Abstract

Visual discrimination of contour curvature was investigated using stimuli having a single point of maximum curvature and a continuous derivative. Curvature discrimination as a function of mean curvature could be described by a power law with an exponent averaging 1.57. Data were also gathered as a function of line width, stimulus orientation, and retinal eccentricity. Finally, masking experiments provided evidence that the mechanisms responsible for curvature discrimination were both orientation and spatial-frequency selective. The data are well fitted by theoretical results derived from a line-element model that has recently been applied to spatial-frequency discrimination.

© 1985 Optical Society of America

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References

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    [CrossRef] [PubMed]
  2. D. E. Knuth, Tex/Metafont (Academic, New York, 1982).
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    [CrossRef] [PubMed]
  4. R. J. Watt, “Further evidence concerning the analysis of curvature in human foveal vision,” Vision Res. 24, 251–253 (1984).
    [CrossRef] [PubMed]
  5. C. Blakemore, R. Over, “Curvature detectors in human vision?” Perception 3, 3–7 (1974).
    [CrossRef] [PubMed]
  6. J. Ogilvie, E. Daicar, “The perception of curvature,” Can. J. Physiol. 21, 521–525 (1967).
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    [CrossRef] [PubMed]
  9. F. B. Hildebrand, Advanced Calculus for Applications, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1976).
  10. H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
    [CrossRef] [PubMed]
  11. G. C. Phillips, H. R. Wilson, “Orientation bandwidths of spatial mechanisms measured by masking,” J. Opt. Soc. Am. A 1, 226–232 (1984).
    [CrossRef] [PubMed]
  12. R. F. Quick, “A vector-magnitude model of contrast detection,” Kybernetik 16, 1299–1302 (1974).
    [CrossRef]
  13. J. Hirsch, R. Hylton, “Limits of spatial frequency discrimination as evidence of neural interpolation,” J. Opt. Soc. Am. 72, 1367–1374 (1982).
    [CrossRef] [PubMed]
  14. H. Leibowitz, “Some observations and theory on the variation of visual acuity with the orientation of the test object,” J. Opt. Soc. Am. 43, 902–905 (1953).
    [CrossRef] [PubMed]
  15. W. H. Swanson, H. R. Wilson, “Eccentricity dependence of contrast matching and oblique masking,” Vision Res. (to be published).
  16. G. Westheimer, “The spatial grain of the perifoveal visual field,” Vision Res. 22, 157–162 (1982).
    [CrossRef] [PubMed]
  17. A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
    [CrossRef] [PubMed]
  18. J. Nachmias, R. V. Sansbury, “Grating contrast: discrimination may be better than detection,” Vision Res. 14, 1039–1042 (1974).
    [CrossRef] [PubMed]
  19. G. E. Legge, J. M. Foley, “Contrast masking in human vision,” J. Opt. Soc. Am. 70, 1458–1470 (1980).
    [CrossRef] [PubMed]
  20. C. R. Carlson, “Thresholds for perceived image sharpness,” Photogr. Sci. Eng. 22, 69–71 (1978).
  21. S. W. Zucker, “Cooperative grouping and early orientation selection,” in Physical and Biological Processing of Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, New York, 1983), pp. 326–334.
    [CrossRef]

1984 (4)

1983 (1)

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

1982 (3)

R. J. Watt, D. P. Andrews, “Contour curvature analysis: hyperacuities in the discrimination of detailed shape,” Vision Res. 22, 449–460 (1982).
[CrossRef] [PubMed]

J. Hirsch, R. Hylton, “Limits of spatial frequency discrimination as evidence of neural interpolation,” J. Opt. Soc. Am. 72, 1367–1374 (1982).
[CrossRef] [PubMed]

G. Westheimer, “The spatial grain of the perifoveal visual field,” Vision Res. 22, 157–162 (1982).
[CrossRef] [PubMed]

1981 (1)

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

1980 (1)

1978 (1)

C. R. Carlson, “Thresholds for perceived image sharpness,” Photogr. Sci. Eng. 22, 69–71 (1978).

1974 (3)

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

C. Blakemore, R. Over, “Curvature detectors in human vision?” Perception 3, 3–7 (1974).
[CrossRef] [PubMed]

R. F. Quick, “A vector-magnitude model of contrast detection,” Kybernetik 16, 1299–1302 (1974).
[CrossRef]

1967 (1)

J. Ogilvie, E. Daicar, “The perception of curvature,” Can. J. Physiol. 21, 521–525 (1967).

1954 (1)

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

1953 (1)

Andrews, D. P.

R. J. Watt, D. P. Andrews, “Contour curvature analysis: hyperacuities in the discrimination of detailed shape,” Vision Res. 22, 449–460 (1982).
[CrossRef] [PubMed]

Attneave, F.

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Blakemore, C.

C. Blakemore, R. Over, “Curvature detectors in human vision?” Perception 3, 3–7 (1974).
[CrossRef] [PubMed]

Carlson, C. R.

C. R. Carlson, “Thresholds for perceived image sharpness,” Photogr. Sci. Eng. 22, 69–71 (1978).

Daicar, E.

J. Ogilvie, E. Daicar, “The perception of curvature,” Can. J. Physiol. 21, 521–525 (1967).

Foley, J. M.

Gelb, D. J.

Hildebrand, F. B.

F. B. Hildebrand, Advanced Calculus for Applications, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1976).

Hirsch, J.

Hylton, R.

Knuth, D. E.

D. E. Knuth, Tex/Metafont (Academic, New York, 1982).

Legge, G. E.

Leibowitz, H.

McFarlane, D. K.

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

Nachmias, J.

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

Ogilvie, J.

J. Ogilvie, E. Daicar, “The perception of curvature,” Can. J. Physiol. 21, 521–525 (1967).

Over, R.

C. Blakemore, R. Over, “Curvature detectors in human vision?” Perception 3, 3–7 (1974).
[CrossRef] [PubMed]

Phillips, G. C.

G. C. Phillips, H. R. Wilson, “Orientation bandwidths of spatial mechanisms measured by masking,” J. Opt. Soc. Am. A 1, 226–232 (1984).
[CrossRef] [PubMed]

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

Quick, R. F.

R. F. Quick, “A vector-magnitude model of contrast detection,” Kybernetik 16, 1299–1302 (1974).
[CrossRef]

Regan, D.

Robson, J. G.

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

Sansbury, R. V.

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

Swanson, W. H.

W. H. Swanson, H. R. Wilson, “Eccentricity dependence of contrast matching and oblique masking,” Vision Res. (to be published).

Watson, A. B.

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

Watt, R. J.

R. J. Watt, “Further evidence concerning the analysis of curvature in human foveal vision,” Vision Res. 24, 251–253 (1984).
[CrossRef] [PubMed]

R. J. Watt, D. P. Andrews, “Contour curvature analysis: hyperacuities in the discrimination of detailed shape,” Vision Res. 22, 449–460 (1982).
[CrossRef] [PubMed]

Westheimer, G.

G. Westheimer, “The spatial grain of the perifoveal visual field,” Vision Res. 22, 157–162 (1982).
[CrossRef] [PubMed]

Wilson, H. R.

Zucker, S. W.

S. W. Zucker, “Cooperative grouping and early orientation selection,” in Physical and Biological Processing of Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, New York, 1983), pp. 326–334.
[CrossRef]

Can. J. Physiol. (1)

J. Ogilvie, E. Daicar, “The perception of curvature,” Can. J. Physiol. 21, 521–525 (1967).

J. Opt. Soc. Am. (3)

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

Kybernetik (1)

R. F. Quick, “A vector-magnitude model of contrast detection,” Kybernetik 16, 1299–1302 (1974).
[CrossRef]

Perception (1)

C. Blakemore, R. Over, “Curvature detectors in human vision?” Perception 3, 3–7 (1974).
[CrossRef] [PubMed]

Photogr. Sci. Eng. (1)

C. R. Carlson, “Thresholds for perceived image sharpness,” Photogr. Sci. Eng. 22, 69–71 (1978).

Psychol. Rev. (1)

F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
[CrossRef] [PubMed]

Vision Res. (6)

R. J. Watt, D. P. Andrews, “Contour curvature analysis: hyperacuities in the discrimination of detailed shape,” Vision Res. 22, 449–460 (1982).
[CrossRef] [PubMed]

R. J. Watt, “Further evidence concerning the analysis of curvature in human foveal vision,” Vision Res. 24, 251–253 (1984).
[CrossRef] [PubMed]

H. R. Wilson, D. K. McFarlane, G. C. Phillips, “Spatial frequency tuning of orientation selective units estimated by oblique masking,” Vision Res. 23, 873–882 (1983).
[CrossRef] [PubMed]

G. Westheimer, “The spatial grain of the perifoveal visual field,” Vision Res. 22, 157–162 (1982).
[CrossRef] [PubMed]

A. B. Watson, J. G. Robson, “Discrimination at threshold: labelled detectors in human vision,” Vision Res. 21, 1115–1122 (1981).
[CrossRef] [PubMed]

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

Other (4)

W. H. Swanson, H. R. Wilson, “Eccentricity dependence of contrast matching and oblique masking,” Vision Res. (to be published).

D. E. Knuth, Tex/Metafont (Academic, New York, 1982).

F. B. Hildebrand, Advanced Calculus for Applications, 2nd ed. (Prentice-Hall, Englewood Cliffs, N.J., 1976).

S. W. Zucker, “Cooperative grouping and early orientation selection,” in Physical and Biological Processing of Images, O. J. Braddick, A. C. Sleigh, eds. (Springer-Verlag, New York, 1983), pp. 326–334.
[CrossRef]

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Figures (10)

Fig. 1
Fig. 1

Examples of curved contours used as stimuli. As shown in A, stimuli were constructed from a portion of a parabolic arc to which straight-line segments at ±45.0° were joined smoothly. The continuation of the parabola (dashed lines) was not visible, and patterns therefore appeared as in B. Curvature in A is greater than that in B.

Fig. 2
Fig. 2

Curvature-increment thresholds plotted as percent of curvature (Weber fractions) versus curvature. Data for both subjects describe U-shaped functions, with optima falling in the 0.02–0.06 range.

Fig. 3
Fig. 3

Curvature-increment thresholds plotted against standard curvature. Filled circles are data for two subjects using 1.6-min-wide contours. These have been replotted from Fig. 2 to emphasize the power-law relationship (straight lines in these coordinates) for curvatures above about 2.0 per deg. Open circles represent data obtained with a 0.4-min-wide contour and indicate that thresholds are uniformly increased for narrower contours.

Fig. 4
Fig. 4

Oblique effect for curvature discrimination. Filled circles for both subjects were obtained by using contours with the tangent at the point of maximum-curvature horizontal, as is shown in Fig. 1. Open circles were measured with patterns that had been rotated 45.0°, so that this tangent was oblique. With the exception of one point for WHS, curvature thresholds were higher for the oblique rather than the horizontal-tangent condition.

Fig. 5
Fig. 5

Effects of cosine-grating masks on curvature discrimination as a function of mask spatial frequency. Threshold elevations are ratios of masked to unmasked curvature thresholds, so that a value of 1.0 indicates no masking. Filled circles are data for horizontal masks, whose bars were parallel to the test contour tangent at its point of maximum curvature, whereas open circles are data obtained with vertical gratings perpendicular to this tangent (cpd, cycles per degree).

Fig. 6
Fig. 6

Effect of eccentricity on curvature thresholds. In the left-hand panel, filled and open symbols are data for two subjects; circles and squares represent foveal curvatures of 4.0 and 8.0 per deg, respectively. Patterns at eccentric locations were scaled up linearly in accord with estimates of variations in the spatial scale of visual-system processing (see text). Points in the right-hand panel represent averages at each eccentricity of all data from the left-hand panel. The dashed line in the right-hand panel is drawn through the foveal point and indicates the expected eccentricity variation if curvature discrimination scaled with eccentricity.

Fig. 7
Fig. 7

Diagram indicating why orientation-selective visual filters will respond differently to contours of differing curvature. A, Three horizontally oriented filters that are vertically displaced to represent the spatial separation of spatial nearest neighbors in the model. The horizontal separation is for clarity only. B and C, Stimulation of a single filter by contours of smaller and larger curvature, respectively. As the larger-curvature stimulus invades more of the inhibitory surround (gray), it will generate a significantly smaller response.

Fig. 8
Fig. 8

Diagram to elucidate the differential responses to curvature of filters at a 15.0° orientation to the horizontal. A, Filters oriented at −15.0°, 0.0°, and 15.0°, which represent the variation in orientation from filter to filter in the model for the high-spatial-frequency mechanisms. (Lower-frequency mechanisms have somewhat greater orientation spacing.) B and C, Stimulation of an oriented filter by contours of smaller and greater curvature, respectively. The contour of greater curvature clearly stimulates less of the center and more of the surround and will therefore generate a significantly smaller response than the contour of lesser curvature.

Fig. 9
Fig. 9

Theoretical curvature thresholds determined by the line-element model compared with data for two subjects. The solid lines indicate the precise theoretical results that were obtained with all constants determined from previous masking studies. Correlations between theory and data were 0.982 for DB and 0.977 for HRW. The dashed lines are vertically (but not horizontally) scaled versions of the theory that emphasize the high correlation between the shape of the theoretical curve and the data.

Fig. 10
Fig. 10

Curve showing theoretical curvature-increment thresholds plotted as a function of the range of orientations present along the stimulus contour. The arrow shows that by an orientation range of 40.0° theoretical thresholds have reached their asymptotic value. This agrees with the observation by Watt and Andrews that curvature-processing mechanisms can only efficiently utilize a 40.0° range of orientations.

Equations (5)

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K = d 2 y / d x 2 [ 1 + ( d y / d x ) 2 ] 3 / 2 .
K = 2 A ( 1 + 4 A 2 x 2 ) 3 / 2 .
LSF ( x , y ) = A [ exp ( x 2 / σ 1 2 ) B exp ( x 2 / σ 2 2 ) + C exp ( x 2 / σ 3 2 ) exp ( y 2 / σ y 2 ) ,
S i ( x , y ) = + L S F i ( x x , y y ) P ( x , y ) P ( x , y ) d x d y .
F i ( S C ) = ( S C ) 2 + K ( S C ) 3 K + ( S C ) 2 ,

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