Abstract

We propose that neural grouping of retinotopically distributed responses in the primary visual cortex (V1) is essential for the determination of apparent tilt, including the tilt illusion. Our psychophysical study shows that apparent tilt is independent of stereo disparity, hue, or contrast of bars, which determine the ownership of their intersection. This leads us to suspect that the neuronal responses within the intersection are excluded from the computation of apparent tilt. To investigate the underlying cortical mechanisms, we developed and examined a V1 network model including the collinear connections observed in the superficial layers. The model shows good agreement with the results of psychophysical experiments, including segmentation independence, contrast dependence, and apparent tilt for various stimuli. The results suggest that collinear connections underlie the neural grouping that excludes the intersection region and establishes the independence of segmentation.

© 2002 Optical Society of America

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  16. A. F. Rossi, R. Desimone, L. G. Ungerleider, “Contextual modulation in primary visual cortex of macaques,” J. Neurosci. 21, 1698–1709 (2001).
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    [CrossRef] [PubMed]
  38. F. W. Campbell, J. G. Gibson, “Application of Fourier analysis to the visibility of gratings,” J. Physiol. 197, 551–566 (1968).
  39. K. Sakai, S. Tanaka, “Preceptual segmentation and neural grouping in tilt illusion,” Neurocomputing 32–33, 979–986 (2000).
    [CrossRef]

2001 (1)

A. F. Rossi, R. Desimone, L. G. Ungerleider, “Contextual modulation in primary visual cortex of macaques,” J. Neurosci. 21, 1698–1709 (2001).
[PubMed]

2000 (7)

A. V. Popple, D. Sagi, “A Fraser illusion without local cues?” Vision Res. 40, 873–878 (2000).
[CrossRef] [PubMed]

M. J. Morgan, A. J. S. Mason, S. Baldassi, “Are there separate first-order and second-order mechanisms for orientation discrimination?” Vision Res. 40, 1751–1763 (2000).
[CrossRef] [PubMed]

J. S. Bakin, K. Nakayama, C. D. Gilbert, “Visual responses in monkey areas V1 and V2 to three-dimensional surface configurations,” J. Neurosci. 20, 8188–8198 (2000).
[PubMed]

H. Zhou, H. S. Friedman, R. von der Heydt, “Coding of border ownership in monkey visual cortex,” J. Neurosci. 20, 6594–6611 (2000).
[PubMed]

M. K. Kapadia, G. Westheimer, C. D. Gilbert, “Spatial distribution of contextual interactions in primary visual cortex and in visual perception,” J. Neurophysiol. 84, 2048–2062 (2000).
[PubMed]

K. Sakai, S. Tanaka, “Spatial pooling in the second-order spatial structure of cortical complex cells,” Vision Res. 40, 855–871 (2000).
[CrossRef] [PubMed]

K. Sakai, S. Tanaka, “Preceptual segmentation and neural grouping in tilt illusion,” Neurocomputing 32–33, 979–986 (2000).
[CrossRef]

1999 (4)

S-C. Yen, E. D. Menschik, L. H. Finkel, “Perceptual grouping in striate cortical networks mediated by synchronization and desynchronization,” Neurocomputing 26–27, 609–616 (1999).
[CrossRef]

M. K. Kapadia, G. Westheimer, C. D. Gilbert, “Dynamics in spatial summation in primary visual cortex of alert monkeys,” Proc. Natl. Acad. Sci. USA 96, 12073–12078 (1999).
[CrossRef]

K. Sakai, S. Tanaka, “Retinotopic coding and neural grouping in tilt illusion,” Neurocomputing 26–27, 837–843 (1999).
[CrossRef]

M. J. Morgan, “The Poggendorff illusion: a bias in the estimation of the orientation of virtual lines by second-stage filters,” Vision Res. 39, 2361–2380 (1999).
[CrossRef] [PubMed]

1998 (2)

S-C. Yen, L. H. Finkel, “Extraction of perceptually salient contours by striate cortical networks,” Vision Res. 38, 719–741 (1998).
[CrossRef] [PubMed]

Z. Li, “A neural model of contour integration in the primary visual cortex,” Neural Comput. 10, 903–940 (1998).
[CrossRef] [PubMed]

1997 (2)

W. H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,” J. Neurosci. 17, 2112–2127 (1997).
[PubMed]

M. J. Morgan, S. Baldassi, “How the human visual system encodes the orientation of a texture, and why it makes mistakes,” Curr. Biol. 7, 999–1002 (1997).
[CrossRef]

1995 (2)

L. Chao-Yi, G. Kun, “Measurement of geometric illusions, illusory contours and stereo-depth at luminance and colour contrast,” Vision Res. 35, 1713–1720 (1995).
[CrossRef]

K. Sakai, L. H. Finkel, “Characterization of spatial-frequency spectrum in the perception of shape from texture,” J. Opt. Soc. Am. A 12, 1208–1224 (1995).
[CrossRef]

1994 (1)

S. Kaski, T. Kohonen, “Winner-take-all networks for physiological models of competitive learning,” Neural Networks 7, 973–984 (1994).
[CrossRef]

1992 (1)

C. von der Malsburg, J. Buhmann, “Sensory segmentation with coupled neural oscillators,” Biol. Cybern. 67, 233–242 (1992).
[CrossRef] [PubMed]

1991 (1)

L. A. Bauman, A. B. Bonds, “Inhibitory refinement of spatial frequency selectivity in single cells of the cat striate cortex,” Vision Res. 31, 933–944 (1991).
[CrossRef] [PubMed]

1990 (2)

M. J. Morgan, C. Casco, “Spatial filtering and spatial primitives in early vision: an explanation of the Zöllner–Judd class of geometrical illusion,” Proc. R. Soc. London Ser. B 242, 1–10 (1990).
[CrossRef]

C. D. Gilbert, T. N. Wiesel, “The influence of contextual stimuli on the orientation selectivity of cells in primary visual cortex of the cat,” Vision Res. 30, 1689–1701 (1990).
[CrossRef] [PubMed]

1989 (4)

A. Yuille, N. M. Grzywacz, “A winner-take-all mechanism based on presynaptic inhibition feedback,” Neural Comput. 1, 334–347 (1989).
[CrossRef]

C. D. Gilbert, T. N. Wiesel, “Columnar specificity of intrinsic horizontal and corticocortical connections in cat visual cortex,” J. Neurosci. 9, 2432–2442 (1989).
[PubMed]

A. B. Bonds, “Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex,” Visual Neurosci. 2, 41–55 (1989).
[CrossRef]

R. von der Heydt, E. Peterhans, “Mechanisms of contour perception in monkey visual cortex. I. Lines of pattern discontinuity,” J. Neurosci. 9, 1731–1748 (1989).
[PubMed]

1988 (1)

M. A. Paradiso, “A theory for the use of visual orientation information which exploits the columnar structure of striate cortex,” Biol. Cybern. 58, 35–49 (1988).
[CrossRef] [PubMed]

1986 (1)

P. Wenderoth, T. O’Connor, M. Johnson, “The tilt illusion as a function of the relative and absolute lengths of test and inducing lines,” Percept. Psychophys. 39, 339–345 (1986).
[CrossRef] [PubMed]

1985 (1)

G. B. Ermentrout, “The behavior of rings of coupled oscillators,” J. Math. Biol. 23, 55–74 (1985).
[CrossRef] [PubMed]

1980 (1)

S. Magnussen, W. Kurtenbach, “Linear summation of tilt illusion and tilt aftereffect,” Vision Res. 20, 39–42 (1980).
[CrossRef] [PubMed]

1978 (1)

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

1975 (1)

T. Oyama, “Determinants of the Zöllner illusion,” Psychol. Res. 37, 261–280 (1975).
[CrossRef] [PubMed]

1973 (1)

R. H. S. Carpenter, C. Blakemore, “Interactions between orientations in human vision,” Exp. Brain Res. 18, 287–303 (1973).
[CrossRef] [PubMed]

1970 (1)

C. Blakemore, R. H. S. Carpenter, M. A. Georgeson, “Lateral inhibition between orientation detectors in the human visual system,” Nature (London) 228, 37–39 (1970).
[CrossRef]

1968 (1)

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

Bakin, J. S.

J. S. Bakin, K. Nakayama, C. D. Gilbert, “Visual responses in monkey areas V1 and V2 to three-dimensional surface configurations,” J. Neurosci. 20, 8188–8198 (2000).
[PubMed]

Baldassi, S.

M. J. Morgan, A. J. S. Mason, S. Baldassi, “Are there separate first-order and second-order mechanisms for orientation discrimination?” Vision Res. 40, 1751–1763 (2000).
[CrossRef] [PubMed]

M. J. Morgan, S. Baldassi, “How the human visual system encodes the orientation of a texture, and why it makes mistakes,” Curr. Biol. 7, 999–1002 (1997).
[CrossRef]

Bauman, L. A.

L. A. Bauman, A. B. Bonds, “Inhibitory refinement of spatial frequency selectivity in single cells of the cat striate cortex,” Vision Res. 31, 933–944 (1991).
[CrossRef] [PubMed]

Blakemore, C.

R. H. S. Carpenter, C. Blakemore, “Interactions between orientations in human vision,” Exp. Brain Res. 18, 287–303 (1973).
[CrossRef] [PubMed]

C. Blakemore, R. H. S. Carpenter, M. A. Georgeson, “Lateral inhibition between orientation detectors in the human visual system,” Nature (London) 228, 37–39 (1970).
[CrossRef]

Bonds, A. B.

L. A. Bauman, A. B. Bonds, “Inhibitory refinement of spatial frequency selectivity in single cells of the cat striate cortex,” Vision Res. 31, 933–944 (1991).
[CrossRef] [PubMed]

A. B. Bonds, “Role of inhibition in the specification of orientation selectivity of cells in the cat striate cortex,” Visual Neurosci. 2, 41–55 (1989).
[CrossRef]

Bosking, W. H.

W. H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,” J. Neurosci. 17, 2112–2127 (1997).
[PubMed]

Buhmann, J.

C. von der Malsburg, J. Buhmann, “Sensory segmentation with coupled neural oscillators,” Biol. Cybern. 67, 233–242 (1992).
[CrossRef] [PubMed]

Campbell, F. W.

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

Carpenter, R. H. S.

R. H. S. Carpenter, C. Blakemore, “Interactions between orientations in human vision,” Exp. Brain Res. 18, 287–303 (1973).
[CrossRef] [PubMed]

C. Blakemore, R. H. S. Carpenter, M. A. Georgeson, “Lateral inhibition between orientation detectors in the human visual system,” Nature (London) 228, 37–39 (1970).
[CrossRef]

Casco, C.

M. J. Morgan, C. Casco, “Spatial filtering and spatial primitives in early vision: an explanation of the Zöllner–Judd class of geometrical illusion,” Proc. R. Soc. London Ser. B 242, 1–10 (1990).
[CrossRef]

Chao-Yi, L.

L. Chao-Yi, G. Kun, “Measurement of geometric illusions, illusory contours and stereo-depth at luminance and colour contrast,” Vision Res. 35, 1713–1720 (1995).
[CrossRef]

Cowan, J. D.

T. Mundel, A. Dimitrov, J. D. Cowan, “Visual cortex circuitry and orientation tuning,” in Advances in Neural Information Processing Systems 9 (MIT Press, Cambridge, Mass., 1997), pp. 887–893.

Desimone, R.

A. F. Rossi, R. Desimone, L. G. Ungerleider, “Contextual modulation in primary visual cortex of macaques,” J. Neurosci. 21, 1698–1709 (2001).
[PubMed]

Dimitrov, A.

T. Mundel, A. Dimitrov, J. D. Cowan, “Visual cortex circuitry and orientation tuning,” in Advances in Neural Information Processing Systems 9 (MIT Press, Cambridge, Mass., 1997), pp. 887–893.

Ermentrout, G. B.

G. B. Ermentrout, “The behavior of rings of coupled oscillators,” J. Math. Biol. 23, 55–74 (1985).
[CrossRef] [PubMed]

Finkel, L. H.

S-C. Yen, E. D. Menschik, L. H. Finkel, “Perceptual grouping in striate cortical networks mediated by synchronization and desynchronization,” Neurocomputing 26–27, 609–616 (1999).
[CrossRef]

S-C. Yen, L. H. Finkel, “Extraction of perceptually salient contours by striate cortical networks,” Vision Res. 38, 719–741 (1998).
[CrossRef] [PubMed]

K. Sakai, L. H. Finkel, “Characterization of spatial-frequency spectrum in the perception of shape from texture,” J. Opt. Soc. Am. A 12, 1208–1224 (1995).
[CrossRef]

Fitzpatrick, D.

W. H. Bosking, Y. Zhang, B. Schofield, D. Fitzpatrick, “Orientation selectivity and the arrangement of horizontal connections in tree shrew striate cortex,” J. Neurosci. 17, 2112–2127 (1997).
[PubMed]

Friedman, H. S.

H. Zhou, H. S. Friedman, R. von der Heydt, “Coding of border ownership in monkey visual cortex,” J. Neurosci. 20, 6594–6611 (2000).
[PubMed]

Georgeson, M. A.

C. Blakemore, R. H. S. Carpenter, M. A. Georgeson, “Lateral inhibition between orientation detectors in the human visual system,” Nature (London) 228, 37–39 (1970).
[CrossRef]

Gibson, J. G.

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

Gilbert, C. D.

J. S. Bakin, K. Nakayama, C. D. Gilbert, “Visual responses in monkey areas V1 and V2 to three-dimensional surface configurations,” J. Neurosci. 20, 8188–8198 (2000).
[PubMed]

M. K. Kapadia, G. Westheimer, C. D. Gilbert, “Spatial distribution of contextual interactions in primary visual cortex and in visual perception,” J. Neurophysiol. 84, 2048–2062 (2000).
[PubMed]

M. K. Kapadia, G. Westheimer, C. D. Gilbert, “Dynamics in spatial summation in primary visual cortex of alert monkeys,” Proc. Natl. Acad. Sci. USA 96, 12073–12078 (1999).
[CrossRef]

C. D. Gilbert, T. N. Wiesel, “The influence of contextual stimuli on the orientation selectivity of cells in primary visual cortex of the cat,” Vision Res. 30, 1689–1701 (1990).
[CrossRef] [PubMed]

C. D. Gilbert, T. N. Wiesel, “Columnar specificity of intrinsic horizontal and corticocortical connections in cat visual cortex,” J. Neurosci. 9, 2432–2442 (1989).
[PubMed]

Grzywacz, N. M.

A. Yuille, N. M. Grzywacz, “A winner-take-all mechanism based on presynaptic inhibition feedback,” Neural Comput. 1, 334–347 (1989).
[CrossRef]

Johnson, M.

P. Wenderoth, T. O’Connor, M. Johnson, “The tilt illusion as a function of the relative and absolute lengths of test and inducing lines,” Percept. Psychophys. 39, 339–345 (1986).
[CrossRef] [PubMed]

Kapadia, M. K.

M. K. Kapadia, G. Westheimer, C. D. Gilbert, “Spatial distribution of contextual interactions in primary visual cortex and in visual perception,” J. Neurophysiol. 84, 2048–2062 (2000).
[PubMed]

M. K. Kapadia, G. Westheimer, C. D. Gilbert, “Dynamics in spatial summation in primary visual cortex of alert monkeys,” Proc. Natl. Acad. Sci. USA 96, 12073–12078 (1999).
[CrossRef]

Kaski, S.

S. Kaski, T. Kohonen, “Winner-take-all networks for physiological models of competitive learning,” Neural Networks 7, 973–984 (1994).
[CrossRef]

Kohonen, T.

S. Kaski, T. Kohonen, “Winner-take-all networks for physiological models of competitive learning,” Neural Networks 7, 973–984 (1994).
[CrossRef]

Kun, G.

L. Chao-Yi, G. Kun, “Measurement of geometric illusions, illusory contours and stereo-depth at luminance and colour contrast,” Vision Res. 35, 1713–1720 (1995).
[CrossRef]

Kuramoto, Y.

Y. Kuramoto, Lecture Notes in Physics 39 (Springer, Berlin, 1975).

Kurtenbach, W.

S. Magnussen, W. Kurtenbach, “Linear summation of tilt illusion and tilt aftereffect,” Vision Res. 20, 39–42 (1980).
[CrossRef] [PubMed]

Li, Z.

Z. Li, “A neural model of contour integration in the primary visual cortex,” Neural Comput. 10, 903–940 (1998).
[CrossRef] [PubMed]

Magnussen, S.

S. Magnussen, W. Kurtenbach, “Linear summation of tilt illusion and tilt aftereffect,” Vision Res. 20, 39–42 (1980).
[CrossRef] [PubMed]

Mason, A. J. S.

M. J. Morgan, A. J. S. Mason, S. Baldassi, “Are there separate first-order and second-order mechanisms for orientation discrimination?” Vision Res. 40, 1751–1763 (2000).
[CrossRef] [PubMed]

Menschik, E. D.

S-C. Yen, E. D. Menschik, L. H. Finkel, “Perceptual grouping in striate cortical networks mediated by synchronization and desynchronization,” Neurocomputing 26–27, 609–616 (1999).
[CrossRef]

Morgan, M. J.

M. J. Morgan, A. J. S. Mason, S. Baldassi, “Are there separate first-order and second-order mechanisms for orientation discrimination?” Vision Res. 40, 1751–1763 (2000).
[CrossRef] [PubMed]

M. J. Morgan, “The Poggendorff illusion: a bias in the estimation of the orientation of virtual lines by second-stage filters,” Vision Res. 39, 2361–2380 (1999).
[CrossRef] [PubMed]

M. J. Morgan, S. Baldassi, “How the human visual system encodes the orientation of a texture, and why it makes mistakes,” Curr. Biol. 7, 999–1002 (1997).
[CrossRef]

M. J. Morgan, C. Casco, “Spatial filtering and spatial primitives in early vision: an explanation of the Zöllner–Judd class of geometrical illusion,” Proc. R. Soc. London Ser. B 242, 1–10 (1990).
[CrossRef]

Mundel, T.

T. Mundel, A. Dimitrov, J. D. Cowan, “Visual cortex circuitry and orientation tuning,” in Advances in Neural Information Processing Systems 9 (MIT Press, Cambridge, Mass., 1997), pp. 887–893.

Nakayama, K.

J. S. Bakin, K. Nakayama, C. D. Gilbert, “Visual responses in monkey areas V1 and V2 to three-dimensional surface configurations,” J. Neurosci. 20, 8188–8198 (2000).
[PubMed]

Näsänen, R.

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

O’Connor, T.

P. Wenderoth, T. O’Connor, M. Johnson, “The tilt illusion as a function of the relative and absolute lengths of test and inducing lines,” Percept. Psychophys. 39, 339–345 (1986).
[CrossRef] [PubMed]

Oyama, T.

T. Oyama, “Determinants of the Zöllner illusion,” Psychol. Res. 37, 261–280 (1975).
[CrossRef] [PubMed]

Paradiso, M. A.

M. A. Paradiso, “A theory for the use of visual orientation information which exploits the columnar structure of striate cortex,” Biol. Cybern. 58, 35–49 (1988).
[CrossRef] [PubMed]

Peterhans, E.

R. von der Heydt, E. Peterhans, “Mechanisms of contour perception in monkey visual cortex. I. Lines of pattern discontinuity,” J. Neurosci. 9, 1731–1748 (1989).
[PubMed]

Popple, A. V.

A. V. Popple, D. Sagi, “A Fraser illusion without local cues?” Vision Res. 40, 873–878 (2000).
[CrossRef] [PubMed]

Rossi, A. F.

A. F. Rossi, R. Desimone, L. G. Ungerleider, “Contextual modulation in primary visual cortex of macaques,” J. Neurosci. 21, 1698–1709 (2001).
[PubMed]

Rovamo, J.

J. Rovamo, V. Virsu, R. Näsänen, “Cortical magnification factor predicts the photopic contrast sensitivity of peripheral vision,” Nature (London) 271, 54–56 (1978).
[CrossRef]

Sagi, D.

A. V. Popple, D. Sagi, “A Fraser illusion without local cues?” Vision Res. 40, 873–878 (2000).
[CrossRef] [PubMed]

Sakai, K.

K. Sakai, S. Tanaka, “Spatial pooling in the second-order spatial structure of cortical complex cells,” Vision Res. 40, 855–871 (2000).
[CrossRef] [PubMed]

K. Sakai, S. Tanaka, “Preceptual segmentation and neural grouping in tilt illusion,” Neurocomputing 32–33, 979–986 (2000).
[CrossRef]

K. Sakai, S. Tanaka, “Retinotopic coding and neural grouping in tilt illusion,” Neurocomputing 26–27, 837–843 (1999).
[CrossRef]

K. Sakai, L. H. Finkel, “Characterization of spatial-frequency spectrum in the perception of shape from texture,” J. Opt. Soc. Am. A 12, 1208–1224 (1995).
[CrossRef]

Schofield, B.

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Figures (12)

Fig. 1
Fig. 1

Examples of stimuli used in the psychophysical experiments, consisting of X-shaped bars and a single bar for a reference (first and third panels). The vertical bar in the X-shaped stimulus is always fixed to the vertical, but the reference bar is tilted randomly ±4° or 6° for each time. Subjects are asked to judge whether the reference bar is tilted to the right or the left of the vertical bar in the X-shaped object. Following this is a distracter consisting of a number of bars whose dimension and contrast were varied randomly (second and fourth panels). A stimulus with a variable reference followed by a distracter is repeatedly shown 80 or 120 times for a single condition.

Fig. 2
Fig. 2

Estimated apparent tilt of the vertical bar in X-shaped stimuli in which the intersection is segmented perceptually by binocular disparity. (A)–(C) show the results of three individuals, and (D) shows the mean of the three. Squares indicate the case where the oblique bar has positive disparity, so that it appears in front of the vertical bar; diamonds show the other case, where the vertical bar appears in front. Circles indicate the apparent tilt without the disparity. Solid and dotted curves are the best-fit curves of up to second order for black squares and open diamonds, respectively. Error bars for (A)–(C) are residual standard deviations showing the divergence of data from the calculated psychometric function, and those for (D) are standard deviations among the three subjects. The result shows the independence of apparent tilt of the perceptual segmentation given by binocular disparity.  

Fig. 3
Fig. 3

Estimated apparent tilt of the vertical bar in X-shaped stimuli in which the intersection is segmented perceptually by hue. (A)–(C) show the results of three individuals, and (D) shows the mean of the three. Squares indicate the case where the oblique bar is placed in front of the vertical bar by means of occlusion; diamonds show the other case, where the vertical bar appears in front. Solid and dotted curves are the best-fit curves of up to second order for squares and diamonds, respectively. Error bars for (A)–(C) are residual standard deviations, and those for (D) are standard deviations among the three subjects. The result shows the independence of apparent tilt of the perceptual segmentation given by hue.

Fig. 4
Fig. 4

Estimated apparent tilt of the vertical bar in X-shaped stimuli consisting of bars with shades of gray. (A)–(C) show the results of three individuals, and (D) shows the mean of the three. Triangles indicate the cases when the vertical bar is light gray and is placed in front of the dark-gray oblique bar by means of occlusion. Circles, squares, and diamonds show other combinations of bar contrast and segmentation as illustrated in the figure. Solid and dotted curves are the best-fit curves of up to second order. Error bars for (A)–(C) are residual standard deviations, and those for (D) are standard deviations among the three subjects. The result shows the independence of apparent tilt of the perceptual segmentation given by contrast and the dependence on bar contrast.

Fig. 5
Fig. 5

Architecture of the model. The first stage of the model consists of simple-cell-like units whose receptive field is given by a Gabor filter with a variety of orientation and spatial-frequency preferences, followed by half-wave rectification. Note that a variety in spatial phase is included in the model presented in Section 4. The second stage of the model determines local orientation. The local orientation is given by the local peak orientation, which is the optimal orientation of the unit that responds most strongly among those units located at the same spatial position. The third stage of the model computes the global orientation of the bars by taking the ensemble average of the local peak orientations within each bar region excluding the intersection region.

Fig. 6
Fig. 6

(A) Needle diagram showing the local peak orientations computed from units tuned to a variety of spatial frequencies. The orientation of each needle indicates the optimal orientation of the unit whose response is the strongest among the units tuned to a variety of spatial frequencies and orientations at the retinotopic position. The input stimulus was two bars intersecting at 30 deg, where the corresponding bar width is indicated by a thick line at the top and the bottom of the needle diagram. A number of needles indicate orientations other than those of the bars, which seems to be an origin of the tilt illusions. (B)–(E) Local orientations computed from each spatial-frequency component. The orientation of each needle indicates the optimal orientation of the unit whose response is the strongest among the units tuned to a variety of orientations but to the same spatial frequency at the retinotopic position. The ratio between the width of the excitatory region of the receptive field (RF) and the bar is (B) 0.25:1, (C) 0.5:1, (D) 1:1, and (E) 2:1.

Fig. 7
Fig. 7

Computed global orientation of the vertical bar in a variety of X-shaped stimuli presented as a function of the crossing angle between the two bars. Corresponding human responses measured psychophysically are also plotted for comparison. The global orientations estimated from the ensemble average of local orientations excluding the intersection show good agreement with the human responses.

Fig. 8
Fig. 8

Example of coupling strength. The tilt of the bars indicates the local peak orientation at the retinotopic position. Given that the local peak orientation at the origin is 30° from the vertical (θ), the unit at the origin is coupled with other units located within the hatched region bounded by the two circles with Min. Radius and the four lines with Max. Angle. A detailed description is given in Appendix B. The coupling strength is strong if (1) the local orientation of the unit at the origin and that of the unit being examined is cocircular, (2) the difference in the local orientations of the two is small, and (3) the Euclidean distance between the two is small. Therefore, in this example, unit 1 has the strongest coupling, followed by units 2 and 3, and unit 4 has the weakest coupling among the four.

Fig. 9
Fig. 9

Two groups formed by the phase model for an X-shaped stimulus with a crossing angle of 30 deg. Three levels of gray indicate a trisected phase at the spatial position: light gray indicates phase in the range of 0±60 degrees, mid-gray for 180±60 degrees, and black for the rest, including no-response units. The width of the bars is indicated by white line segments at the top and the bottom of the vertical bar. Most units on and around the vertical bar are bound into a single group, and so is the oblique bar. The intersection region is not bound to either group.

Fig. 10
Fig. 10

Global orientation of the vertical bar computed by the phase model. Corresponding human responses measured psychophysically are also plotted for comparison. The same conventions as those in Fig. 2 are used. The computed global orientations show agreement with the human responses in the sense that the illusory tilt increases as the crossing angle decreases, although the amount of the tilt is less than half that of the human responses.

Fig. 11
Fig. 11

Global orientation of the vertical bar, computed by the phase model, in the multiple-gray stimuli. The same conventions as those in Fig. 4 are used. The model reproduces the independence of apparent tilt of perceptual segmentation and the dependence on bar contrast, showing good agreement with the psychophysical experiments.

Fig. 12
Fig. 12

Global orientation of the vertical bar in the stimuli with variable bar widths. The abscissa is the relative width of the oblique bar with respect to the width of the vertical bar. The width of the vertical bar is identical to those used above, and the crossing angle between the two bars is fixed to 30 deg. Squares show the global orientation computed by the phase model. Circles show the human responses to the same stimuli measured similarly to those described in Section 2. Solid and dotted curves are the linear regression curves for the global orientation obtained from the phase model and the psychophysical experiments, respectively. The result of the simulation indicates that the global orientation decreases as the width of the oblique bar increases, showing agreement with the human responses.

Equations (29)

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

dφidt=ωi+ijKijsin(φj-φi),
Oo,fo(x0, y0)=(I0*Mo,f)(x, y),
Oo,f1 (x0, y0)=QLT(Oo,f0 (x0, y0)),
QLT(s)=0ifs<cthresholds,otherwise.
O2(x0, y0)=arg maxo(Oo,f1(x0, y0)).
O3(x0, y0)=1nI1x,yI1(x0, y0)O2(x, y),
dφidt=ωi+1Inijκijsin(φj-φi),
κij=kijcirkijorikijdistδcond,
kijcir=12πσcirexp-12θj-ϕijσcir,
ϕij=θi-2 tan-1yj-yixj-xi-θi.
kijori=12πσoriexp-12θj-θiσori.
kijdist=12πσdistexp-12ρijσdist,
ρij=[(xj-xi)2+(yj-yi)2]1/2.
δcond=δangleδradius.
Ifθj-θi<max angle
δ angle=1;
otherwise,
δ angle=0.
Ifψi1>minradius andψi2>minradius
δradius=1,
otherwise,
δradius=0.
ψi1=[(xj-xc1)2+(yj-yc1)2]1/2,
ψi2=[(xj-xc2)2+(yj-yc2)2]1/2,
xc1=xi+minradiuscos θi,
yc1=yi-minradiussin θi,
xc2=xi-minradiuscos θi,
yc2=yi+minradiussin θi.
GOvert=1nφ0φ0°±60°argθ φi

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