Abstract

Visual processing of contour curvature was investigated by measuring increment thresholds for curvatures from 0.31 to 25.4 deg−1. Curvature discrimination was assessed for three classes of stimuli: simple curved contours, high-frequency bandpass-filtered contours, and low-pass-filtered contours. High-frequency bandpass filtering had no effect on discrimination at low curvatures and only a modest effect at high curvatures. In contrast, low-pass filtering caused substantial threshold elevations at all curvatures. Thus the data lead to the surprising conclusion that high-spatial-frequency, orientation-selective mechanisms dominate curvature processing over the entire range of curvatures tested, a conclusion at odds with previous suggestions that large, low-spatial-frequency filters are involved in analyzing low curvatures. The data are explained accurately by a two-process model for curvature extraction: at high curvatures the local-processing model proposed by Wilson [ J. Opt. Soc. Am. A. 2, 1191 ( 1985)] fits the data well, whereas at low curvatures orientations are compared at points displaced a fixed distance along the tangent to the curve.

© 1989 Optical Society of America

Full Article  |  PDF Article

References

  • View by:
  • |
  • |
  • |

  1. F. Attneave, “Some informational aspects of visual perception,” Psychol. Rev. 61, 183–193 (1954).
    [CrossRef] [PubMed]
  2. D. D. Hoffman, W. Richards, “Parts of recognition,” Cognition 18, 65–96 (1984).
    [CrossRef] [PubMed]
  3. W. Richards, B. Dawson, D. Whittington, “Encoding contour shape by curvature extrema,” J. Opt. Soc. Am. A 3, 1483–1491 (1986).
    [CrossRef] [PubMed]
  4. H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A 1, 124-131 (1984).
    [CrossRef] [PubMed]
  5. H. R. Wilson, D. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A1, 1091–1096 (1984).
  6. H. R. Wilson, “Discrimination of contour curvature: data and theory,” J. Opt. Soc. Am. A 2, 1191–1199 (1985).
    [CrossRef] [PubMed]
  7. R. J. Watt, D. P. Andrews, “Contour curvature analysis: hyperacuities in the discrimination of detailed shape,” Vision Res. 22, 449–460 (1982).
    [CrossRef] [PubMed]
  8. R. F. Quick, “A vector-magnitude model of contrast detection,” Kybernetik 16, 1299–1302 (1974).
    [CrossRef]
  9. R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence for spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
    [CrossRef] [PubMed]
  10. H. R. Wilson, “Responses of spatial mechanisms can explain hyperacuity,” Vision Res. 26, 453–469 (1986).
    [CrossRef] [PubMed]
  11. M. J. Morgan, D. Regan, “Opponent model for line interval discrimination: interval and vernier performance compared,” Vision Res. 27, 107–118 (1987).
    [CrossRef] [PubMed]
  12. 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]
  13. 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]
  14. W. Richards, A. Polit, “Texture matching,” Kybernetik 16, 155–162 (1974).
    [CrossRef] [PubMed]
  15. W. Richards, “Quantifying sensory channels: generalizing colorimetry to orientation and texture, touch and tones,” Sensory Processes 3, 207–229 (1979).
  16. R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
    [CrossRef]
  17. H. R. Wilson, “Development of spatiotemporal mechanisms in infant vision,” Vision Res. 28, 611–628 (1988).
    [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. C. R. Carlson, “Thresholds for perceived image sharpness,” Photogr. Sci. Eng. 22, 69–71 (1978).
  20. G. E. Legge, J. M. Foley, “Contrast masking in human vision,”J. Opt. Soc. Am. 70, 1458–1470 (1980).
    [CrossRef] [PubMed]
  21. D. R. Williams, “Aliasing in human foveal vision,” Vision Res. 25, 195–205 (1985).
    [CrossRef] [PubMed]
  22. D. R. Williams, “Seeing through the photoreceptor mosaic,” Trends Neurosci. 9, 193–198 (1986).
    [CrossRef]
  23. H. R. Wilson, W. Richards, “Curvature and separation discrimination at texture boundaries,” Invest. Ophthalmol. Vis. Sci. Suppl. 29, 408 (1988).
  24. J. J. Koenderink, W. Richards, “Two-dimensional curvature operators,” J. Opt. Soc. Am. A 5, 1136–1141 (1988).
    [CrossRef]
  25. J. J. Koenderink, A. J. Van Doorn, “Representations of local geometry in the visual system,” Biol. Cybern. 55, 1–9 (1986).
  26. T. Pavlidis, Algorithms for Graphics and Image Processing (Computer Science, Rockville, Md., 1982).
    [CrossRef]
  27. M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” in Proceedings of the IEEE 1st International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 259–268.
  28. P. Parent, S. W. Zucker, “Trace inference, curvature consistency, and curve detection,” Pub. CIM-86-3 (McGill Research Center for Intelligent Machines, Montreal, Quebec, Canada, 1985).
  29. M. Brady, H. Asada, “Smoothed local symmetries and their implementation,” Robotics Res. 3, 37–61 (1984).
  30. A. Dobbins, S. W. Zucker, M. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
    [CrossRef] [PubMed]
  31. D. Marr, “Early processing of visual information,” Philos. Trans. R. Soc. London Ser. B 275, 483–524 (1976).
    [CrossRef]
  32. M. Ferraro, D. H. Foster, “Discrete and continuous modes of curved line discrimination controlled by effective stimulus duration,” Spatial Vision 1, 219–230 (1986).
    [CrossRef]
  33. C. Blakemore, R. Over, “Curvature detectors in human vision?” Perception 3, 3–7 (1974).
    [CrossRef] [PubMed]
  34. D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
    [CrossRef]

1988 (3)

H. R. Wilson, “Development of spatiotemporal mechanisms in infant vision,” Vision Res. 28, 611–628 (1988).
[CrossRef] [PubMed]

H. R. Wilson, W. Richards, “Curvature and separation discrimination at texture boundaries,” Invest. Ophthalmol. Vis. Sci. Suppl. 29, 408 (1988).

J. J. Koenderink, W. Richards, “Two-dimensional curvature operators,” J. Opt. Soc. Am. A 5, 1136–1141 (1988).
[CrossRef]

1987 (2)

A. Dobbins, S. W. Zucker, M. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

M. J. Morgan, D. Regan, “Opponent model for line interval discrimination: interval and vernier performance compared,” Vision Res. 27, 107–118 (1987).
[CrossRef] [PubMed]

1986 (5)

W. Richards, B. Dawson, D. Whittington, “Encoding contour shape by curvature extrema,” J. Opt. Soc. Am. A 3, 1483–1491 (1986).
[CrossRef] [PubMed]

H. R. Wilson, “Responses of spatial mechanisms can explain hyperacuity,” Vision Res. 26, 453–469 (1986).
[CrossRef] [PubMed]

M. Ferraro, D. H. Foster, “Discrete and continuous modes of curved line discrimination controlled by effective stimulus duration,” Spatial Vision 1, 219–230 (1986).
[CrossRef]

J. J. Koenderink, A. J. Van Doorn, “Representations of local geometry in the visual system,” Biol. Cybern. 55, 1–9 (1986).

D. R. Williams, “Seeing through the photoreceptor mosaic,” Trends Neurosci. 9, 193–198 (1986).
[CrossRef]

1985 (2)

1984 (5)

H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A 1, 124-131 (1984).
[CrossRef] [PubMed]

H. R. Wilson, D. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A1, 1091–1096 (1984).

D. D. Hoffman, W. Richards, “Parts of recognition,” Cognition 18, 65–96 (1984).
[CrossRef] [PubMed]

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]

M. Brady, H. Asada, “Smoothed local symmetries and their implementation,” Robotics Res. 3, 37–61 (1984).

1983 (2)

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]

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence for spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
[CrossRef] [PubMed]

1982 (2)

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. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

1980 (1)

1979 (2)

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

W. Richards, “Quantifying sensory channels: generalizing colorimetry to orientation and texture, touch and tones,” Sensory Processes 3, 207–229 (1979).

1978 (1)

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

1976 (1)

D. Marr, “Early processing of visual information,” Philos. Trans. R. Soc. London Ser. B 275, 483–524 (1976).
[CrossRef]

1974 (4)

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

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

W. Richards, A. Polit, “Texture matching,” Kybernetik 16, 155–162 (1974).
[CrossRef] [PubMed]

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

1954 (1)

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

Albrecht, D. G.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

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]

Asada, H.

M. Brady, H. Asada, “Smoothed local symmetries and their implementation,” Robotics Res. 3, 37–61 (1984).

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]

Brady, M.

M. Brady, H. Asada, “Smoothed local symmetries and their implementation,” Robotics Res. 3, 37–61 (1984).

Carlson, C. R.

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

Cynader, M.

A. Dobbins, S. W. Zucker, M. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

Dawson, B.

DeValois, R. L.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

Dobbins, A.

A. Dobbins, S. W. Zucker, M. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

Ferraro, M.

M. Ferraro, D. H. Foster, “Discrete and continuous modes of curved line discrimination controlled by effective stimulus duration,” Spatial Vision 1, 219–230 (1986).
[CrossRef]

Foley, J. M.

Foster, D. H.

M. Ferraro, D. H. Foster, “Discrete and continuous modes of curved line discrimination controlled by effective stimulus duration,” Spatial Vision 1, 219–230 (1986).
[CrossRef]

Gelb, D. J.

Hoffman, D. D.

D. D. Hoffman, W. Richards, “Parts of recognition,” Cognition 18, 65–96 (1984).
[CrossRef] [PubMed]

Kass, M.

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” in Proceedings of the IEEE 1st International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 259–268.

Koenderink, J. J.

J. J. Koenderink, W. Richards, “Two-dimensional curvature operators,” J. Opt. Soc. Am. A 5, 1136–1141 (1988).
[CrossRef]

J. J. Koenderink, A. J. Van Doorn, “Representations of local geometry in the visual system,” Biol. Cybern. 55, 1–9 (1986).

Legge, G. E.

Marr, D.

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

D. Marr, “Early processing of visual information,” Philos. Trans. R. Soc. London Ser. B 275, 483–524 (1976).
[CrossRef]

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]

Morgan, M. J.

M. J. Morgan, D. Regan, “Opponent model for line interval discrimination: interval and vernier performance compared,” Vision Res. 27, 107–118 (1987).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence for spatial primitives in human vision,” Vision Res. 23, 1465–1477 (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]

Over, R.

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

Parent, P.

P. Parent, S. W. Zucker, “Trace inference, curvature consistency, and curve detection,” Pub. CIM-86-3 (McGill Research Center for Intelligent Machines, Montreal, Quebec, Canada, 1985).

Pavlidis, T.

T. Pavlidis, Algorithms for Graphics and Image Processing (Computer Science, Rockville, Md., 1982).
[CrossRef]

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]

Poggio, T.

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

Polit, A.

W. Richards, A. Polit, “Texture matching,” Kybernetik 16, 155–162 (1974).
[CrossRef] [PubMed]

Quick, R. F.

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

Regan, D.

M. J. Morgan, D. Regan, “Opponent model for line interval discrimination: interval and vernier performance compared,” Vision Res. 27, 107–118 (1987).
[CrossRef] [PubMed]

H. R. Wilson, D. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A1, 1091–1096 (1984).

Richards, W.

H. R. Wilson, W. Richards, “Curvature and separation discrimination at texture boundaries,” Invest. Ophthalmol. Vis. Sci. Suppl. 29, 408 (1988).

J. J. Koenderink, W. Richards, “Two-dimensional curvature operators,” J. Opt. Soc. Am. A 5, 1136–1141 (1988).
[CrossRef]

W. Richards, B. Dawson, D. Whittington, “Encoding contour shape by curvature extrema,” J. Opt. Soc. Am. A 3, 1483–1491 (1986).
[CrossRef] [PubMed]

D. D. Hoffman, W. Richards, “Parts of recognition,” Cognition 18, 65–96 (1984).
[CrossRef] [PubMed]

W. Richards, “Quantifying sensory channels: generalizing colorimetry to orientation and texture, touch and tones,” Sensory Processes 3, 207–229 (1979).

W. Richards, A. Polit, “Texture matching,” Kybernetik 16, 155–162 (1974).
[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]

Terzopoulos, D.

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” in Proceedings of the IEEE 1st International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 259–268.

Thorell, L. G.

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

Van Doorn, A. J.

J. J. Koenderink, A. J. Van Doorn, “Representations of local geometry in the visual system,” Biol. Cybern. 55, 1–9 (1986).

Watt, R. J.

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence for spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
[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]

Whittington, D.

Williams, D. R.

D. R. Williams, “Seeing through the photoreceptor mosaic,” Trends Neurosci. 9, 193–198 (1986).
[CrossRef]

D. R. Williams, “Aliasing in human foveal vision,” Vision Res. 25, 195–205 (1985).
[CrossRef] [PubMed]

Wilson, H. R.

H. R. Wilson, W. Richards, “Curvature and separation discrimination at texture boundaries,” Invest. Ophthalmol. Vis. Sci. Suppl. 29, 408 (1988).

H. R. Wilson, “Development of spatiotemporal mechanisms in infant vision,” Vision Res. 28, 611–628 (1988).
[CrossRef] [PubMed]

H. R. Wilson, “Responses of spatial mechanisms can explain hyperacuity,” Vision Res. 26, 453–469 (1986).
[CrossRef] [PubMed]

H. R. Wilson, “Discrimination of contour curvature: data and theory,” J. Opt. Soc. Am. A 2, 1191–1199 (1985).
[CrossRef] [PubMed]

H. R. Wilson, D. J. Gelb, “Modified line-element theory for spatial-frequency and width discrimination,” J. Opt. Soc. Am. A 1, 124-131 (1984).
[CrossRef] [PubMed]

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. Regan, “Spatial-frequency adaptation and grating discrimination: predictions of a line-element model,” J. Opt. Soc. Am. A1, 1091–1096 (1984).

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]

Witkin, A.

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” in Proceedings of the IEEE 1st International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 259–268.

Zucker, S. W.

A. Dobbins, S. W. Zucker, M. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

P. Parent, S. W. Zucker, “Trace inference, curvature consistency, and curve detection,” Pub. CIM-86-3 (McGill Research Center for Intelligent Machines, Montreal, Quebec, Canada, 1985).

Biol. Cybern. (1)

J. J. Koenderink, A. J. Van Doorn, “Representations of local geometry in the visual system,” Biol. Cybern. 55, 1–9 (1986).

Cognition (1)

D. D. Hoffman, W. Richards, “Parts of recognition,” Cognition 18, 65–96 (1984).
[CrossRef] [PubMed]

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

H. R. Wilson, W. Richards, “Curvature and separation discrimination at texture boundaries,” Invest. Ophthalmol. Vis. Sci. Suppl. 29, 408 (1988).

J. Opt. Soc. Am. (1)

J. Opt. Soc. Am. A (6)

Kybernetik (2)

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

W. Richards, A. Polit, “Texture matching,” Kybernetik 16, 155–162 (1974).
[CrossRef] [PubMed]

Nature (1)

A. Dobbins, S. W. Zucker, M. Cynader, “Endstopped neurons in the visual cortex as a substrate for calculating curvature,” Nature 329, 438–441 (1987).
[CrossRef] [PubMed]

Perception (1)

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

Philos. Trans. R. Soc. London Ser. B (1)

D. Marr, “Early processing of visual information,” Philos. Trans. R. Soc. London Ser. B 275, 483–524 (1976).
[CrossRef]

Photogr. Sci. Eng. (1)

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

Proc. R. Soc. London Ser. B (1)

D. Marr, T. Poggio, “A computational theory of human stereo vision,” Proc. R. Soc. London Ser. B 204, 301–328 (1979).
[CrossRef]

Psychol. Rev. (1)

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

Robotics Res. (1)

M. Brady, H. Asada, “Smoothed local symmetries and their implementation,” Robotics Res. 3, 37–61 (1984).

Sensory Processes (1)

W. Richards, “Quantifying sensory channels: generalizing colorimetry to orientation and texture, touch and tones,” Sensory Processes 3, 207–229 (1979).

Spatial Vision (1)

M. Ferraro, D. H. Foster, “Discrete and continuous modes of curved line discrimination controlled by effective stimulus duration,” Spatial Vision 1, 219–230 (1986).
[CrossRef]

Trends Neurosci. (1)

D. R. Williams, “Seeing through the photoreceptor mosaic,” Trends Neurosci. 9, 193–198 (1986).
[CrossRef]

Vision Res. (9)

R. L. DeValois, D. G. Albrecht, L. G. Thorell, “Spatial frequency selectivity of cells in macaque visual cortex,” Vision Res. 22, 545–559 (1982).
[CrossRef]

H. R. Wilson, “Development of spatiotemporal mechanisms in infant vision,” Vision Res. 28, 611–628 (1988).
[CrossRef] [PubMed]

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

D. R. Williams, “Aliasing in human foveal vision,” Vision Res. 25, 195–205 (1985).
[CrossRef] [PubMed]

R. J. Watt, M. J. Morgan, “The recognition and representation of edge blur: evidence for spatial primitives in human vision,” Vision Res. 23, 1465–1477 (1983).
[CrossRef] [PubMed]

H. R. Wilson, “Responses of spatial mechanisms can explain hyperacuity,” Vision Res. 26, 453–469 (1986).
[CrossRef] [PubMed]

M. J. Morgan, D. Regan, “Opponent model for line interval discrimination: interval and vernier performance compared,” Vision Res. 27, 107–118 (1987).
[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]

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

Other (3)

T. Pavlidis, Algorithms for Graphics and Image Processing (Computer Science, Rockville, Md., 1982).
[CrossRef]

M. Kass, A. Witkin, D. Terzopoulos, “Snakes: active contour models,” in Proceedings of the IEEE 1st International Conference on Computer Vision (Institute of Electrical and Electronics Engineers, New York, 1987), pp. 259–268.

P. Parent, S. W. Zucker, “Trace inference, curvature consistency, and curve detection,” Pub. CIM-86-3 (McGill Research Center for Intelligent Machines, Montreal, Quebec, Canada, 1985).

Cited By

OSA participates in CrossRef's Cited-By Linking service. Citing articles from OSA journals and other participating publishers are listed here.

Alert me when this article is cited.


Figures (11)

Fig. 1
Fig. 1

Schematic of curvature processing by units with orientation-selective receptive fields. Receptive fields are represented as having an excitatory center (white) and inhibitory flanking regions (gray). Although not shown, units that have other orientations but are centered at the same point would also be involved in curvature processing. A, A small receptive field tuned to high spatial frequencies would give a differential response to sharp curvatures, as the curved contour crosses from the excitatory region into the inhibitory region within the length of the receptive field. B, At lower curvatures, larger receptive fields tuned to lower spatial frequencies would be required to cover the same extent of the curve as the smaller receptive field in A. Note that the small receptive field in A would respond well to the contour in B but would be too short to provide accurate curvature discrimination. C, As an alternative strategy for processing low curvatures, the visual system might compare contour orientations estimated by units displaced along the curve.

Fig. 2
Fig. 2

Examples of curved contours used in these experiments. A and B are examples of simple contours containing a wide range of spatial frequencies, with the curvature in B being greater than that in A. C and D illustrate two bandpass-filtered curves (D is the greater curvature) in which the width of the white center is exactly balanced by that of the black flanks to produce a stimulus with an average luminance equal to the gray background. The arrangement of the white and black portions of the bandpass-filtered curve is depicted more clearly in the magnified portion of the screen in E. Note that the total widths of the white regions and the black regions are the same. As shown in A–D, the heights of the curves on the viewing screen were varied randomly from trial to trial to eliminate absolute position as a possible spurious cue for discrimination.

Fig. 3
Fig. 3

A, Curvature increment thresholds for two subjects, HRW and WAR, as a function of base curvature. The data plateau for curvatures below about 2.0 deg−1 but increase rapidly at higher curvatures. The standard deviation (shown as bars above and below a data point) in the upper right-hand corner was typical of the data for both subjects and all curvatures. B, The same data were divided by the curvature and plotted to show curvature Weber fractions. Weber fractions reach a minimum of ~5.0% at intermediate values but increase at both low and high curvatures. The increment-threshold data in A form the baseline against which threshold elevations produced by low-pass and bandpass filtering were measured. The theoretical curves (solid curves) in A and B reflect the operation of the two curvature-discrimination processes shown in Fig. 9 and discussed in the text.

Fig. 4
Fig. 4

Threshold elevations, for subjects HRW and WAR, produced by the bandpass-filtered curves shown in Figs. 2C and 2D. The peak frequency of the bandpass filter was 25.0 cpd. Threshold elevation is defined as the ratio of the increment threshold for the bandpass-filtered curve to the threshold for the simple unfiltered curve (data in Fig. 3A), and thus a value of 1.0 indicates that the filtering had no effect. Bandpass filtering elevated thresholds at high curvatures but had no effect at low curvatures. The error bars show standard deviations for one subject; those for the second were comparable. The theoretical curve (solid curve) was calculated by using the two modes of curvature processing described in the text.

Fig. 5
Fig. 5

Threshold elevations relative to the data in Fig. 3A produced by low-pass filtering. The low-pass characteristic of the ground-glass filter was adjusted so that the grating acuity of each subject was reduced to 7.5 cpd. Average threshold elevations ranged from a factor of ~2.0 at low curvatures to ~6.5 at high ones. Thus it is clear that visual mechanisms tuned to low spatial frequencies are less effective at curvature discrimination at all curvatures than are high-frequency mechanisms. The curve shows threshold elevations calculated with the two-process curvature model.

Fig. 6
Fig. 6

Contrast dependence of curvature discrimination for two subjects at curvatures of 10.66 deg−1 (squares) and 0.61 deg−1 (triangles). In the tested contrast range from 25 to 100%, regression lines fitted to the data on logarithmic coordinates had slopes averaging −0.33 and −0.26, respectively, at the higher and lower curvatures.

Fig. 7
Fig. 7

Predictions of the local-curvature-discrimination model (solid curves) described by Wilson6 compared with data replotted from Figs. 3A and 4B. In agreement with a previous application of this model to curvature data, the fit in A is accurate at curvatures of ≳2.0 deg−1 but systematically too high at low curvatures. That the systematic deviation of the low-curvature predictions in A reflects a major failing of the model is shown by the comparison in B, in which the model again accurately predicts threshold elevations at high curvatures but incorrectly predicts large threshold elevations at lower curvatures as well.

Fig. 8
Fig. 8

Synopsis of a curvature-encoding scheme operative at low curvatures. Small, high-frequency central units at C compute the local orientation or tangent to the curve. The high-frequency units D are displaced a fixed distance ±Δ from C along the direction of the tangent to the curve. Comparison of the difference Θ in contour orientation between C and D can be used to provide an estimate of the radius of curvature, as is evident from the simple geometry in the lower portion of the diagram [see Eq. (6)]. As the same fixed distance Δ along the estimated tangent to the curve is used at all curvatures, this scheme does not require a priori knowledge of the curve to be processed.

Fig. 9
Fig. 9

Curvature increment-threshold data replotted from Fig. 3A compared with model calculations for two curvature-discrimination processes. The solid curve shows curvature discrimination for units centered at the curvature extremum as described previously by Wilson6 and plotted in Fig. 7A. The dashed curve indicates increment thresholds calculated with the displaced high-frequency units shown in Fig. 8. Overall visual performance is given by the lower envelope of these two curves, which is shown by the solid curves in Fig. 3.

Fig. 10
Fig. 10

Response of a linear combination [see Eq. (7)] of five high-spatial-frequency, orientation-selective units as a function of contour curvature. Filled symbols indicate the response to downward curvature, and open symbols indicate the much weaker response to upward curvature. The filled symbols provide a neural signal that is correlated very highly (coefficient of 0.99) with the actual curvature (straight diagonal line). The hypothetical receptive field of this combination of units is shown in Fig. 11A.

Fig. 11
Fig. 11

Excitatory and inhibitory regions of two hypothetical receptive fields that might be involved in curvature encoding. Excitatory zones are shown as white (strongest) and light-gray areas, and inhibitory regions are indicated by black (strongest) and dark-gray areas. Each panel is 10.0 arcmin on a side. A, Receptive field produced by the linear combination of five high-spatial frequency, orientation-selective receptive fields [see Eq. 7)]. Because of the shape of the excitatory zone, this receptive field is optimally sensitive to downward-curving contours. B, Receptive field for curvature extraction derived from differential geometry.24,25 Note the similarity between this and the lower two thirds of the field in A. As there is currently no physiological evidence for receptive fields such as these, it is possible that the visual system encodes curvature only implicitly as a pattern of activity in a localized population of neurons.

Equations (7)

Equations on this page are rendered with MathJax. Learn more.

y = { - A x 2 for x 1 2 A 1 4 A - x for x > 1 2 A .
RF ( 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 ) = - - RF i ( x - x , y - y ) P ( x , y ) d x d y .
F ( S C ) = ( S C ) 2 + K ( S C ) 3 - K + ( S C ) 2 ,
Δ F = [ i = 1 6 Θ = 0 180 Δ x = - 1 + 1 Δ F i ( S i C ) 2 ] 0.5 ,
R = 1 K = Δ sin ( Θ D - Θ C ) ,
K = 3.2 ( R + 30 + R - 30 ) + 2.25 ( R + 45 + R - 45 ) - 3.5 R neighbor ,

Metrics