Abstract

Several studies have investigated the effect of signal location uncertainty on the detectability of simple visual signals in uncorrelated Gaussian noise with a deterministic background. For this case, human performance in locating a signal in a forced-choice experiment has been successfully predicted for 2–1800 alternative locations with the use of signal detection theory and the usual assumption that the observer’s internal response is Gaussian distributed. Gaussian uncorrelated noise is far from realistic medical image noise, which includes not only fluctuations in intensity of quantum origin but also other anatomical objects lying in the x-ray path (structured backgrounds). Our goal is to determine whether signal detection theory with the Gaussian assumption is adequate for the case of structured backgrounds, or whether other more complex models need to be developed to predict human performance as a function of the number of possible signal locations in structured backgrounds. We present experimental data suggesting that an assumed Gaussian internal response accurately predicts the decrease in observer performance as the number of alternative locations is increased. The one exception is a lower-than-predicted performance for the detection of low-contrast signals for two alternative locations. Performance as measured by the index of detectability d′ is also found to be linear with signal contrast. Together these findings extend the applicability of signal detection theory with Gaussian internal response functions to the case of complex structured backgrounds.

© 1996 Optical Society of America

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    [CrossRef] [PubMed]

1995 (4)

M. P. Eckstein, J. S. Whiting, “Lesion detection in structured noise,” Acad. Radiol. 3, 249–253 (1995).
[CrossRef]

A. E. Burgess, “Comparison of receiver operating characteristics and forced choice observer performance measurement methods,” Med. Phys. 22, 643–655 (1995).
[CrossRef] [PubMed]

J. Palmer, “Attention in visual search: distinguishing four cases of a set-size effect,” Cur. Dir. Psychol. Sci. 4, 118–123 (1995).
[CrossRef]

W. S. Geisler, D. G. Albrecht, “Cortical neurons: isolation of contrast gain control,” Vision Res. 32, 1409–1410 (1995).
[CrossRef]

1994 (3)

G. Zelinski, D. Sheinberg, H. Butloff, “Eye movements during visual search,” Invest. Ophthalmol. Vis. Sci. (Suppl.) 34, 2617 (1994).

M. P. Eckstein, J. S. Whiting, A. E. Burgess, J. P. Thomas, S. S. Shimozaki, “Signal positional uncertainty and signals in dynamic noise,” Opt. Photon. News 5(8), 128 (1994) (OSA Annual Meeting Suppl.).

N. Eigler, M. P. Eckstein, D. Honig, J. S. Whiting, “Improving detection of coronary morphologic features from digital angiograms: effect of stenosis stabilized display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

1993 (1)

C. Hermann, E. Buhr, D. Hoeschen, S. Y. Fan, “Comparison of ROC and AFC methods in a visual detection task,” Med. Phys. 3, 805–812 (1993).
[CrossRef]

1992 (2)

1988 (2)

A. E. Burgess, B. Colborne, “Visual signal detection IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617–627 (1988).
[CrossRef] [PubMed]

J. A. Hanley, “The robustness of the “binormal” assumption used in fitting ROC curves,” Med. Decis. Making 8, 197–203 (1988).
[CrossRef] [PubMed]

1987 (1)

1986 (1)

K. Ohara, H. P. Chan, K. Doi, M. L. Giger, H. Fujita, “Investigation of basic imaging properties in digital radiography: detection of simulated low objects in digital subtraction angiographic images,” Med. Phys. 13, 304–311 (1986).
[CrossRef] [PubMed]

1985 (2)

1984 (2)

A. E. Burgess, H. Ghandeharian, “Visual signal detection. I. Ability to use phase information,” J. Opt. Soc. Am. A 1, 900–905 (1984).
[CrossRef] [PubMed]

A. E. Burgess, H. Ghandeharian, “Visual signal detection. II. Signal location identification,” J. Opt. Soc. Am. A 1, 905–910 (1984).
[CrossRef]

1983 (1)

M. S. Chester, A. H. Hay, “Quantitative relation between detectability and noise power,” Phys. Med. Biol. 28, 1113–1125 (1983).
[CrossRef]

1982 (1)

D. J. Tolhurst, J. A. Movshson, 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 (3)

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

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]

J. M. Foley, G. E. Legge, “Contrast detection and near threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef]

1980 (1)

1978 (1)

B. Wandell, R. D. Luce, “Pooling peripheral information: averages versus extreme values,” J. Math Psychol. 17, 220–235 (1978).
[CrossRef]

1974 (2)

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

J. Nachmias, R. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

1968 (1)

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

1954 (1)

W. W. Peterson, T. G. Birsdall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theor. PGIT-4, 171–212 (1954).
[CrossRef]

Albrecht, D. G.

W. S. Geisler, D. G. Albrecht, “Cortical neurons: isolation of contrast gain control,” Vision Res. 32, 1409–1410 (1995).
[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.

Birsdall, T. G.

W. W. Peterson, T. G. Birsdall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theor. PGIT-4, 171–212 (1954).
[CrossRef]

Bochud, F. O.

F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. Kundel, ed., Proc. SPIE2436, 156–164 (1995).
[CrossRef]

Buhr, E.

C. Hermann, E. Buhr, D. Hoeschen, S. Y. Fan, “Comparison of ROC and AFC methods in a visual detection task,” Med. Phys. 3, 805–812 (1993).
[CrossRef]

Burgess, A.

Burgess, A. E.

A. E. Burgess, “Comparison of receiver operating characteristics and forced choice observer performance measurement methods,” Med. Phys. 22, 643–655 (1995).
[CrossRef] [PubMed]

M. P. Eckstein, J. S. Whiting, A. E. Burgess, J. P. Thomas, S. S. Shimozaki, “Signal positional uncertainty and signals in dynamic noise,” Opt. Photon. News 5(8), 128 (1994) (OSA Annual Meeting Suppl.).

A. E. Burgess, B. Colborne, “Visual signal detection IV. Observer inconsistency,” J. Opt. Soc. Am. A 5, 617–627 (1988).
[CrossRef] [PubMed]

A. E. Burgess, “Visual signal detection. III. On Bayesian use of prior knowledge and cross correlation,” J. Opt. Soc. Am. A 2, 1498–1507 (1985).
[CrossRef] [PubMed]

A. E. Burgess, H. Ghandeharian, “Visual signal detection. I. Ability to use phase information,” J. Opt. Soc. Am. A 1, 900–905 (1984).
[CrossRef] [PubMed]

A. E. Burgess, H. Ghandeharian, “Visual signal detection. II. Signal location identification,” J. Opt. Soc. Am. A 1, 905–910 (1984).
[CrossRef]

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]

A. E. Burgess, “On observer internal noise,” in Application of Optical Instrumentation in Medicine XIV and Picture Archiving and Communication Systems (PACS IV) for Medical Applications IV, S. J. Dwyer, R. H. Schneider, eds. Proc. SPIE626, 208–213 (1986).

A. E. Burgess, “Detection and identification efficiency: an update,” in Application of Optical Instrumentation in Medicine XIII: Medical Image Production, Processing, and Display, S. J. Dwyer, R. H. Schneider, eds., Proc. SPIE535, 50–56 (1985).
[CrossRef]

Butloff, H.

G. Zelinski, D. Sheinberg, H. Butloff, “Eye movements during visual search,” Invest. Ophthalmol. Vis. Sci. (Suppl.) 34, 2617 (1994).

Chan, H. P.

K. Ohara, H. P. Chan, K. Doi, M. L. Giger, H. Fujita, “Investigation of basic imaging properties in digital radiography: detection of simulated low objects in digital subtraction angiographic images,” Med. Phys. 13, 304–311 (1986).
[CrossRef] [PubMed]

Chester, M. S.

M. S. Chester, A. H. Hay, “Quantitative relation between detectability and noise power,” Phys. Med. Biol. 28, 1113–1125 (1983).
[CrossRef]

Colborne, B.

Dean, A. F.

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

Doi, K.

K. Ohara, H. P. Chan, K. Doi, M. L. Giger, H. Fujita, “Investigation of basic imaging properties in digital radiography: detection of simulated low objects in digital subtraction angiographic images,” Med. Phys. 13, 304–311 (1986).
[CrossRef] [PubMed]

Eckstein, M. P.

M. P. Eckstein, J. S. Whiting, “Lesion detection in structured noise,” Acad. Radiol. 3, 249–253 (1995).
[CrossRef]

N. Eigler, M. P. Eckstein, D. Honig, J. S. Whiting, “Improving detection of coronary morphologic features from digital angiograms: effect of stenosis stabilized display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

M. P. Eckstein, J. S. Whiting, A. E. Burgess, J. P. Thomas, S. S. Shimozaki, “Signal positional uncertainty and signals in dynamic noise,” Opt. Photon. News 5(8), 128 (1994) (OSA Annual Meeting Suppl.).

M. P. Eckstein, C. A. Morioka, J. S. Whiting, N. Eigler, “Psychophysical evaluation of the effect of JPEG, full-frame DCT and wavelet image compression on signal detection in medical image noise,” in Medical Imaging 1995: Image Perception, H. Kundel, ed., Proc. SPIE2436, 79–89 (1995).
[CrossRef]

J. S. Whiting, M. P. Eckstein, S. Einav, N. Eigler, “Perceptual evaluation of JPEG compression for medical dynamic image sequences,” in Annual Meeting, Vol. 23 of 1992 OSA, Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 160.

M. P. Eckstein, J. S. Whiting, J. P. Thomas, S. S. Shimozaki, “Are response times a reliable measure of information accrual in data-limited tasks?” submitted to J. Exp. Psychol. Hum. Perform. Percept.

Eigler, N.

N. Eigler, M. P. Eckstein, D. Honig, J. S. Whiting, “Improving detection of coronary morphologic features from digital angiograms: effect of stenosis stabilized display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [PubMed]

M. P. Eckstein, C. A. Morioka, J. S. Whiting, N. Eigler, “Psychophysical evaluation of the effect of JPEG, full-frame DCT and wavelet image compression on signal detection in medical image noise,” in Medical Imaging 1995: Image Perception, H. Kundel, ed., Proc. SPIE2436, 79–89 (1995).
[CrossRef]

J. S. Whiting, M. P. Eckstein, S. Einav, N. Eigler, “Perceptual evaluation of JPEG compression for medical dynamic image sequences,” in Annual Meeting, Vol. 23 of 1992 OSA, Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 160.

Einav, S.

J. S. Whiting, M. P. Eckstein, S. Einav, N. Eigler, “Perceptual evaluation of JPEG compression for medical dynamic image sequences,” in Annual Meeting, Vol. 23 of 1992 OSA, Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 160.

Fan, S. Y.

C. Hermann, E. Buhr, D. Hoeschen, S. Y. Fan, “Comparison of ROC and AFC methods in a visual detection task,” Med. Phys. 3, 805–812 (1993).
[CrossRef]

Foley, J. M.

J. M. Foley, G. E. Legge, “Contrast detection and near threshold discrimination in human vision,” Vision Res. 21, 1041–1053 (1981).
[CrossRef]

Foley, M.

Fox, W. C.

W. W. Peterson, T. G. Birsdall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theor. PGIT-4, 171–212 (1954).
[CrossRef]

Fujita, H.

K. Ohara, H. P. Chan, K. Doi, M. L. Giger, H. Fujita, “Investigation of basic imaging properties in digital radiography: detection of simulated low objects in digital subtraction angiographic images,” Med. Phys. 13, 304–311 (1986).
[CrossRef] [PubMed]

Geisler, W. S.

W. S. Geisler, D. G. Albrecht, “Cortical neurons: isolation of contrast gain control,” Vision Res. 32, 1409–1410 (1995).
[CrossRef]

Ghandeharian, H.

A. E. Burgess, H. Ghandeharian, “Visual signal detection. II. Signal location identification,” J. Opt. Soc. Am. A 1, 905–910 (1984).
[CrossRef]

A. E. Burgess, H. Ghandeharian, “Visual signal detection. I. Ability to use phase information,” J. Opt. Soc. Am. A 1, 900–905 (1984).
[CrossRef] [PubMed]

Giger, M. L.

K. Ohara, H. P. Chan, K. Doi, M. L. Giger, H. Fujita, “Investigation of basic imaging properties in digital radiography: detection of simulated low objects in digital subtraction angiographic images,” Med. Phys. 13, 304–311 (1986).
[CrossRef] [PubMed]

Graber, M. A.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

Green, D. M.

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

Gumbell, E. J.

E. J. Gumbell, Statistics of Extremes (Columbia U. Press, New York, 1958).

Hanley, J. A.

J. A. Hanley, “The robustness of the “binormal” assumption used in fitting ROC curves,” Med. Decis. Making 8, 197–203 (1988).
[CrossRef] [PubMed]

Harris, R. J.

R. J. Harris, A Primer of Multivariate Statistics (Academic, New York, 1985).

Hay, A. H.

M. S. Chester, A. H. Hay, “Quantitative relation between detectability and noise power,” Phys. Med. Biol. 28, 1113–1125 (1983).
[CrossRef]

Heeger, D. J.

D. J. Heeger, “Normalization of cell responses in cat visual cortex,” Vis. Neurosci. 9, 181–197 (1992).
[CrossRef] [PubMed]

Hermann, C.

C. Hermann, E. Buhr, D. Hoeschen, S. Y. Fan, “Comparison of ROC and AFC methods in a visual detection task,” Med. Phys. 3, 805–812 (1993).
[CrossRef]

Hessler, C.

F. O. Bochud, F. R. Verdun, C. Hessler, J. F. Valley, “Detectability on radiological images: the influence of anatomical noise,” in Medical Imaging 1995: Image Perception, H. Kundel, ed., Proc. SPIE2436, 156–164 (1995).
[CrossRef]

Hoeschen, D.

C. Hermann, E. Buhr, D. Hoeschen, S. Y. Fan, “Comparison of ROC and AFC methods in a visual detection task,” Med. Phys. 3, 805–812 (1993).
[CrossRef]

Honig, D.

N. Eigler, M. P. Eckstein, D. Honig, J. S. Whiting, “Improving detection of coronary morphologic features from digital angiograms: effect of stenosis stabilized display,” Circulation 89, 2700–2709 (1994).
[CrossRef] [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]

Judy, P. F.

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

R. G. Swensson, P. F. Judy, “Measuring performance efficiency and consistency in visual discriminations with noisy images,” J. Exp. Psychol. Hum. Percept. Perform. (to be published).
[PubMed]

R. G. Swensson, P. F. Judy, “Background area effects on feature detectability in CT and uncorrelated noise,” presented at the 73rd Annual Meeting of the Radiological Society of North America, Chicago, III., December 4, 1987.

Kersten, D.

Kundel, H. L.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

H. L. Kundel, “Medical image perception,” in Proceedings of the Conference on Developing a Long-Term Plan for Imaging Research, B. L. Holman, S. H. Edwards, eds. (National Institutes of Health, Bethesda, MD, 1994), pp. 39–42.

Legge, G. E.

Luce, R. D.

B. Wandell, R. D. Luce, “Pooling peripheral information: averages versus extreme values,” J. Math Psychol. 17, 220–235 (1978).
[CrossRef]

Morioka, C. A.

M. P. Eckstein, C. A. Morioka, J. S. Whiting, N. Eigler, “Psychophysical evaluation of the effect of JPEG, full-frame DCT and wavelet image compression on signal detection in medical image noise,” in Medical Imaging 1995: Image Perception, H. Kundel, ed., Proc. SPIE2436, 79–89 (1995).
[CrossRef]

Movshson, J. A.

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

Nachmias, J.

J. Nachmias, R. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
[CrossRef] [PubMed]

Ohara, K.

K. Ohara, H. P. Chan, K. Doi, M. L. Giger, H. Fujita, “Investigation of basic imaging properties in digital radiography: detection of simulated low objects in digital subtraction angiographic images,” Med. Phys. 13, 304–311 (1986).
[CrossRef] [PubMed]

Palmer, J.

J. Palmer, “Attention in visual search: distinguishing four cases of a set-size effect,” Cur. Dir. Psychol. Sci. 4, 118–123 (1995).
[CrossRef]

Pelli, D. G.

Peterson, W. W.

W. W. Peterson, T. G. Birsdall, W. C. Fox, “The theory of signal detectability,” IRE Trans. Inf. Theor. PGIT-4, 171–212 (1954).
[CrossRef]

Revesz, G.

G. Revesz, H. L. Kundel, M. A. Graber, “The influence of structured noise on detection of radiological abnormalities,” Invest. Radiol. 9, 479–486 (1974).
[CrossRef] [PubMed]

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J. S. Whiting, M. P. Eckstein, S. Einav, N. Eigler, “Perceptual evaluation of JPEG compression for medical dynamic image sequences,” in Annual Meeting, Vol. 23 of 1992 OSA, Technical Digest Series (Optical Society of America, Washington, D.C., 1992), p. 160.

M. P. Eckstein, C. A. Morioka, J. S. Whiting, N. Eigler, “Psychophysical evaluation of the effect of JPEG, full-frame DCT and wavelet image compression on signal detection in medical image noise,” in Medical Imaging 1995: Image Perception, H. Kundel, ed., Proc. SPIE2436, 79–89 (1995).
[CrossRef]

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Figures (13)

Fig. 1
Fig. 1

Left, Gaussian uncorrelated noise, which can be taken as an approximation of image noise of quantum origin that is Poisson distributed. Previous studies have investigated human performance in detecting and identifying signals in Gaussian uncorrelated noise as a function of basic image properties. Center, sample of noise spatially correlated by convolution with a Gaussian point-spread function. Right, sample of structured background from a digital x-ray coronary image consisting of other anatomical objects and texture that is irrelevant to the task.

Fig. 2
Fig. 2

Index of detectability based on the Gaussian assumption d′ as a function of possible number of signal locations M for an underlying rectangular response distribution. If the distribution were close to Gaussian, then the functions would be constant as a function of number of possible locations. Departures from the d′ constancy can be used to assess the adequacy of the Gaussian assumption.

Fig. 3
Fig. 3

Percent correct [based on Eqs. (1)(4)] as a function of number of possible signal locations for three different underlying distributions: Gaussian, Laplace, and rectangular. Theoretical plots shown correspond to a distance of 2.0 (in standard deviation units) between the signal and noise distributions.

Fig. 4
Fig. 4

Index of detectability based on the Gaussian assumption d′ as a function of possible number of signal locations M for an underlying Laplace distribution. The decrease in d′ is due to the fact that percent correct drops faster for the Laplace distribution; therefore an index of detectability using the Gaussian assumption will produce increasing d′ with increasing number of possible locations.

Fig. 5
Fig. 5

Index of detectability based on the Gaussian assumption d′ versus signal contrast for an underlying rectangular response distribution. For the Gaussian distribution, d′ is linear with signal contrast. For the rectangular distribution the function is also close to linear (R2 ≈ 0.98).

Fig. 6
Fig. 6

Index of detectability based on the Gaussian assumption d′ versus signal contrast for underlying Laplace distributions. For a Gaussian distribution, d′ is linear with signal contrast. For the Laplace distribution the function is also close to linear (R2 ≈ 0.98).

Fig. 7
Fig. 7

Computer simulation of arterial segments and a hemispherical lesion embedded in a structured background. We start with three-dimensional models of the arterial segments and the filling defect (hemispherical lesion attached to the wall of the artery). The filling-contrast material that is injected into the artery attenuates the x-ray beam. The attenuation is exponentially related to the contrast-filled path length transversed by the x rays. The image contrast of the lesion is created because of the shorter path lengths transversed by the x-ray beam incident on the lesion.

Fig. 8
Fig. 8

Example of the test image used for the psychophysical study. Each test image included 16 simulated arterial segments. Each artery consisted of two right cylinders and a sinusoidally modulated narrowing area in the center. The signal could appear at the vertical and horizontal center of the top or the bottom lobes of one artery. For the image shown, the signal (bright spot) is located at the bottom lobe of the fifth arterial segment from the left.

Fig. 9
Fig. 9

Percent correct detection of the signal as a function of signal contrast for five different numbers of possible locations (M = 2, 4, 8, 16, and 32 locations) for both observers. Each data point is based on 300 trials. The solid curves are fits on each individual MAFC condition based on signal detection theory with the Gaussian assumption.

Fig. 10
Fig. 10

Index of detectability with the Gaussian assumption d′ transformed from percent correct [Eq. (2)] as a function of signal contrast. Different symbols correspond to different numbers of possible locations M. The solid curves are regressions on all the data. The correlation coefficient is 0.98.

Fig. 11
Fig. 11

Index of detectability with the Gaussian assumption d′ as a function of number of possible locations or alternatives M, for both observers. Error bars correspond to 95% confidence intervals based on binomial error. Constancy of d′ as a function of number of possible locations is a measure of the adequacy of the Gaussian model. Asterisks indicate significant difference (p < 0.05) in a paired comparison with respect to the 32AFC task.

Fig. 12
Fig. 12

Response times for both observers as a function of number of possible locations for five different levels of signal contrast. Error bars correspond to 95% confidence intervals based session-to-session variability.

Fig. 13
Fig. 13

Response times for both observers as a function of error rate in the task for different numbers of possible locations. A comparison of points of similar error rates shows that the difference in response times across number of possible locations is reduced.

Equations (11)

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P c = - + s ( x ) i = 1 M - 1 N i ( x ) d x ,
N i ( x ) = - x n i ( y ) d y .
s ( x ) = ( 1 2 ) π σ 2 exp [ - ( x - u 0 ) 2 2 σ 2 ] ,
n ( x ) = 1 2 π σ 2 exp ( - x 2 2 σ 2 ) ,
r ( x - u 0 ) = ½ a for - a + u 0 < x < a + u 0 , r ( x - u 0 ) = 0 elsewhere , r ( x ) = ½ a for - a < x < a , r ( x ) = 0 elsewhere .
E = - + - + [ S ( x , y ) / I 0 ] 2 d x d y ,
I a ( x , y ) = I 0 ( x , y ) exp [ - μ 0 t a ( x , y ) ] ,
I s ( x , y ) = I 0 ( x , y ) exp [ - μ 0 t ( x , y ) ] , t ( x , y ) = [ t a ( x , y ) - D t s ( x , y ) > 0 ] ,
E = [ I s ( x , y ) - I a ( x , y ) I a ( x , y ) ] 2 d x d y = { exp [ μ 0 D t s ( x , y ) ] - 1 } 2 d x d y ,
σ d 2 = ( d P c ) 2 σ p c 2 , where ( d P c ) 2 = ( P c d ) - 2 ,
σ d 2 = σ P C 2 { [ P ( c ) d ] - g ( x - d ) G M - 1 ( x ) x d x } 2 ,

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