Abstract

We propose a two-parameter model for the perceived size (spatial extent) of a Gaussian-windowed, drifting sinusoidal luminance pattern (a Gabor patch) based on the simple assumption that perceived size is determined by detection threshold for the sinusoidal carrier. Psychophysical measures of perceived size vary with peak contrast, Gaussian standard deviation, and carrier spatial frequency in a manner predicted by the model. At suprathreshold peak contrasts Gabor perceived size is relatively unaffected by systemic noise but varies in a manner that is consistent with the influence of local contrast gain control. However, at and near threshold, systemic noise plays a major role in determining perceived size. The data and the model indicate that measures of contrast threshold using Gaussian-windowed stimuli (or any other nonflat contrast window) are determined not just by contrast response of the neurons activated by the stimulus but also by integration of that activation over a noisy, contrast-dependent extent of the stimulus in space and time. Thus, when we wish to measure precisely the influence of spatial and temporal integration on threshold, we cannot do so by combining contrast threshold measures with Gaussian-windowed stimuli.

© 1997 Optical Society of America

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References

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  1. S. J. Anderson, D. C. Burr, “Receptive field size of human motion detection units,” Vis. Res. 27, 621–635 (1987).
    [CrossRef] [PubMed]
  2. S. J. Anderson, D. C. Burr, “Spatial summation properties of directionally selective mechanisms in human vision,” J. Opt. Soc. Am. A 8, 1330–1339 (1991).
    [CrossRef] [PubMed]
  3. A. B. Watson, K. Turano, “The optimal motion stimulus,” Vis. Res. 35, 325–336 (1995).
    [CrossRef] [PubMed]
  4. D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 24, 891–910 (1946).
  5. J. G. Daugman, “Spatial visual channels in the Fourier plane,” Vis. Res. 24, 891–910 (1984).
    [CrossRef] [PubMed]
  6. D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple-cell receptive-field profiles in the cat’s visual cortex,” Phys. Rev. B 228, 379–400 (1986).
  7. R. A. Young, “The Gaussian derivative theory of vision: Analysis of cortical cell receptive field line-weighting profiles,” (1985).
  8. R. A. Young, “The Gaussian derivative model for spatial vision: I. Retinal mechanisms,” Spatial Vision 2, 273–293 (1987).
    [PubMed]
  9. Current discussions of this topic can be found with a World Wide Web browser such as Mosaic or Netscape. See the topic “Tutorials, FAQs, and proceedings” at the Universal Resource Locator (URL) designated by http://vision.arc.nasa.gov/VisionScience/VisionScience.html .
  10. R. B. Tootell, M. S. Silverman, R. L. De Valois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
    [CrossRef] [PubMed]
  11. R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “Deoxyglucose analysis of retinotopic organization in primate striate cortex,” Science 218, 902–904 (1982).
    [CrossRef] [PubMed]
  12. E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
    [CrossRef] [PubMed]
  13. C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: The detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
    [CrossRef] [PubMed]
  14. We interchangeably refer to this measurement as perceived radius, perceived size, and perceived spatial extent throughout the manuscript.
  15. W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).
  16. The contour level is determined as that change in squared error (i.e., error variance) that is different from the minimum error with 95%; likelihood. The amount of squared-error change is determined with the standard F-ratio test. For details see the handbook for the Scientist® for Experimental Data Fitting software package (MicroMath Scientific Software, P.O. Box 71550, Salt Lake City, Utah 84171-0509).
  17. For definitions of analysis-of-fit values see the handbook for the Scientist® for Experimental Data Fitting software package (MicroMath Scientific Software, P.O. Box 71550, Salt Lake City, Utah 84171-0509).
  18. H. Akaike, “A new look at statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).
  19. I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
    [PubMed]
  20. D. J. Gelb, H. R. Wilson, “Shifts in perceived size as a function of contrast and temporal modulation,” Vis. Res. 23, 71–82 (1983).
    [CrossRef] [PubMed]
  21. This situation is made more complicated by the fact that contrast gain control processes occur during the first 500 ms of stimulus presentation [H. R. Wilson, R. Humanski, “Spatial frequency adaptation and contrast gain control,” Vis. Res. 33, 1133–1149 (1993)].
    [CrossRef] [PubMed]
  22. A. J. van Doorn, J. J. Koenderink, “Spatiotemporal integration in the detection of coherent motion,” Vis. Res. 24, 47–53 (1984).
    [CrossRef] [PubMed]
  23. W. A. van de Grind, J. J. Koenderink, A. J. van Doorn, “The distribution of human motion detector properties in the monocular visual field,” Vis. Res. 26, 797–810 (1986).
    [CrossRef] [PubMed]
  24. R. E. Fredericksen, F. A. J. Verstraten, W. A. van de Grind, “Spatial summation and its interaction with the temporal integration mechanism in human motion perception,” Vis. Res. 34, 3171–3188 (1994).
    [CrossRef]
  25. J. T. Todd, J. F. Norman, “The effects of spatiotemporal integration on maximum displacement thresholds for the detection of coherent motion,” Vis. Res. 35, 2287–2302 (1995).
    [CrossRef] [PubMed]
  26. R. G. Brown, Introduction to Random Signal Analysis and Kalman Filtering (Wiley, New York, 1983), p. 347.
  27. A. B. Bonds, “Temporal dynamics of contrast gain in single cells of the cat striate cortex,” 6, 239–255 (1991).
    [PubMed]
  28. M. W. Cannon, S. C. Fullenkamp, “Spatial interactions in apparent contrast: Inhibitory effects among grating patterns of different spatial frequencies, spatial positions and orientations,” Vis. Res. 31, 1985–1998 (1991).
    [CrossRef] [PubMed]
  29. D. Ariely, C. Burbeck, “Statistical encoding of multiple stimuli: a theory of distributed representation,” Invest. Ophthalmol. Vis. Sci. Suppl. 36, 472 (1995).

1995 (3)

A. B. Watson, K. Turano, “The optimal motion stimulus,” Vis. Res. 35, 325–336 (1995).
[CrossRef] [PubMed]

J. T. Todd, J. F. Norman, “The effects of spatiotemporal integration on maximum displacement thresholds for the detection of coherent motion,” Vis. Res. 35, 2287–2302 (1995).
[CrossRef] [PubMed]

D. Ariely, C. Burbeck, “Statistical encoding of multiple stimuli: a theory of distributed representation,” Invest. Ophthalmol. Vis. Sci. Suppl. 36, 472 (1995).

1994 (1)

R. E. Fredericksen, F. A. J. Verstraten, W. A. van de Grind, “Spatial summation and its interaction with the temporal integration mechanism in human motion perception,” Vis. Res. 34, 3171–3188 (1994).
[CrossRef]

1993 (1)

This situation is made more complicated by the fact that contrast gain control processes occur during the first 500 ms of stimulus presentation [H. R. Wilson, R. Humanski, “Spatial frequency adaptation and contrast gain control,” Vis. Res. 33, 1133–1149 (1993)].
[CrossRef] [PubMed]

1991 (3)

A. B. Bonds, “Temporal dynamics of contrast gain in single cells of the cat striate cortex,” 6, 239–255 (1991).
[PubMed]

M. W. Cannon, S. C. Fullenkamp, “Spatial interactions in apparent contrast: Inhibitory effects among grating patterns of different spatial frequencies, spatial positions and orientations,” Vis. Res. 31, 1985–1998 (1991).
[CrossRef] [PubMed]

S. J. Anderson, D. C. Burr, “Spatial summation properties of directionally selective mechanisms in human vision,” J. Opt. Soc. Am. A 8, 1330–1339 (1991).
[CrossRef] [PubMed]

1987 (2)

S. J. Anderson, D. C. Burr, “Receptive field size of human motion detection units,” Vis. Res. 27, 621–635 (1987).
[CrossRef] [PubMed]

R. A. Young, “The Gaussian derivative model for spatial vision: I. Retinal mechanisms,” Spatial Vision 2, 273–293 (1987).
[PubMed]

1986 (2)

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple-cell receptive-field profiles in the cat’s visual cortex,” Phys. Rev. B 228, 379–400 (1986).

W. A. van de Grind, J. J. Koenderink, A. J. van Doorn, “The distribution of human motion detector properties in the monocular visual field,” Vis. Res. 26, 797–810 (1986).
[CrossRef] [PubMed]

1985 (2)

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
[CrossRef] [PubMed]

1984 (2)

J. G. Daugman, “Spatial visual channels in the Fourier plane,” Vis. Res. 24, 891–910 (1984).
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Spatiotemporal integration in the detection of coherent motion,” Vis. Res. 24, 47–53 (1984).
[CrossRef] [PubMed]

1983 (1)

D. J. Gelb, H. R. Wilson, “Shifts in perceived size as a function of contrast and temporal modulation,” Vis. Res. 23, 71–82 (1983).
[CrossRef] [PubMed]

1982 (1)

R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “Deoxyglucose analysis of retinotopic organization in primate striate cortex,” Science 218, 902–904 (1982).
[CrossRef] [PubMed]

1981 (1)

R. B. Tootell, M. S. Silverman, R. L. De Valois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

1975 (1)

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: The detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
[CrossRef] [PubMed]

1974 (1)

H. Akaike, “A new look at statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).

1946 (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 24, 891–910 (1946).

Akaike, H.

H. Akaike, “A new look at statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).

Anderson, S. J.

Ariely, D.

D. Ariely, C. Burbeck, “Statistical encoding of multiple stimuli: a theory of distributed representation,” Invest. Ophthalmol. Vis. Sci. Suppl. 36, 472 (1995).

Bonds, A. B.

A. B. Bonds, “Temporal dynamics of contrast gain in single cells of the cat striate cortex,” 6, 239–255 (1991).
[PubMed]

Brown, R. G.

R. G. Brown, Introduction to Random Signal Analysis and Kalman Filtering (Wiley, New York, 1983), p. 347.

Burbeck, C.

D. Ariely, C. Burbeck, “Statistical encoding of multiple stimuli: a theory of distributed representation,” Invest. Ophthalmol. Vis. Sci. Suppl. 36, 472 (1995).

Burr, D. C.

Cannon, M. W.

M. W. Cannon, S. C. Fullenkamp, “Spatial interactions in apparent contrast: Inhibitory effects among grating patterns of different spatial frequencies, spatial positions and orientations,” Vis. Res. 31, 1985–1998 (1991).
[CrossRef] [PubMed]

Daugman, J. G.

J. G. Daugman, “Spatial visual channels in the Fourier plane,” Vis. Res. 24, 891–910 (1984).
[CrossRef] [PubMed]

De Valois, R. L.

E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “Deoxyglucose analysis of retinotopic organization in primate striate cortex,” Science 218, 902–904 (1982).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, R. L. De Valois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

Field, D. J.

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple-cell receptive-field profiles in the cat’s visual cortex,” Phys. Rev. B 228, 379–400 (1986).

Flannery, B. P.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Fredericksen, R. E.

R. E. Fredericksen, F. A. J. Verstraten, W. A. van de Grind, “Spatial summation and its interaction with the temporal integration mechanism in human motion perception,” Vis. Res. 34, 3171–3188 (1994).
[CrossRef]

Freeman, R. D.

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

Fullenkamp, S. C.

M. W. Cannon, S. C. Fullenkamp, “Spatial interactions in apparent contrast: Inhibitory effects among grating patterns of different spatial frequencies, spatial positions and orientations,” Vis. Res. 31, 1985–1998 (1991).
[CrossRef] [PubMed]

Gabor, D.

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 24, 891–910 (1946).

Gelb, D. J.

D. J. Gelb, H. R. Wilson, “Shifts in perceived size as a function of contrast and temporal modulation,” Vis. Res. 23, 71–82 (1983).
[CrossRef] [PubMed]

Humanski, R.

This situation is made more complicated by the fact that contrast gain control processes occur during the first 500 ms of stimulus presentation [H. R. Wilson, R. Humanski, “Spatial frequency adaptation and contrast gain control,” Vis. Res. 33, 1133–1149 (1993)].
[CrossRef] [PubMed]

Klein, S.

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: The detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
[CrossRef] [PubMed]

Koenderink, J. J.

W. A. van de Grind, J. J. Koenderink, A. J. van Doorn, “The distribution of human motion detector properties in the monocular visual field,” Vis. Res. 26, 797–810 (1986).
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Spatiotemporal integration in the detection of coherent motion,” Vis. Res. 24, 47–53 (1984).
[CrossRef] [PubMed]

Norman, J. F.

J. T. Todd, J. F. Norman, “The effects of spatiotemporal integration on maximum displacement thresholds for the detection of coherent motion,” Vis. Res. 35, 2287–2302 (1995).
[CrossRef] [PubMed]

Ohzawa, I.

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

Press, W. H.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Schwartz, E.

E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
[CrossRef] [PubMed]

Sclar, G.

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

Silverman, M. S.

E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “Deoxyglucose analysis of retinotopic organization in primate striate cortex,” Science 218, 902–904 (1982).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, R. L. De Valois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

Stromeyer, C. F.

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: The detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
[CrossRef] [PubMed]

Switkes, E.

E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “Deoxyglucose analysis of retinotopic organization in primate striate cortex,” Science 218, 902–904 (1982).
[CrossRef] [PubMed]

Teukolsky, A. A.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Todd, J. T.

J. T. Todd, J. F. Norman, “The effects of spatiotemporal integration on maximum displacement thresholds for the detection of coherent motion,” Vis. Res. 35, 2287–2302 (1995).
[CrossRef] [PubMed]

Tolhurst, D. J.

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple-cell receptive-field profiles in the cat’s visual cortex,” Phys. Rev. B 228, 379–400 (1986).

Tootell, R. B.

E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “Deoxyglucose analysis of retinotopic organization in primate striate cortex,” Science 218, 902–904 (1982).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, R. L. De Valois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

Turano, K.

A. B. Watson, K. Turano, “The optimal motion stimulus,” Vis. Res. 35, 325–336 (1995).
[CrossRef] [PubMed]

van de Grind, W. A.

R. E. Fredericksen, F. A. J. Verstraten, W. A. van de Grind, “Spatial summation and its interaction with the temporal integration mechanism in human motion perception,” Vis. Res. 34, 3171–3188 (1994).
[CrossRef]

W. A. van de Grind, J. J. Koenderink, A. J. van Doorn, “The distribution of human motion detector properties in the monocular visual field,” Vis. Res. 26, 797–810 (1986).
[CrossRef] [PubMed]

van Doorn, A. J.

W. A. van de Grind, J. J. Koenderink, A. J. van Doorn, “The distribution of human motion detector properties in the monocular visual field,” Vis. Res. 26, 797–810 (1986).
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Spatiotemporal integration in the detection of coherent motion,” Vis. Res. 24, 47–53 (1984).
[CrossRef] [PubMed]

Verstraten, F. A. J.

R. E. Fredericksen, F. A. J. Verstraten, W. A. van de Grind, “Spatial summation and its interaction with the temporal integration mechanism in human motion perception,” Vis. Res. 34, 3171–3188 (1994).
[CrossRef]

Vetterling, W. T.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

Watson, A. B.

A. B. Watson, K. Turano, “The optimal motion stimulus,” Vis. Res. 35, 325–336 (1995).
[CrossRef] [PubMed]

Wilson, H. R.

This situation is made more complicated by the fact that contrast gain control processes occur during the first 500 ms of stimulus presentation [H. R. Wilson, R. Humanski, “Spatial frequency adaptation and contrast gain control,” Vis. Res. 33, 1133–1149 (1993)].
[CrossRef] [PubMed]

D. J. Gelb, H. R. Wilson, “Shifts in perceived size as a function of contrast and temporal modulation,” Vis. Res. 23, 71–82 (1983).
[CrossRef] [PubMed]

Young, R. A.

R. A. Young, “The Gaussian derivative model for spatial vision: I. Retinal mechanisms,” Spatial Vision 2, 273–293 (1987).
[PubMed]

R. A. Young, “The Gaussian derivative theory of vision: Analysis of cortical cell receptive field line-weighting profiles,” (1985).

IEEE Trans. Autom. Control (1)

H. Akaike, “A new look at statistical model identification,” IEEE Trans. Autom. Control 19, 716–723 (1974).

Invest. Ophthalmol. Vis. Sci. Suppl. (1)

D. Ariely, C. Burbeck, “Statistical encoding of multiple stimuli: a theory of distributed representation,” Invest. Ophthalmol. Vis. Sci. Suppl. 36, 472 (1995).

J. Inst. Electr. Eng. (1)

D. Gabor, “Theory of communication,” J. Inst. Electr. Eng. 24, 891–910 (1946).

J. Neurophysiol. (1)

I. Ohzawa, G. Sclar, R. D. Freeman, “Contrast gain control in the cat’s visual system,” J. Neurophysiol. 54, 651–667 (1985).
[PubMed]

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

Phys. Rev. B (1)

D. J. Field, D. J. Tolhurst, “The structure and symmetry of simple-cell receptive-field profiles in the cat’s visual cortex,” Phys. Rev. B 228, 379–400 (1986).

Science (3)

R. B. Tootell, M. S. Silverman, R. L. De Valois, “Spatial frequency columns in primary visual cortex,” Science 214, 813–815 (1981).
[CrossRef] [PubMed]

R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “Deoxyglucose analysis of retinotopic organization in primate striate cortex,” Science 218, 902–904 (1982).
[CrossRef] [PubMed]

E. Schwartz, R. B. Tootell, M. S. Silverman, E. Switkes, R. L. De Valois, “On the mathematical structure of the visuotopic mapping of macaque striate cortex,” Science 227, 1065–1066 (1985).
[CrossRef] [PubMed]

Spatial Vision (1)

R. A. Young, “The Gaussian derivative model for spatial vision: I. Retinal mechanisms,” Spatial Vision 2, 273–293 (1987).
[PubMed]

Temporal dynamics of contrast gain in single cells of the cat striate cortex (1)

A. B. Bonds, “Temporal dynamics of contrast gain in single cells of the cat striate cortex,” 6, 239–255 (1991).
[PubMed]

Vis. Res. (11)

M. W. Cannon, S. C. Fullenkamp, “Spatial interactions in apparent contrast: Inhibitory effects among grating patterns of different spatial frequencies, spatial positions and orientations,” Vis. Res. 31, 1985–1998 (1991).
[CrossRef] [PubMed]

S. J. Anderson, D. C. Burr, “Receptive field size of human motion detection units,” Vis. Res. 27, 621–635 (1987).
[CrossRef] [PubMed]

D. J. Gelb, H. R. Wilson, “Shifts in perceived size as a function of contrast and temporal modulation,” Vis. Res. 23, 71–82 (1983).
[CrossRef] [PubMed]

This situation is made more complicated by the fact that contrast gain control processes occur during the first 500 ms of stimulus presentation [H. R. Wilson, R. Humanski, “Spatial frequency adaptation and contrast gain control,” Vis. Res. 33, 1133–1149 (1993)].
[CrossRef] [PubMed]

A. J. van Doorn, J. J. Koenderink, “Spatiotemporal integration in the detection of coherent motion,” Vis. Res. 24, 47–53 (1984).
[CrossRef] [PubMed]

W. A. van de Grind, J. J. Koenderink, A. J. van Doorn, “The distribution of human motion detector properties in the monocular visual field,” Vis. Res. 26, 797–810 (1986).
[CrossRef] [PubMed]

R. E. Fredericksen, F. A. J. Verstraten, W. A. van de Grind, “Spatial summation and its interaction with the temporal integration mechanism in human motion perception,” Vis. Res. 34, 3171–3188 (1994).
[CrossRef]

J. T. Todd, J. F. Norman, “The effects of spatiotemporal integration on maximum displacement thresholds for the detection of coherent motion,” Vis. Res. 35, 2287–2302 (1995).
[CrossRef] [PubMed]

A. B. Watson, K. Turano, “The optimal motion stimulus,” Vis. Res. 35, 325–336 (1995).
[CrossRef] [PubMed]

J. G. Daugman, “Spatial visual channels in the Fourier plane,” Vis. Res. 24, 891–910 (1984).
[CrossRef] [PubMed]

C. F. Stromeyer, S. Klein, “Evidence against narrow-band spatial frequency channels in human vision: The detectability of frequency modulated gratings,” Vis. Res. 15, 899–910 (1975).
[CrossRef] [PubMed]

Other (7)

We interchangeably refer to this measurement as perceived radius, perceived size, and perceived spatial extent throughout the manuscript.

W. H. Press, A. A. Teukolsky, W. T. Vetterling, B. P. Flannery, Numerical Recipes in C, 2nd ed. (Cambridge U. Press, Cambridge, 1992).

The contour level is determined as that change in squared error (i.e., error variance) that is different from the minimum error with 95%; likelihood. The amount of squared-error change is determined with the standard F-ratio test. For details see the handbook for the Scientist® for Experimental Data Fitting software package (MicroMath Scientific Software, P.O. Box 71550, Salt Lake City, Utah 84171-0509).

For definitions of analysis-of-fit values see the handbook for the Scientist® for Experimental Data Fitting software package (MicroMath Scientific Software, P.O. Box 71550, Salt Lake City, Utah 84171-0509).

Current discussions of this topic can be found with a World Wide Web browser such as Mosaic or Netscape. See the topic “Tutorials, FAQs, and proceedings” at the Universal Resource Locator (URL) designated by http://vision.arc.nasa.gov/VisionScience/VisionScience.html .

R. A. Young, “The Gaussian derivative theory of vision: Analysis of cortical cell receptive field line-weighting profiles,” (1985).

R. G. Brown, Introduction to Random Signal Analysis and Kalman Filtering (Wiley, New York, 1983), p. 347.

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Figures (9)

Fig. 1
Fig. 1

Demonstration of the phenomenon of perceived Gabor size dependence on contrast. Gaussian standard deviation decreases from top to bottom, and peak contrast decreases from left to right. Contrast values (85%, 21%, 6%) were selected to produce approximately equal steps in perceived radius. Note that patches on rightward diagonals (e.g., bottom left to top right) have approximately equal apparent sizes.

Fig. 2
Fig. 2

Abstract representation of how visual space is mapped to area V1 in primates. (a) Half-bull’s-eye with logarithmically spaced concentric rings. Assuming fixation at the small black circle at the center of the half-bull’s-eye, (b) shows how the bull’s-eye is mapped to V1 as indicated by 2-deoxyglucose experiments. The paths in visual space represented by the arrows in (a) follow cortical paths as indicated in (b). The numbers provide a key to those paths, while digit size indicates schematically how visual area away from fixation is relatively expanded or compressed in the cortical representation.

Fig. 3
Fig. 3

Illustration of the working hypothesis. The Gaussian contrast envelope and the contrast-modulated, drifting sinusoidal carrier are shown together with a hypothetical contrast level (Ce) that limits visibility of the carrier. The perceived radius (Pr) indicated in the graph and in Eq. (2) is shown as the point at which the carrier contrast falls below Ce.

Fig. 4
Fig. 4

Perceived Gabor radius plotted against Gaussian standard deviation, with peak contrast (values shown in the legend) as a parameter in each graph. The data for both subjects are shown in graph pairs, with the spatial frequency for each condition indicated in each graph. The typical 95% confidence interval for all data points ranged from ±1 arcmin for the smallest standard deviation to ±2 arcmin for the largest standard deviation. No error bars are shown because they are all smaller than the size of the symbols.

Fig. 5
Fig. 5

Data from Fig. 4 replotted against peak contrast, together with the curve fits from the use of Eq. (5). The data for subject EF are shown with open symbols, while the data for subject PB are shown with filled symbols. The fits for EF are shown with solid curves, while the fits for PB are shown with dashed curves.

Fig. 6
Fig. 6

Perceived Gabor radius data for contrast values of 3.5%, 7%, 14%, 28%, and 56% for a standard deviation of 21 pixels (39.4 arcmin) across spatial frequency. Some of the data are taken from Fig. 2, while the data for 0.3 and 0.6 cpd were additionally measured for completion of the curves. The graph clearly shows that perceived radius of a Gabor of otherwise fixed parameter values depends on the spatial frequency of the carrier.

Fig. 7
Fig. 7

Measured contrast detection thresholds (cross-filled symbols) together with model predictions (open symbols). 95% confidence intervals are added to the model predictions with the use of the 95% confidence intervals for K0 and K1. Subject PB’s data are shifted downward by 10 dB for clarification of the presentation. Only one point per subject can be rejected as different from the prediction. Those two points occur for the two spatial frequencies that contain fewer low-peak-contrast data to constrain the steepest-sloping part of the model, the segment of the model curve that must be accurate if it is to predict threshold well.

Fig. 8
Fig. 8

Derivative of perceived radius with respect to contrast (Pr/Cp, shown by the solid curve) for a standard deviation of 21 and derivative of perceived radius with respect to standard deviation [Pr/σ given by Eq. (6), shown by the dashed curve]. In both cases we have set K0=1 and K1=0 for convenience.

Fig. 9
Fig. 9

Plot (a) shows how systemic noise combined with the perceived radius curve, Pr(Cp), determines perceived size at threshold. The curve at the bottom of (a) is an example of a PDF, ρn˜, which is Gaussian distributed on a logarithmic Cp axis. The PDF and Pr(Cp) parameters are given in (a); K0 and K1 are from the 0.1-cpd condition for subject EF (see Table 1). The PDF for perceived size, ρP˜r, is shown at the left of (a). Only the portion of ρn˜ to the right of the vertical line marking Cth is transformed to produce ρP˜r. The portion of ρn˜ to the left of Cth represents conditions that are not visible and map to Pr=0 (not shown for clarity). Both ρn˜ and ρP˜r have been scaled in magnitude for purposes of clarity. Note that the ρP˜r curve in (a) and the left-hand curves in (b) and (c) are identical [except for a scaling in (a)] because they are the same condition: Cp=Cth and σn=2 dB. Plot (b) shows how ρP˜r changes as Cp increases for fixed variance of ρn˜. Note that as Cp increases, the variance of ρP˜r decreases. Plot (c) shows how ρP˜r changes as the variance of ρn˜ increases for Cp fixed at threshold. Note that the PDF’s shown in (b) and (c) may not have unit area, because they do not include portions of ρP˜r that fall below threshold; that case would be represented by a discrete probability category at Pr=0, but inclusion of that category complicates the plots.

Tables (1)

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Table 1 Fitted Parameters for Eq. (4) for Each Subject for Each Spatial Frequency, Together with a Statistical Analysis of the Goodness of Fita

Equations (12)

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L(x, y, t)=Lm1+Cp cos2πxfc+θc(t)×exp-12xσx2-12yσy2,
Ce=Cp exp-12Prσ2,
Pr=σ-2 lnCeCp(arcmin).
Ce=K0+CpK1=Cp exp-12Prσ2
Pr=σ-2 lnK0+CpK1Cparcmin.
Prσ=-2 lnK0+CpK1Cparcmin/%contrast.
fp=0.2564σ(cpd).
Pa=πσ2-2 lnK0+CpK1Cparcmin2,
PaCp=2πσ2K0CpK0+Cp2K1(arcmin)2/(%contrast).
Ce=Cp+n˜exp-12Prσ2,
Pr=σ-2 lnCeCp+n˜arcmin.
ρP˜r=|Cp/Pr|ρn˜[Cp(Pr)],

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