much as the ideal observer by signal-location uncertainty even though the ideal observer outperforms the human observer for both tasks. Three different simplified pinhole imaging systems are simulated, and the humans and the model observers rank the systems in the same order for both the SKE and the SKS tasks.

© 2005 Optical Society of America

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    [CrossRef] [PubMed]

2004 (2)

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Search for lesions in mammograms: statistical characterization of observer responses,” Med. Phys. 31, 24–36 (2004).
[CrossRef] [PubMed]

Y. Zhang, B. T. Pham, M. P. Eckstein, “Evaluation of JPEG 2000 encoder options: human and model observer detection of variable signals in X-ray coronary angiograms,” IEEE Trans. Med. Imaging 23, 613–632 (2004).
[CrossRef] [PubMed]

2003 (2)

2001 (4)

R. M. Manjeshwar, D. L. Wilson, “Effect of inherent location uncertainty on detection of stationary targets in noisy image sequences,” J. Opt. Soc. Am. A 18, 78–85 (2001).
[CrossRef]

A. E. Burgess, F. L. Jacobson, P. F. Judy, “Human observer detection experiments with mammograms and power-law noise,” Med. Phys. 28, 419–437 (2001).
[CrossRef] [PubMed]

F. A. Wichmann, N. J. Hill, “The psychometric function: I. Fitting, sampling, and goodness of fit,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

F. A. Wichmann, N. J. Hill, “The psychometric function: II. Bootstrap-based confidence intervals and sampling,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

1999 (1)

C. E. Metz, X. Pan, “Proper binormal ROC curves: theory and maximum-likelihood estimation,” J. Math. Psychol. 43, 1–33 (1999).
[CrossRef] [PubMed]

1998 (2)

C. E. Metz, B. A. Herman, J. H. Shen, “Maximum likelihood estimation of receiver operating characteristic (ROC) curves from continuously-distributed data,” Stat. Med. 17, 1033–1053 (1998).
[CrossRef] [PubMed]

H. H. Barrett, C. K. Abbey, E. Clarkson, “Objective assessment of image quality III: ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
[CrossRef]

1997 (1)

1992 (1)

1985 (1)

1984 (1)

1982 (1)

A. E. Burgess, R. F. Wagner, R. J. J. Jennings, “Statistical efficiency: a measure of human visual signal detection performance,” J. Appl. Photogr. Eng. 8, 76–78 (1982).

1981 (2)

R. G. Swensson, P. F. Judy, “Detection of noisy visual targets: Models for the effects of spatial uncertainty and signal-to-noise ratio,” Percept. Psychophys. 29, 521–534 (1981).
[CrossRef] [PubMed]

J. Nachmias, “On the psychometric function for contrast detection,” Vision Res. 21, 215–223 (1981).
[CrossRef] [PubMed]

1974 (1)

R. F. Quick, “A vector magnitude model of contrast detection,” Kybernetik 16, 65–67 (1974).
[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]

1951 (1)

W. Weibull, “Statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

Abbey, C.

H. H. Barrett, C. Abbey, B. Gallas, M. Eckstein, “Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 27–43 (1998).
[CrossRef]

Abbey, C. K.

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Search for lesions in mammograms: statistical characterization of observer responses,” Med. Phys. 31, 24–36 (2004).
[CrossRef] [PubMed]

H. H. Barrett, C. K. Abbey, E. Clarkson, “Objective assessment of image quality III: ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
[CrossRef]

A. E. Burgess, Xing Li, C. K. Abbey, “Visual signal detectability with two noise components: anomalous masking effects,” J. Opt. Soc. Am. A 14, 2420–2442 (1997).
[CrossRef]

M. P. Eckstein, C. K. Abbey, “Model observers for signal-known-statistically tasks (SKS),” in Medical Imaging 2001: Image Perception, E. A. Krupinski, D. P. Chakraborty, eds., Proc. SPIE4324, 91–102 (2001).
[CrossRef]

Barrett, H. H.

B. D. Gallas, H. H. Barrett, “Validating the use of channels to estimate the ideal linear observer,” J. Opt. Soc. Am. A 20, 1725–1738 (2003).
[CrossRef]

M. A. Kupinski, J. W. Hoppin, E. Clarkson, H. H. Barrett, “Ideal observer computation using Markov-chain Monte Carlo,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[CrossRef]

H. H. Barrett, C. K. Abbey, E. Clarkson, “Objective assessment of image quality III: ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
[CrossRef]

J. P. Rolland, H. H. Barrett, “Effect of random background inhomogeneity on observer detection performance,” J. Opt. Soc. Am. A 9, 649–658 (1992).
[CrossRef] [PubMed]

H. H. Barrett, C. Abbey, B. Gallas, M. Eckstein, “Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 27–43 (1998).
[CrossRef]

S. Park, M. A. Kupinski, E. Clarkson, H. H. Barrett, “Ideal-observer performance under signal and background uncertainty,” in Information Processing in Medical Imaging, Vol. 2732 of Lecture Notes in Computer Science, C. J. Taylor, J. A. Noble, eds. (Springer-Verlag, New York, 2003), pp. 342–353.

H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

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]

Bochud, F. O.

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Search for lesions in mammograms: statistical characterization of observer responses,” Med. Phys. 31, 24–36 (2004).
[CrossRef] [PubMed]

Burgess, A. E.

A. E. Burgess, F. L. Jacobson, P. F. Judy, “Human observer detection experiments with mammograms and power-law noise,” Med. Phys. 28, 419–437 (2001).
[CrossRef] [PubMed]

A. E. Burgess, Xing Li, C. K. Abbey, “Visual signal detectability with two noise components: anomalous masking effects,” J. Opt. Soc. Am. A 14, 2420–2442 (1997).
[CrossRef]

A. E. Burgess, H. Chandharian, “Visual signal detection. II. Signal-location identification,” J. Opt. Soc. Am. A 1, 906–910 (1984).
[CrossRef] [PubMed]

A. E. Burgess, R. F. Wagner, R. J. J. Jennings, “Statistical efficiency: a measure of human visual signal detection performance,” J. Appl. Photogr. Eng. 8, 76–78 (1982).

Chandharian, H.

Clarkson, E.

M. A. Kupinski, J. W. Hoppin, E. Clarkson, H. H. Barrett, “Ideal observer computation using Markov-chain Monte Carlo,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[CrossRef]

H. H. Barrett, C. K. Abbey, E. Clarkson, “Objective assessment of image quality III: ROC metrics, ideal observers, and likelihood-generating functions,” J. Opt. Soc. Am. A 15, 1520–1535 (1998).
[CrossRef]

S. Park, M. A. Kupinski, E. Clarkson, H. H. Barrett, “Ideal-observer performance under signal and background uncertainty,” in Information Processing in Medical Imaging, Vol. 2732 of Lecture Notes in Computer Science, C. J. Taylor, J. A. Noble, eds. (Springer-Verlag, New York, 2003), pp. 342–353.

Eckstein, M.

H. H. Barrett, C. Abbey, B. Gallas, M. Eckstein, “Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 27–43 (1998).
[CrossRef]

Eckstein, M. P.

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Search for lesions in mammograms: statistical characterization of observer responses,” Med. Phys. 31, 24–36 (2004).
[CrossRef] [PubMed]

Y. Zhang, B. T. Pham, M. P. Eckstein, “Evaluation of JPEG 2000 encoder options: human and model observer detection of variable signals in X-ray coronary angiograms,” IEEE Trans. Med. Imaging 23, 613–632 (2004).
[CrossRef] [PubMed]

M. P. Eckstein, C. K. Abbey, “Model observers for signal-known-statistically tasks (SKS),” in Medical Imaging 2001: Image Perception, E. A. Krupinski, D. P. Chakraborty, eds., Proc. SPIE4324, 91–102 (2001).
[CrossRef]

Gallas, B.

H. H. Barrett, C. Abbey, B. Gallas, M. Eckstein, “Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 27–43 (1998).
[CrossRef]

Gallas, B. D.

Gifford, H. C.

H. C. Gifford, P. H. Pretorius, M. A. King, “Comparison of human- and model-observer LROC studies,” in Medical Imaging 2003: Image Perception, D. P. Chakraborty, E. A. Krupinski, eds., Proc. SPIE5034, 112–122 (2003).
[CrossRef]

Herman, B. A.

C. E. Metz, B. A. Herman, J. H. Shen, “Maximum likelihood estimation of receiver operating characteristic (ROC) curves from continuously-distributed data,” Stat. Med. 17, 1033–1053 (1998).
[CrossRef] [PubMed]

Hill, N. J.

F. A. Wichmann, N. J. Hill, “The psychometric function: I. Fitting, sampling, and goodness of fit,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

F. A. Wichmann, N. J. Hill, “The psychometric function: II. Bootstrap-based confidence intervals and sampling,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

Hoppin, J. W.

Jacobson, F. L.

A. E. Burgess, F. L. Jacobson, P. F. Judy, “Human observer detection experiments with mammograms and power-law noise,” Med. Phys. 28, 419–437 (2001).
[CrossRef] [PubMed]

Jennings, R. J. J.

A. E. Burgess, R. F. Wagner, R. J. J. Jennings, “Statistical efficiency: a measure of human visual signal detection performance,” J. Appl. Photogr. Eng. 8, 76–78 (1982).

Judy, P. F.

A. E. Burgess, F. L. Jacobson, P. F. Judy, “Human observer detection experiments with mammograms and power-law noise,” Med. Phys. 28, 419–437 (2001).
[CrossRef] [PubMed]

R. G. Swensson, P. F. Judy, “Detection of noisy visual targets: Models for the effects of spatial uncertainty and signal-to-noise ratio,” Percept. Psychophys. 29, 521–534 (1981).
[CrossRef] [PubMed]

P. F. Judy, M. F. Kijewski, R. G. Swensson, “Observer detection performance loss: target size uncertainty,” in Medical Imaging 1997: Image Perception, E. A. Krupinski, D. P. Chakraborty, eds., Proc. SPIE3036, 39–47 (1997).
[CrossRef]

Kijewski, M. F.

P. F. Judy, M. F. Kijewski, R. G. Swensson, “Observer detection performance loss: target size uncertainty,” in Medical Imaging 1997: Image Perception, E. A. Krupinski, D. P. Chakraborty, eds., Proc. SPIE3036, 39–47 (1997).
[CrossRef]

King, M. A.

H. C. Gifford, P. H. Pretorius, M. A. King, “Comparison of human- and model-observer LROC studies,” in Medical Imaging 2003: Image Perception, D. P. Chakraborty, E. A. Krupinski, eds., Proc. SPIE5034, 112–122 (2003).
[CrossRef]

Kupinski, M. A.

M. A. Kupinski, J. W. Hoppin, E. Clarkson, H. H. Barrett, “Ideal observer computation using Markov-chain Monte Carlo,” J. Opt. Soc. Am. A 20, 430–438 (2003).
[CrossRef]

S. Park, M. A. Kupinski, E. Clarkson, H. H. Barrett, “Ideal-observer performance under signal and background uncertainty,” in Information Processing in Medical Imaging, Vol. 2732 of Lecture Notes in Computer Science, C. J. Taylor, J. A. Noble, eds. (Springer-Verlag, New York, 2003), pp. 342–353.

Li, Xing

Manjeshwar, R. M.

Metz, C. E.

C. E. Metz, X. Pan, “Proper binormal ROC curves: theory and maximum-likelihood estimation,” J. Math. Psychol. 43, 1–33 (1999).
[CrossRef] [PubMed]

C. E. Metz, B. A. Herman, J. H. Shen, “Maximum likelihood estimation of receiver operating characteristic (ROC) curves from continuously-distributed data,” Stat. Med. 17, 1033–1053 (1998).
[CrossRef] [PubMed]

Myers, K. J.

H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

Nachmias, J.

J. Nachmias, “On the psychometric function for contrast detection,” Vision Res. 21, 215–223 (1981).
[CrossRef] [PubMed]

Pan, X.

C. E. Metz, X. Pan, “Proper binormal ROC curves: theory and maximum-likelihood estimation,” J. Math. Psychol. 43, 1–33 (1999).
[CrossRef] [PubMed]

Park, S.

S. Park, M. A. Kupinski, E. Clarkson, H. H. Barrett, “Ideal-observer performance under signal and background uncertainty,” in Information Processing in Medical Imaging, Vol. 2732 of Lecture Notes in Computer Science, C. J. Taylor, J. A. Noble, eds. (Springer-Verlag, New York, 2003), pp. 342–353.

Pelli, D. G.

Pham, B. T.

Y. Zhang, B. T. Pham, M. P. Eckstein, “Evaluation of JPEG 2000 encoder options: human and model observer detection of variable signals in X-ray coronary angiograms,” IEEE Trans. Med. Imaging 23, 613–632 (2004).
[CrossRef] [PubMed]

Pretorius, P. H.

H. C. Gifford, P. H. Pretorius, M. A. King, “Comparison of human- and model-observer LROC studies,” in Medical Imaging 2003: Image Perception, D. P. Chakraborty, E. A. Krupinski, eds., Proc. SPIE5034, 112–122 (2003).
[CrossRef]

Quick, R. F.

R. F. Quick, “A vector magnitude model of contrast detection,” Kybernetik 16, 65–67 (1974).
[CrossRef]

Rolland, J. P.

Shen, J. H.

C. E. Metz, B. A. Herman, J. H. Shen, “Maximum likelihood estimation of receiver operating characteristic (ROC) curves from continuously-distributed data,” Stat. Med. 17, 1033–1053 (1998).
[CrossRef] [PubMed]

Swensson, R. G.

R. G. Swensson, P. F. Judy, “Detection of noisy visual targets: Models for the effects of spatial uncertainty and signal-to-noise ratio,” Percept. Psychophys. 29, 521–534 (1981).
[CrossRef] [PubMed]

P. F. Judy, M. F. Kijewski, R. G. Swensson, “Observer detection performance loss: target size uncertainty,” in Medical Imaging 1997: Image Perception, E. A. Krupinski, D. P. Chakraborty, eds., Proc. SPIE3036, 39–47 (1997).
[CrossRef]

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]

Wagner, R. F.

A. E. Burgess, R. F. Wagner, R. J. J. Jennings, “Statistical efficiency: a measure of human visual signal detection performance,” J. Appl. Photogr. Eng. 8, 76–78 (1982).

Weibull, W.

W. Weibull, “Statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

Wichmann, F. A.

F. A. Wichmann, N. J. Hill, “The psychometric function: I. Fitting, sampling, and goodness of fit,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

F. A. Wichmann, N. J. Hill, “The psychometric function: II. Bootstrap-based confidence intervals and sampling,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

Wilson, D. L.

Zhang, Y.

Y. Zhang, B. T. Pham, M. P. Eckstein, “Evaluation of JPEG 2000 encoder options: human and model observer detection of variable signals in X-ray coronary angiograms,” IEEE Trans. Med. Imaging 23, 613–632 (2004).
[CrossRef] [PubMed]

IEEE Trans. Med. Imaging (1)

Y. Zhang, B. T. Pham, M. P. Eckstein, “Evaluation of JPEG 2000 encoder options: human and model observer detection of variable signals in X-ray coronary angiograms,” IEEE Trans. Med. Imaging 23, 613–632 (2004).
[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. Appl. Mech. (1)

W. Weibull, “Statistical distribution function of wide applicability,” J. Appl. Mech. 18, 292–297 (1951).

J. Appl. Photogr. Eng. (1)

A. E. Burgess, R. F. Wagner, R. J. J. Jennings, “Statistical efficiency: a measure of human visual signal detection performance,” J. Appl. Photogr. Eng. 8, 76–78 (1982).

J. Math. Psychol. (1)

C. E. Metz, X. Pan, “Proper binormal ROC curves: theory and maximum-likelihood estimation,” J. Math. Psychol. 43, 1–33 (1999).
[CrossRef] [PubMed]

J. Opt. Soc. Am. A (8)

Kybernetik (1)

R. F. Quick, “A vector magnitude model of contrast detection,” Kybernetik 16, 65–67 (1974).
[CrossRef]

Med. Phys. (2)

A. E. Burgess, F. L. Jacobson, P. F. Judy, “Human observer detection experiments with mammograms and power-law noise,” Med. Phys. 28, 419–437 (2001).
[CrossRef] [PubMed]

F. O. Bochud, C. K. Abbey, M. P. Eckstein, “Search for lesions in mammograms: statistical characterization of observer responses,” Med. Phys. 31, 24–36 (2004).
[CrossRef] [PubMed]

Percept. Psychophys. (3)

F. A. Wichmann, N. J. Hill, “The psychometric function: I. Fitting, sampling, and goodness of fit,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

F. A. Wichmann, N. J. Hill, “The psychometric function: II. Bootstrap-based confidence intervals and sampling,” Percept. Psychophys. 63, 1314–1329 (2001).
[CrossRef]

R. G. Swensson, P. F. Judy, “Detection of noisy visual targets: Models for the effects of spatial uncertainty and signal-to-noise ratio,” Percept. Psychophys. 29, 521–534 (1981).
[CrossRef] [PubMed]

Stat. Med. (1)

C. E. Metz, B. A. Herman, J. H. Shen, “Maximum likelihood estimation of receiver operating characteristic (ROC) curves from continuously-distributed data,” Stat. Med. 17, 1033–1053 (1998).
[CrossRef] [PubMed]

Vision Res. (1)

J. Nachmias, “On the psychometric function for contrast detection,” Vision Res. 21, 215–223 (1981).
[CrossRef] [PubMed]

Other (7)

H. C. Gifford, P. H. Pretorius, M. A. King, “Comparison of human- and model-observer LROC studies,” in Medical Imaging 2003: Image Perception, D. P. Chakraborty, E. A. Krupinski, eds., Proc. SPIE5034, 112–122 (2003).
[CrossRef]

R. H. S. Carpenter, J. G. Robson, eds., Vision Research: A Practical Guide to Laboratory Methods (Oxford U. Press, New York, 1998).

H. H. Barrett, C. Abbey, B. Gallas, M. Eckstein, “Stabilized estimates of Hotelling-observer detection performance in patient-structured noise,” in Medical Imaging 1998: Image Perception, H. L. Kundel, ed., Proc. SPIE3340, 27–43 (1998).
[CrossRef]

H. H. Barrett, K. J. Myers, Foundations of Image Science (Wiley, New York, 2004).

S. Park, M. A. Kupinski, E. Clarkson, H. H. Barrett, “Ideal-observer performance under signal and background uncertainty,” in Information Processing in Medical Imaging, Vol. 2732 of Lecture Notes in Computer Science, C. J. Taylor, J. A. Noble, eds. (Springer-Verlag, New York, 2003), pp. 342–353.

P. F. Judy, M. F. Kijewski, R. G. Swensson, “Observer detection performance loss: target size uncertainty,” in Medical Imaging 1997: Image Perception, E. A. Krupinski, D. P. Chakraborty, eds., Proc. SPIE3036, 39–47 (1997).
[CrossRef]

M. P. Eckstein, C. K. Abbey, “Model observers for signal-known-statistically tasks (SKS),” in Medical Imaging 2001: Image Perception, E. A. Krupinski, D. P. Chakraborty, eds., Proc. SPIE4324, 91–102 (2001).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Gray-level histograms of three different lumpy backgrounds through the imaging systems A, B, and C.

Fig. 2
Fig. 2

Images of a fixed signal through the three imaging systems A, B, and C.

Fig. 3
Fig. 3

2AFC signal-detection task.

Fig. 4
Fig. 4

Psychometric curves for the ideal observer on the SKE and SKS tasks for each imaging system. The measured data points of observer performance and their error bars are provided.

Fig. 5
Fig. 5

Psychometric curves for the human observer on the SKE and SKS tasks for each imaging system. The measured data points of observer performance and their error bars are provided.

Fig. 6
Fig. 6

Psychometric curves for the eCHO on the SKE and SKS tasks for each imaging system. The measured data points of observer performance are provided.

Fig. 7
Fig. 7

Psychometric curves for each observer on the SKE tasks for imaging systems A, B, and C.

Fig. 8
Fig. 8

Psychometric curves for the ideal observer and the human observer on the SKS tasks for imaging systems A, B, and C.

Fig. 9
Fig. 9

Psychometric curves for each imaging system on the SKE tasks.

Fig. 10
Fig. 10

Psychometric curves for each imaging system on the SKS tasks.

Fig. 11
Fig. 11

Observer detectability for the SKE tasks. The measured data points of observer performance and their error bars are provided.

Fig. 12
Fig. 12

Observer detectability for the SKS tasks. The measured data points of observer performance and their error bars are provided.

Fig. 13
Fig. 13

The human-observer efficiency for the SKE and SKS tasks and the eCHO efficiency for the SKE tasks for imaging systems A, B, and C. (a) SKE, (b) SKS, (c) SKE.

Tables (1)

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Table 1 Characteristics of the Three Imaging Systems

Equations (45)

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g=Hf+n,
pr(g|g¯)=m=1Mexp(-g¯m) g¯mgmgm!.
gm=Sdrhm(r)f(r)+nm,
hm(r)=h2πw2exp-(r-pm)(r-pm)2w2.
H0 : g=Hfb+n,
H1 : g=H(fb+fs)+n.
fb=fb(r)=n=1NL(r-cn|a, s),
L(r-cn|a, s)=a exp-(r-c)(r-c)2s2.
fs=fs(r)=as exp{-[R(r-c)]D-1[R(r-c)]},
R=cosθsinθ-sinθcosθ,
D=2σ12002σ22.
Λ(g)=pr(g|H1)pr(g|H0),
Λ(g)=dbdspr(g|b, s, H1)pr(b)pr(s)dbpr(g|b, H0)pr(b).
Λ(g)=dαdθΛBSKE(g|b(θ), s(α))pr(θ|g, H0)pr(α),
ΛBSKE(g|b(θ), s(α))=pr(g|b(θ), s(α), H1)pr(g|b(θ), H0),
pr(θ|g, H0)=pr(g|b(θ), H0)pr(θ)dθpr(g|b(θ), H0)pr(θ).
Λˆ(g)=1J j=1JΛBSKE[g|b(θ(j)), s(α(j))],
α((θ(i), α(i)), (θ˜, α˜))
=min1, [pr(θ˜|g, H0)pr(α˜)][qb(θ(i)|θ˜)qs(α(i)|α˜)][pr(θ(i)|g, H0)pr(α(i))][qb(θ˜|θ(i))qs(α˜|α(i))].
pr(g|b(θ˜), H0)pr(θ˜)pr(g|b(θ(i)), H0)pr(θ(i)).
pr(g|b(θ), H0)pr(θ)=pr(g|b(θ), H0)pr(N)pr({cn})=m=1Mexp[-bm(θ)] bm(θ)gmgm!×exp(-N¯) N¯NN!N!MN.
AUC=dMgdMgpr(g|H0)pr(g|H1)step[θ(g)-θ(g)]
Pr(correct)=Pr[θ(g)>θ(g)]
=dMgdMgpr(g|H0)pr(g|H1)×step[θ(g)-θ(g)],
wg=Kg-1s,
s=g¯1-g¯0,  g¯j=g|Hj,  j=0, 1,
Kg=12[K0+K1],
Kj=(g-g¯j)(g-g¯j)t|Hj.
t=wgtg,
SNRHot2=stKg-1s.
v=Tg=T(Hf+n),
wv=Kv-1s¯v,
SNReCHO2=s¯vtKv-1s¯v,
Kv=TKgTt,
s¯v=svs,  sv=Ts.
Kv=12[Kv,0+Kv,1],
Kv,0=T[b¯I+Kb]Tt,
Kv,1=T[(b¯+s¯)I+Kb+Ks]Tt
=Kv,0+s¯TTt+TKsTt,
da2 erf-1(2AUC-1),
e=da()da(ideal)2.
p=ψ(as; α, β, γ, λ)=γ+(1-λ-γ)F(as; α, β),
L(Ω; y)=i=1Kniyiniψ(as,i; Ω)yini[1-ψ(as,i; Ω)](1-yi)ni.
F(x; α, β)=1-exp[-(x/α)β],  0x<,
W(λ)=1  on[0, 0.05]0  otherwise.

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