Abstract

We compared human detection of visual targets in noisy images with that of a theoretically optimum matched filter. Using a small thin target with vertically aligned markers, we obtained hyperefficient detection as high as 91% as compared with the theoretical optimum, a value far exceeding the 30–50% value typically reported. When the markers were removed, detection efficiencies degraded to an average of 27%, even though subjects were aware that the target was always placed in the center of a reasonably small panel. Using a nine-alternative forced-choice experiment, we compared detection by human observers with a matched-filter computational observer on a trial-by-trial basis. With the markers present, when humans missed the correct panel, they most often chose the panel with the second-highest decision variable output from the computational observer, suggesting that the template-matching model is a good one. To model results without the markers, we included location uncertainty and additional noise sources in the template matching of the computational observer. A location uncertainty of only 1 pixel, corresponding to a retinal distance of ≈12 μm, a dimension of the order of the size of the receptive field of photoreceptors, explained the psychometric data. With the marker present, the model suggests that hyperefficient detection is obtained by limiting target location uncertainty to <6 μm. Together these results give important new insights into human visual detection mechanisms.

© 2001 Optical Society of America

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  34. M. P. Eckstein, A. J. Ahumada, A. B. Watson, “Visual signal detection in structured backgrounds. II. Effects of contrast gain control, background variations, and white noise,” J. Opt. Soc. Am. A 14, 2406–2419 (1997).
    [CrossRef]
  35. Z.-L. Lu, B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764–778 (1999).
    [CrossRef]

2001

1999

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

Z.-L. Lu, B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764–778 (1999).
[CrossRef]

1998

P. Xue, D. L. Wilson, “Detection of moving objects in pulsed x-ray fluoroscopy,” J. Opt. Soc. Am. A 15, 375–388 (1998).
[CrossRef]

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

1997

1996

P. Xue, D. L. Wilson, “Pulsed fluoroscopy detectability from interspersed adaptive forced choice measurements,” Med. Phys. 23, 1833–1843 (1996).
[CrossRef] [PubMed]

R. Hubner, “The efficiency of different cue types for reducing spatial frequency uncertainty,” Vision Res. 36, 410–408 (1996).

1995

L.-P. Shiu, H. Pashler, “Spatial attention and vernier acuity,” Vision Res. 35, 337–343 (1995).
[CrossRef] [PubMed]

1994

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “Perceptual comparison of pulsed and continuous fluoroscopy,” Med. Phys. 21, 245–256 (1994).
[CrossRef] [PubMed]

1991

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

L. B. Stelmach, P. J. Hearty, “Requirements for static and dynamic spatial resolution in advanced television systems: a psychophysical evaluation,” J. Soc. Motion Pict. Television Eng. 100, 5–9 (1991).

1988

1987

1985

1984

1983

R. J. Watt, M. J. Morgan, R. M. Ward, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Perception Psychophys. 33, 20–28 (1983).
[CrossRef]

1982

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1982).
[CrossRef]

1981

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

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

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London 213, 451–477 (1981).
[CrossRef]

R. G. Swensson, P. 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]

1975

G. Westheimer, S. P. McKee, “Visual acuity in the presence of retinal-image motion,” J. Opt. Soc. Am. A 65, 847–850 (1975).
[CrossRef]

1968

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,” J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

Ahumada, A. J.

M. P. Eckstein, A. J. Ahumada, A. B. Watson, “Visual signal detection in structured backgrounds. II. Effects of contrast gain control, background variations, and white noise,” J. Opt. Soc. Am. A 14, 2406–2419 (1997).
[CrossRef]

B. L. Beard, A. J. Ahumada, “A technique to extract relevant image features for visual tasks,” in Human Vision and Electronic Imaging III, B. E. Rogowitz, T. N. Pappas, eds., Proc. SPIE3299, 79–85 (1998).
[CrossRef]

Aufrichtig, R.

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “Perceptual comparison of pulsed and continuous fluoroscopy,” Med. Phys. 21, 245–256 (1994).
[CrossRef] [PubMed]

Balz, G. Z.

G. Z. Balz, H. S. Hock, “The effect of attentional spread on spatial resolution,” Vision Res. 37, 1499–1510 (1997).
[CrossRef] [PubMed]

Barlow, H. B.

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

Beard, B. L.

B. L. Beard, A. J. Ahumada, “A technique to extract relevant image features for visual tasks,” in Human Vision and Electronic Imaging III, B. E. Rogowitz, T. N. Pappas, eds., Proc. SPIE3299, 79–85 (1998).
[CrossRef]

Burgess, A. E.

Cohn, T. E.

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

Colborne, B.

Davis, E. T.

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Perception Psychophys. 33, 20–28 (1983).
[CrossRef]

Dean, A. F.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1982).
[CrossRef]

Dosher, B. A.

Eckstein, M. P.

M. P. Eckstein, A. J. Ahumada, A. B. Watson, “Visual signal detection in structured backgrounds. II. Effects of contrast gain control, background variations, and white noise,” J. Opt. Soc. Am. A 14, 2406–2419 (1997).
[CrossRef]

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Detection and contrast discrimination of moving signals in uncorrelated Gaussian noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

Fahle, M.

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London 213, 451–477 (1981).
[CrossRef]

Geisler, W. S.

Ghandeharian, H.

Gilmore, G. C.

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “Perceptual comparison of pulsed and continuous fluoroscopy,” Med. Phys. 21, 245–256 (1994).
[CrossRef] [PubMed]

Graham, N.

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Perception Psychophys. 33, 20–28 (1983).
[CrossRef]

Green, D.

D. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, New York, 1974).

Hearty, P. J.

L. B. Stelmach, P. J. Hearty, “Requirements for static and dynamic spatial resolution in advanced television systems: a psychophysical evaluation,” J. Soc. Motion Pict. Television Eng. 100, 5–9 (1991).

Hock, H. S.

G. Z. Balz, H. S. Hock, “The effect of attentional spread on spatial resolution,” Vision Res. 37, 1499–1510 (1997).
[CrossRef] [PubMed]

Hubner, R.

R. Hubner, “The efficiency of different cue types for reducing spatial frequency uncertainty,” Vision Res. 36, 410–408 (1996).

Humphreys, G. W.

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

Jabri, K. N.

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

Jennings, R.

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

Judy, P.

R. G. Swensson, P. 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]

Kersten, D.

Kramer, P.

E. T. Davis, P. Kramer, N. Graham, “Uncertainty about spatial frequency, spatial position or contrast of visual patterns,” Perception Psychophys. 33, 20–28 (1983).
[CrossRef]

Kundel, H. L.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).

Lasley, D. J.

D. J. Lasley, T. E. Cohn, “Why luminance discrimination may be better than detection,” Vision Res. 21, 273–278 (1981).
[CrossRef] [PubMed]

Lauver, S.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).

Lavie, N.

Legge, G. E.

Lu, Z.-L.

Manjeshwar, R. M.

McDonough, R. N.

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise, 2nd ed. (Academic, San Diego, 1995).

McKee, S. P.

G. Westheimer, S. P. McKee, “Visual acuity in the presence of retinal-image motion,” J. Opt. Soc. Am. A 65, 847–850 (1975).
[CrossRef]

Morgan, M. J.

J. A. Solomon, N. Lavie, M. J. Morgan, “Contrast discrimination function: spatial cuing effects,” J. Opt. Soc. Am. A 14, 2443–2448 (1997).
[CrossRef]

R. J. Watt, M. J. Morgan, R. M. Ward, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

Movshon, J. A.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1982).
[CrossRef]

Muller, H. J.

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

Nodine, C. F.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).

Pashler, H.

L.-P. Shiu, H. Pashler, “Spatial attention and vernier acuity,” Vision Res. 35, 337–343 (1995).
[CrossRef] [PubMed]

Pelli, D. G.

Poggio, T.

M. Fahle, T. Poggio, “Visual hyperacuity: spatiotemporal interpolation in human vision,” Proc. R. Soc. London 213, 451–477 (1981).
[CrossRef]

Shiu, L.-P.

L.-P. Shiu, H. Pashler, “Spatial attention and vernier acuity,” Vision Res. 35, 337–343 (1995).
[CrossRef] [PubMed]

Solomon, J. A.

Stelmach, L. B.

L. B. Stelmach, P. J. Hearty, “Requirements for static and dynamic spatial resolution in advanced television systems: a psychophysical evaluation,” J. Soc. Motion Pict. Television Eng. 100, 5–9 (1991).

Swensson, R. G.

R. G. Swensson, P. 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]

Swets, J. A.

D. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Krieger, New York, 1974).

Thomas, C. W.

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “Perceptual comparison of pulsed and continuous fluoroscopy,” Med. Phys. 21, 245–256 (1994).
[CrossRef] [PubMed]

Thomas, J. P.

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Detection and contrast discrimination of moving signals in uncorrelated Gaussian noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

Tolhurst, D. J.

D. J. Tolhurst, J. A. Movshon, A. F. Dean, “The statistical reliability of signals in single neurons in cat and monkey visual cortex,” Vision Res. 23, 775–785 (1982).
[CrossRef]

Toto, L.

H. L. Kundel, C. F. Nodine, L. Toto, S. Lauver, “A circle cue enhances detection of simulated masses on mammogram backgrounds,” in Medical Imaging: Image Perception, H. L. Kundel, ed., Proc. SPIE3036, 81–84 (1997).

Wagner, R. F.

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

Ward, R. M.

R. J. Watt, M. J. Morgan, R. M. Ward, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

Watson, A. B.

Watt, R. J.

R. J. Watt, M. J. Morgan, R. M. Ward, “The use of different cues in vernier acuity,” Vision Res. 23, 991–995 (1983).
[CrossRef] [PubMed]

Westheimer, G.

G. Westheimer, S. P. McKee, “Visual acuity in the presence of retinal-image motion,” J. Opt. Soc. Am. A 65, 847–850 (1975).
[CrossRef]

Whalen, A. D.

R. N. McDonough, A. D. Whalen, Detection of Signals in Noise, 2nd ed. (Academic, San Diego, 1995).

Whiting, J. S.

M. P. Eckstein, J. S. Whiting, J. P. Thomas, “Detection and contrast discrimination of moving signals in uncorrelated Gaussian noise,” in Medical Imaging 1996: Image Perception, H. L. Kundel, ed., Proc. SPIE2712, 9–25 (1996).
[CrossRef]

Wickelgren, W. A.

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,” J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

Wilson, D. L.

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

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

P. Xue, D. L. Wilson, “Detection of moving objects in pulsed x-ray fluoroscopy,” J. Opt. Soc. Am. A 15, 375–388 (1998).
[CrossRef]

P. Xue, D. L. Wilson, “Pulsed fluoroscopy detectability from interspersed adaptive forced choice measurements,” Med. Phys. 23, 1833–1843 (1996).
[CrossRef] [PubMed]

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “Perceptual comparison of pulsed and continuous fluoroscopy,” Med. Phys. 21, 245–256 (1994).
[CrossRef] [PubMed]

Xue, P.

P. Xue, D. L. Wilson, “Detection of moving objects in pulsed x-ray fluoroscopy,” J. Opt. Soc. Am. A 15, 375–388 (1998).
[CrossRef]

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

P. Xue, D. L. Wilson, “Pulsed fluoroscopy detectability from interspersed adaptive forced choice measurements,” Med. Phys. 23, 1833–1843 (1996).
[CrossRef] [PubMed]

D. L. Wilson, P. Xue, R. Aufrichtig, “Perception of fluoroscopy last-image-hold,” Med. Phys. 21, 1875–1883 (1994).
[CrossRef] [PubMed]

R. Aufrichtig, P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “Perceptual comparison of pulsed and continuous fluoroscopy,” Med. Phys. 21, 245–256 (1994).
[CrossRef] [PubMed]

Behav. Res. Methods Instrum. Comput.

P. Xue, C. W. Thomas, G. C. Gilmore, D. L. Wilson, “An adaptive reference/test paradigm: application to pulsed fluoroscopy perception,” Behav. Res. Methods Instrum. Comput. 30, 332–348 (1998).
[CrossRef]

IEEE Trans. Med. Imaging

D. L. Wilson, K. N. Jabri, R. Aufrichtig, “Perception of temporally filtered x-ray fluoroscopy images,” IEEE Trans. Med. Imaging 18, 22–31 (1999).
[CrossRef] [PubMed]

J. Exp. Psychol.

H. J. Muller, G. W. Humphreys, “Luminance-increment detection: capacity limited or not?” J. Exp. Psychol. 17, 107–124 (1991).

J. Math. Psychol.

W. A. Wickelgren, “Unidimensional strength theory and component analysis of noise in absolute and comparative judgments,” J. Math. Psychol. 5, 102–122 (1968).
[CrossRef]

J. Opt. Soc. Am. A

G. Westheimer, S. P. McKee, “Visual acuity in the presence of retinal-image motion,” J. Opt. Soc. Am. A 65, 847–850 (1975).
[CrossRef]

Z.-L. Lu, B. A. Dosher, “Characterizing human perceptual inefficiencies with equivalent internal noise,” J. Opt. Soc. Am. A 16, 764–778 (1999).
[CrossRef]

W. S. Geisler, “Physical limits of acuity and hyperacuity,” J. Opt. Soc. Am. A 1, 775–782 (1984).
[CrossRef] [PubMed]

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Figures (7)

Fig. 1
Fig. 1

Detection model predictions. The ideal observer (1) has a linear psychometric function passing through the origin with unit slope. With location uncertainty, the curve is parallel but shifted to the right (2). There is characteristic nonlinear approach to the origin. The suboptimal observer with internal noise and sampling inefficiency has a linear psychometric function through the origin with a slope less than one (3). The suboptimal observer with the same internal noise, and sampling inefficiency, as well as location uncertainty, has a nonlinear psychometric function that is parallel to curve 3 at high signal energies, but shifted to the right (4). Details of computational methods are described in Subsection 3.C.2.

Fig. 2
Fig. 2

9AFC display. The display is a single image of 384×384 pixels divided into nine panels of equal area. The target for detection is placed randomly in one of the nine panels. When present, location markers appear in all nine panels as shown.

Fig. 3
Fig. 3

Contrast sensitivities with and without location markers. There is a significant (p<0.01) increase in contrast sensitivity in the presence of location markers with an average increase of 68±2%. Data with standard errors are plotted for three subjects at the three performance levels of 40%, 60%, and 80% probability correct. Standard errors were calculated as described previously.2

Fig. 4
Fig. 4

Average detection efficiencies as compared with the matched-filter computational observer. Data for three subjects with and without location markers are shown at three levels of probability correct. Collapsing data across subjects, the average detection efficiencies were 82±9% and 27±5% with and without the location markers, respectively.

Fig. 5
Fig. 5

Comparison of human responses to those from the ideal computational observer. Frequency of incorrect responses by the human observers are plotted as a function of the rank-ordered decision variable values from the computational observer. An analysis of variance followed by the Student–Neuman–Kuel test showed that subjects chose the panel with rank 2 significantly (p<0.05) more often than the rest of the panels. Without the marker (b), the analysis of variance was performed without considering the data with rank 1. The Student–Neuman–Kuel test showed that subjects were equally likely (p>0.05) to choose panels with ranks 2–7. However, these panels were chosen significantly (p<0.05) more often than the panels with ranks 8 and 9.

Fig. 6
Fig. 6

Linear fits to human psychometric data. Target signal-to-noise ratios, with (filled symbols) and without (open symbols) markers, were computed from the final target contrasts in the adaptive experiments. As a result, data points are spread along the x axis rather than the y axis. Regression lines are fit by minimizing the squared deviations from the target signal-to-noise ratio estimates. Fitting parameters are given in the text.

Fig. 7
Fig. 7

Model comparison to data with and without location uncertainty. The detectability index, d, is plotted as a function of target signal-to-noise ratio. The model was fit to data by manually adjusting parameters. Solid curves are for nominal parameters; dashed curves are for upper and lower bounds on parameter values. With the markers, curves are described by δ=0.82±0.02 and zero location uncertainty. Without the markers, the nominal curve is obtained with δ=0.49 and 1-pixel location uncertainty. Upper dashed curve, obtained with δ=0.51 and 1-pixel location uncertainty. Lower dashed curved parameters, δ=0.47 and 3-pixel location uncertainty.

Equations (5)

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dMF=(E/σext2)1/2,
dSO=(δE/σext2)1/2,
δ=Jσext2(1+K2)σext2+σint2,
L=Hj=1Pexp1σext2xySj(x, y)D(x, y),
C=gb-gt,

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