Abstract

Previous studies have specified the foveal pattern that is seen most efficiently, with the assumption that the waveform of the best pattern matches the impulse response of the most sensitive visual filter. We measured the threshold contrast for circular, collinear, and orthogonal Gabor stimuli of 6 Hz temporal frequency presented 7 deg above the fixation point. We found that the threshold contrast energy is minimal for a class of stimuli whose Fourier-spectra bandwidth is less than ∼1 octave. These findings suggest that an energy algorithm might underlie spatial summation of peripheral Gabor patches. The different behavior of spatial summation in fovea and periphery might reflect the differences in pattern detectability across space in the central and peripheral visual fields. It is also possible that a coherent (cross-correlation) algorithm is employed in detection of foveal stimuli and that an incoherent (energy) algorithm is employed in detection of peripheral stimuli.

© 2001 Optical Society of America

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1999 (4)

U. Polat, C. W. Tyler, “What pattern the eye sees best,” Vision Res. 39, 887–895 (1999).
[CrossRef] [PubMed]

A. E. Burgess, “The Rose model, revisited,” J. Opt. Soc. Am. A 16, 633–646 (1999).
[CrossRef]

V. Manahilov, W. Simpson, “Energy model for contrast detection: spatiotemporal characteristics of threshold vision,” Biol. Cybern. 81, 61–71 (1999).
[CrossRef] [PubMed]

H. S. Orbach, H. R. Wilson, “Factors limiting peripheral pattern discrimination,” Spatial Vis. 12, 83–106 (1999).
[CrossRef]

1998 (1)

V. Manahilov, “Triphasic temporal impulse responses and Mach bands in time,” Vision Res. 38, 447–458 (1998).
[CrossRef] [PubMed]

1995 (1)

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

1993 (1)

P. Bijl, J. J. Koenderink, “Visibility of elliptical Gaussian blobs,” Vision Res. 33, 243–255 (1993).
[CrossRef] [PubMed]

1992 (1)

G. L. Savage, M. S. Banks, “Scotopic visual efficiency: constraints by optics, receptor properties, and rod pooling,” Vision Res. 32, 645–656 (1992).
[CrossRef] [PubMed]

1991 (2)

1989 (1)

J. S. Pointer, R. F. Hess, “The contrast sensitivity gradient across the human visual field with emphasis on the low spatial frequency range,” Vision Res. 29, 1133–1151 (1989).
[CrossRef]

1987 (1)

N. Graham, J. G. Robson, “Summation of very close spatial frequencies: the importance of spatial probability summation,” Vision Res. 27, 1997–2007 (1987).
[CrossRef] [PubMed]

1985 (2)

L. A. Temme, L. Malcus, W. K. Noell, “Peripheral visual field is radially organized,” Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

W. H. Swanson, H. R. Wilson, “Eccentricity dependence of contrast matching and oblique masking,” Vision Res. 25, 1285–1295 (1985).
[CrossRef] [PubMed]

1984 (2)

1983 (1)

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

1982 (2)

J. Rovamo, V. Virsu, P. Laurinen, L. Hyvarinen, “Resolution of gratings oriented along and across meridians in peripheral vision,” Invest. Ophthalmol. Visual Sci. 23, 666–670 (1982).

A. B. Watson, “Summation of grating patches indicates many types of detector at one retinal location,” Vision Res. 22, 17–25 (1982).
[CrossRef] [PubMed]

1981 (1)

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

1979 (1)

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

1978 (4)

R. F. Quick, W. W. Mullins, T. A. Reichert, “Spatial summation effects on two-component grating thresholds,” J. Opt. Soc. Am. 68, 116–124 (1978).
[CrossRef] [PubMed]

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Detectability of power fluctuations of temporal visual noise,” Vision Res. 18, 191–195 (1978).
[CrossRef] [PubMed]

1974 (1)

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

1970 (2)

C. Rashbass, “The visibility of transient changes of luminance,” J. Physiol. (London) 210, 165–186 (1970).

H. Levitt, “Transformed up-down methods in psychoacoustics,” J. Acoust. Soc. Am. 49, 467–477 (1970).
[CrossRef]

1968 (1)

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

1958 (2)

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

W. P. Tanner, T. G. Birdsall, “Definition of d’ and h as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

1935 (1)

C. H. Graham, R. Margaria, “Area and the intensity time relation in the peripheral retina,” Am. J. Physiol. 113, 299–305 (1935).

Anderson, S. J.

Banks, M. S.

G. L. Savage, M. S. Banks, “Scotopic visual efficiency: constraints by optics, receptor properties, and rod pooling,” Vision Res. 32, 645–656 (1992).
[CrossRef] [PubMed]

M. S. Banks, A. B. Sekuler, S. J. Anderson, “Peripheral spatial vision: limits imposed by optics, photoreceptors, and receptor pooling,” J. Opt. Soc. Am. A 8, 1775–1787 (1991).
[CrossRef] [PubMed]

Barlow, H. B.

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[CrossRef] [PubMed]

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

Bijl, P.

P. Bijl, J. J. Koenderink, “Visibility of elliptical Gaussian blobs,” Vision Res. 33, 243–255 (1993).
[CrossRef] [PubMed]

Birdsall, T. G.

W. P. Tanner, T. G. Birdsall, “Definition of d’ and h as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Braddick, O.

O. Braddick, “Is spatial phase degraded in peripheral vision and visual pathology?” in Documenta Ophthalmologica Proceedings Series, L. Maffei, ed. (W. Junk, The Hague, 1981), pp. 255–262.

Burgess, A.

Burgess, A. E.

A. E. Burgess, “The Rose model, revisited,” J. Opt. Soc. Am. A 16, 633–646 (1999).
[CrossRef]

A. E. Burgess, “High level visual decision efficiencies,” in Vision: Codding and Efficiency, C. B. Blakemore, ed. (Cambridge U. Press, New York, 1990), pp. 431–440.

Campbell, F. W.

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Ghandeharian, H.

Graham, C. H.

C. H. Graham, R. Margaria, “Area and the intensity time relation in the peripheral retina,” Am. J. Physiol. 113, 299–305 (1935).

Graham, N.

N. Graham, J. G. Robson, “Summation of very close spatial frequencies: the importance of spatial probability summation,” Vision Res. 27, 1997–2007 (1987).
[CrossRef] [PubMed]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

Green, D. B.

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

Hess, R. F.

J. S. Pointer, R. F. Hess, “The contrast sensitivity gradient across the human visual field with emphasis on the low spatial frequency range,” Vision Res. 29, 1133–1151 (1989).
[CrossRef]

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

Howell, E. R.

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

Hyvarinen, L.

J. Rovamo, V. Virsu, P. Laurinen, L. Hyvarinen, “Resolution of gratings oriented along and across meridians in peripheral vision,” Invest. Ophthalmol. Visual Sci. 23, 666–670 (1982).

Kersten, D.

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

Koenderink, J. J.

P. Bijl, J. J. Koenderink, “Visibility of elliptical Gaussian blobs,” Vision Res. 33, 243–255 (1993).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Detectability of power fluctuations of temporal visual noise,” Vision Res. 18, 191–195 (1978).
[CrossRef] [PubMed]

Laurinen, P.

J. Rovamo, V. Virsu, P. Laurinen, L. Hyvarinen, “Resolution of gratings oriented along and across meridians in peripheral vision,” Invest. Ophthalmol. Visual Sci. 23, 666–670 (1982).

Levitt, H.

H. Levitt, “Transformed up-down methods in psychoacoustics,” J. Acoust. Soc. Am. 49, 467–477 (1970).
[CrossRef]

Malcus, L.

L. A. Temme, L. Malcus, W. K. Noell, “Peripheral visual field is radially organized,” Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Manahilov, V.

V. Manahilov, W. Simpson, “Energy model for contrast detection: spatiotemporal characteristics of threshold vision,” Biol. Cybern. 81, 61–71 (1999).
[CrossRef] [PubMed]

V. Manahilov, “Triphasic temporal impulse responses and Mach bands in time,” Vision Res. 38, 447–458 (1998).
[CrossRef] [PubMed]

Margaria, R.

C. H. Graham, R. Margaria, “Area and the intensity time relation in the peripheral retina,” Am. J. Physiol. 113, 299–305 (1935).

Mullins, W. W.

Noell, W. K.

L. A. Temme, L. Malcus, W. K. Noell, “Peripheral visual field is radially organized,” Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Orbach, H. S.

H. S. Orbach, H. R. Wilson, “Factors limiting peripheral pattern discrimination,” Spatial Vis. 12, 83–106 (1999).
[CrossRef]

Pelli, G. D.

G. D. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
[CrossRef] [PubMed]

Pointer, J. S.

J. S. Pointer, R. F. Hess, “The contrast sensitivity gradient across the human visual field with emphasis on the low spatial frequency range,” Vision Res. 29, 1133–1151 (1989).
[CrossRef]

Polat, U.

U. Polat, C. W. Tyler, “What pattern the eye sees best,” Vision Res. 39, 887–895 (1999).
[CrossRef] [PubMed]

Quick, R. F.

Rashbass, C.

C. Rashbass, “The visibility of transient changes of luminance,” J. Physiol. (London) 210, 165–186 (1970).

Reichert, T. A.

Robson, J. G.

N. Graham, J. G. Robson, “Summation of very close spatial frequencies: the importance of spatial probability summation,” Vision Res. 27, 1997–2007 (1987).
[CrossRef] [PubMed]

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Rovamo, J.

J. Rovamo, V. Virsu, P. Laurinen, L. Hyvarinen, “Resolution of gratings oriented along and across meridians in peripheral vision,” Invest. Ophthalmol. Visual Sci. 23, 666–670 (1982).

Savage, G. L.

G. L. Savage, M. S. Banks, “Scotopic visual efficiency: constraints by optics, receptor properties, and rod pooling,” Vision Res. 32, 645–656 (1992).
[CrossRef] [PubMed]

Sekuler, A. B.

Simpson, W.

V. Manahilov, W. Simpson, “Energy model for contrast detection: spatiotemporal characteristics of threshold vision,” Biol. Cybern. 81, 61–71 (1999).
[CrossRef] [PubMed]

Swanson, W. H.

W. H. Swanson, H. R. Wilson, “Eccentricity dependence of contrast matching and oblique masking,” Vision Res. 25, 1285–1295 (1985).
[CrossRef] [PubMed]

Swets, J. A.

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

Tanner, W. P.

W. P. Tanner, T. G. Birdsall, “Definition of d’ and h as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

Temme, L. A.

L. A. Temme, L. Malcus, W. K. Noell, “Peripheral visual field is radially organized,” Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Turano, K.

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

Tyler, C. W.

U. Polat, C. W. Tyler, “What pattern the eye sees best,” Vision Res. 39, 887–895 (1999).
[CrossRef] [PubMed]

van Doorn, A. J.

J. J. Koenderink, A. J. van Doorn, “Detectability of power fluctuations of temporal visual noise,” Vision Res. 18, 191–195 (1978).
[CrossRef] [PubMed]

Virsu, V.

J. Rovamo, V. Virsu, P. Laurinen, L. Hyvarinen, “Resolution of gratings oriented along and across meridians in peripheral vision,” Invest. Ophthalmol. Visual Sci. 23, 666–670 (1982).

Watson, A. B.

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

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

A. B. Watson, “Summation of grating patches indicates many types of detector at one retinal location,” Vision Res. 22, 17–25 (1982).
[CrossRef] [PubMed]

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

A. B. Watson, “Temporal sensitivity,” in Handbook of Perception and Human Performance I: Sensory Processes and Perception, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986), pp. 6.1–6.41.

Wilson, H. R.

H. S. Orbach, H. R. Wilson, “Factors limiting peripheral pattern discrimination,” Spatial Vis. 12, 83–106 (1999).
[CrossRef]

W. H. Swanson, H. R. Wilson, “Eccentricity dependence of contrast matching and oblique masking,” Vision Res. 25, 1285–1295 (1985).
[CrossRef] [PubMed]

Zhang, L.

G. D. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
[CrossRef] [PubMed]

Am. J. Optom. Physiol. Opt. (1)

L. A. Temme, L. Malcus, W. K. Noell, “Peripheral visual field is radially organized,” Am. J. Optom. Physiol. Opt. 62, 545–554 (1985).
[CrossRef] [PubMed]

Am. J. Physiol. (1)

C. H. Graham, R. Margaria, “Area and the intensity time relation in the peripheral retina,” Am. J. Physiol. 113, 299–305 (1935).

Biol. Cybern. (1)

V. Manahilov, W. Simpson, “Energy model for contrast detection: spatiotemporal characteristics of threshold vision,” Biol. Cybern. 81, 61–71 (1999).
[CrossRef] [PubMed]

Invest. Ophthalmol. Visual Sci. (1)

J. Rovamo, V. Virsu, P. Laurinen, L. Hyvarinen, “Resolution of gratings oriented along and across meridians in peripheral vision,” Invest. Ophthalmol. Visual Sci. 23, 666–670 (1982).

J. Acoust. Soc. Am. (2)

H. Levitt, “Transformed up-down methods in psychoacoustics,” J. Acoust. Soc. Am. 49, 467–477 (1970).
[CrossRef]

W. P. Tanner, T. G. Birdsall, “Definition of d’ and h as psychophysical measures,” J. Acoust. Soc. Am. 30, 922–928 (1958).
[CrossRef]

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (3)

J. Physiol. (London) (3)

C. Rashbass, “The visibility of transient changes of luminance,” J. Physiol. (London) 210, 165–186 (1970).

H. B. Barlow, “Temporal and spatial summation in human vision at different background intensities,” J. Physiol. (London) 141, 337–350 (1958).

F. W. Campbell, J. G. Robson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. (London) 197, 551–566 (1968).

Kybernetik (1)

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

Nature (1)

A. B. Watson, H. B. Barlow, J. G. Robson, “What does the eye see best?” Nature 302, 419–422 (1983).
[CrossRef] [PubMed]

Spatial Vis. (1)

H. S. Orbach, H. R. Wilson, “Factors limiting peripheral pattern discrimination,” Spatial Vis. 12, 83–106 (1999).
[CrossRef]

Vision Res. (16)

G. L. Savage, M. S. Banks, “Scotopic visual efficiency: constraints by optics, receptor properties, and rod pooling,” Vision Res. 32, 645–656 (1992).
[CrossRef] [PubMed]

J. S. Pointer, R. F. Hess, “The contrast sensitivity gradient across the human visual field with emphasis on the low spatial frequency range,” Vision Res. 29, 1133–1151 (1989).
[CrossRef]

V. Manahilov, “Triphasic temporal impulse responses and Mach bands in time,” Vision Res. 38, 447–458 (1998).
[CrossRef] [PubMed]

W. H. Swanson, H. R. Wilson, “Eccentricity dependence of contrast matching and oblique masking,” Vision Res. 25, 1285–1295 (1985).
[CrossRef] [PubMed]

H. B. Barlow, “The efficiency of detecting changes of density in random dot patterns,” Vision Res. 18, 637–650 (1978).
[CrossRef] [PubMed]

G. D. Pelli, L. Zhang, “Accurate control of contrast on microcomputer displays,” Vision Res. 31, 1337–1350 (1991).
[CrossRef] [PubMed]

D. Kersten, “Spatial summation in visual noise,” Vision Res. 24, 1977–1990 (1984).
[CrossRef] [PubMed]

U. Polat, C. W. Tyler, “What pattern the eye sees best,” Vision Res. 39, 887–895 (1999).
[CrossRef] [PubMed]

E. R. Howell, R. F. Hess, “The functional area for summation to threshold for sinusoidal gratings,” Vision Res. 18, 369–374 (1978).
[CrossRef] [PubMed]

J. G. Robson, N. Graham, “Probability summation and regional variation in contrast sensitivity across the visual field,” Vision Res. 21, 409–418 (1981).
[CrossRef] [PubMed]

A. B. Watson, “Probability summation over time,” Vision Res. 19, 515–522 (1979).
[CrossRef] [PubMed]

A. B. Watson, “Summation of grating patches indicates many types of detector at one retinal location,” Vision Res. 22, 17–25 (1982).
[CrossRef] [PubMed]

N. Graham, J. G. Robson, “Summation of very close spatial frequencies: the importance of spatial probability summation,” Vision Res. 27, 1997–2007 (1987).
[CrossRef] [PubMed]

J. J. Koenderink, A. J. van Doorn, “Detectability of power fluctuations of temporal visual noise,” Vision Res. 18, 191–195 (1978).
[CrossRef] [PubMed]

P. Bijl, J. J. Koenderink, “Visibility of elliptical Gaussian blobs,” Vision Res. 33, 243–255 (1993).
[CrossRef] [PubMed]

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

Other (5)

A. B. Watson, “Visual detection of spatial contrast patterns: evaluation of five simple models,” Opt. Express6, 12–33 (2000), http://epubs.osa.org/opticsexpress/topbiframe.htm .
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D. B. Green, J. A. Swets, Signal Detection Theory and Psychophysics (Wiley, New York, 1974).

A. B. Watson, “Temporal sensitivity,” in Handbook of Perception and Human Performance I: Sensory Processes and Perception, K. R. Boff, L. Kaufman, J. P. Thomas, eds. (Wiley, New York, 1986), pp. 6.1–6.41.

O. Braddick, “Is spatial phase degraded in peripheral vision and visual pathology?” in Documenta Ophthalmologica Proceedings Series, L. Maffei, ed. (W. Junk, The Hague, 1981), pp. 255–262.

A. E. Burgess, “High level visual decision efficiencies,” in Vision: Codding and Efficiency, C. B. Blakemore, ed. (Cambridge U. Press, New York, 1990), pp. 431–440.

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Figures (9)

Fig. 1
Fig. 1

Model predictions of threshold contrast for detection of circular Gabor stimuli as a function of stimulus area (discussed in the text). Diamonds, predictions of a mismatched cross-correlation model (MC); squares and triangles, predictions of a probability-summation model (PS) with β=3.5 and β=5, respectively; circles, predictions of an energy model (E); solid symbols, model calculations with use of a filter whose impulse response has width and height of 1.2 cycles; open symbols, model calculations with use of a filter whose impulse response has width and height of 2.4 cycles. The predictions of the mismatched cross-correlation and probability-summation models are vertically shifted.

Fig. 2
Fig. 2

Model predictions of threshold stimulus energy for detection of circular Gabor stimuli as a function of stimulus area. The other designations are as in Fig. 1.

Fig. 3
Fig. 3

Examples of static Gabor stimuli used in the present experiments. The stimuli are cosine gratings of 2 c/deg enveloped by a two-dimensional Gaussian function. A, circular Gabor patch with width and height of 1.2 deg (2.4 grating cycles); B, collinear Gabor patch, horizontal grating with width of 2.4 deg (4.8 grating cycles) and height of 0.6 deg (1.2 grating cycles); C, orthogonal Gabor patch, vertical grating, with width and height as in B. The width and height of the stimuli are defined as twice the relevant Gaussian half-width. The stimuli shown have equal area.

Fig. 4
Fig. 4

Mean contrast sensitivity to circular Gabor stimuli as a function of carrier spatial frequency. Stimulus width and height were 4 deg, duration was 0.32 s, and temporal frequency was 6 Hz. The data are from the five observers tested.

Fig. 5
Fig. 5

Mean threshold contrast for detection of collinear (gray circles) and orthogonal (black circles) Gabor stimuli as a function of stimulus area. All stimuli had a height of 0.6 deg. Vertical bars, 95% confidence intervals.

Fig. 6
Fig. 6

Mean threshold contrast for detection of circular (open circles) Gabor stimuli as a function of stimulus area. Gray and black circles, data for collinear and orthogonal Gabor stimuli, respectively, for each observer as in Fig. 5; solid curves, predictions of the energy model as discussed in the text.

Fig. 7
Fig. 7

Mean threshold stimulus energy for detection of circular, collinear, and orthogonal Gabor stimuli as a function of stimulus area. The other designations are as in Fig. 6.

Fig. 8
Fig. 8

Stimulus (outlines of the gray areas) and response (outlines of the white areas) spatial frequency power spectra for three threshold circular Gabor stimuli calculated by Eqs. (20) and (22), respectively. Both stimulus width and stimulus height had 0.26 (A), 1.72 (B), and 8 grating cycles (C). Dotted curves are the power spectra of the transfer function in relative units.

Fig. 9
Fig. 9

Effect of the variations in contrast sensitivity in space on the threshold energy of Gabor patches, calculated by the energy model. The spatial waveform of the stimuli was multiplied by a two-dimensional Gaussian whose spatial constant s was ∝ (constant sensitivity across space), 4, 2, and 1 deg.

Equations (27)

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F=Eid/Es.
h(x, y)=cos(2πyvy) exp(-x2/s02-y2/s02).
R=Cg(x, y)h(x, y)dxdy,
C=K/g(x, y)h(x, y)dxdy.
Es=C2[g(x, y)]2dxdy=K2[g(x, y)]2dxdy/g(x, y)h(x, y)dxdy2.
P=1-(1-γ)i(1-Pi).
Pi=1-exp[-|Cg(x, y)*h(x, y)|β],
P=1-(1-γ)exp-|Cg(x, y)*h(x, y)|βdxdy.
Cβ|g(x, y)*h(x, y)|βdxdy=L,
C=L/|h(x, y)*g(x, y)|βdxdy1/β.
Es=L2/β[g(x, y)]2dxdy/|h(x, y)*g(x, y)|βdxdy2/β.
F(fx, fy, ft)=S(fx, fy, ft)*H(fx, fy, ft).
Er=|S(fx, fy, ft)H(fx, fy, ft)|2d fxd fyd ft.
Er=C2|G(fx, fy, ft)H(fx, fy, ft)|2d fxd fyd ft.
P=1-(1-γ)exp(-Erη),
C=E01/2/|G(fx, fy, ft)×H(fx, fy, ft)|2d fxd fyd ft1/2.
|H(fx, fy, vt)|=E01/2CSF(fx, fy, vt),
C=1/|G(fx, fy, ft)×CSF(fx, fy, vt)|2d fxd fyd ft1/2.
Es=C2|G(fs, fy, ft)|2d fxd fyd ft,
Es=|G(fs, fy, ft)|2d fxd fyd ft/|G(fx, fy, ft)CSF(fx, fy, vt)|2d fxd fyd ft.
CSF=1.8-2.5{log [(fx2+fy2)1/2]-log(2)}2.
g(x, y, t)=cos(2πyvy)×exp[-(x-xc)2/sx2-(y-yc)2/sy2]cos(2πtvt)exp[-(t-2.5st)2/st2],
Ps(fy)=C2 |G(fx, fy, ft)|2d fx,d ft,
Pr(fy)=C2|G(fx, fy, ft)H(fx, fy, ft)|2d fxd ft,
Pr(fy)=E0C2|G(fx, fy, ft)CSF(fx, fy, vt)|2d fxd ft.
Es=Er+El.
W(x, y)=exp[-(x2-y2)/s2]

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