Abstract

Current illusory contour models do not predict the disappearance of the Kanizsa illusion due to specific spatial luminance distributions within the inducers. We suggest that these stimulus conditions are characterized by an insufficient amount of induced brightness. Our model’s core assumption is that contour edge detection of the Kanizsa illusion and the simultaneous contrast (brightness induction) effect are triggered by the same mechanism. The simultaneous contrast can immunize the occlusion detection mechanism against spatial and temporal noise. Our model contains physiologically inspired building blocks that detect the oriented contour edges, complete the illusory contours, and enhance them. The model succeeds in predicting the appearance and the disappearance of many different Kanizsa illusion variants.

© 2011 Optical Society of America

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References

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    [CrossRef] [PubMed]
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    [CrossRef] [PubMed]
  4. P. J. Kellman, “Interpolation processes in the visual perception of objects,” Neural Netw. 16, 915–923 (2003).
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    [CrossRef] [PubMed]
  8. R. Shapley and J. Gordon, “Nonlinearity in the perception of form,” Percept. Psychophys. 37, 84–88 (1985).
    [CrossRef] [PubMed]
  9. S. Grossberg and E. Mingolla, “Neural dynamics of form perception: Boundary completion, illusory figures, and neon spreading,” Psychol. Rev. 92, 173–211 (1985).
    [CrossRef] [PubMed]
  10. S. Grossberg and E. Mingolla, “Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations,” Percept. Psyhophys. 38, 141–171 (1985).
    [CrossRef]
  11. T. Banton and D. Levy, “The perceived strength of illusory contours,” Percept. Psychophys. 52, 676–684 (1992).
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  12. R. Von der Heydt, E. Peterhans, and G. Baumgartner, “Illusory contours and cortical neural responses,” Science 224 (4654), 1260–1262 (1984).
    [CrossRef] [PubMed]
  13. F. Heitger, R. Von der Heydt, E. Peterhans, L. Rosenthaler, and O. Kübler, “Simulation of neural contour mechanisms: representing anomalous contours,” Image Vision Comput. 16, 407–421 (1998).
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  23. S. Grossberg, E. Mingolla, and W. D. Ross, “How does the cortex do perceptual grouping?” Trends Neurosci. 20, 106–111 (1997).
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  24. S. Grossberg, “Cortical dynamics of three-dimensional figure-ground Perception of two-dimensional pictures,” Psychol. Rev. 104, 618–658 (1997).
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  34. L. R. Williams and D. W. Jacobs, “Local parallel computation of stochastic completion fields,” Neural Comput. 9, 859–881(1997).
    [CrossRef]
  35. S. Ullman, “Filling-in gaps—shape of subjective contours and a model for their generation,” Biol. Cybern. 25, 1–6 (1976).
  36. M. Kass, A. Witkin, and D. Terzopoulos, “Snakes—active contour models,” Int. J. Comput. Vision 1, 321–331 (1988).
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  37. S. Zucker, “Two stages of curve detection suggest two styles of visual computation,” Neural Comput. 1, 68–81 (1989).
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  38. G. Guy and G. Medioni, “Inferring global perceptual contours from local features,” Int. J. Comput. Vision 20, 113–133 (1996).
    [CrossRef]
  39. D. Hubel and T. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. 195, 215–243(1968).
    [PubMed]
  40. B. Spehar, “Degraded illusory contour formation with non-uniform inducers in Kanizsa configurations: the role of contrast polarity,” Vision Res. 40, 2653–2659 (2000).
    [CrossRef] [PubMed]
  41. E. Heinemann, “Simultaneous brightness induction as a function of inducing-and test-field luminances,” J. Exp. Psychol. 50, 89–96 (1955).
    [CrossRef] [PubMed]
  42. E. Heinemann, “Simultaneous brightness induction,” in Handbook of Sensory Physiology, D.Jameson and L.M.Hurvich, eds. (Springer, 1972).
  43. J. Kinney, “Factors affecting induced color,” Vision Res. 2, 503–525 (1962).
    [CrossRef]
  44. M. F. Wesner and S. K. Shevell, “Color-perception within a chromatic context—changes in red green equilibria caused by noncontiguous light,” Vision Res. 32, 1623–1634 (1992).
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  45. R. Dahari and H. Spitzer, “Spatiotemporal adaptation model for retinal ganglion cells,” J. Opt. Soc. Am. A 13, 419–435 (1996).
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    [PubMed]
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    [PubMed]
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    [CrossRef] [PubMed]
  63. H. Spitzer and S. Hochstein, “A complex-cell receptive-field model,” J. Neurophysiol. 53, 1266–1286 (1985).
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    [CrossRef] [PubMed]
  66. S. Grossberg and A. Yazdanbakhsh, “Laminar cortical dynamics of 3D surface perception: Stratification, transparency, and neon color spreading,” Vision Res. 45, 1725–1743 (2005).
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  67. T. F. Shipley and P. J. Kellman, “The role of discontinuities in the perception of subjective figures,” Percept. Psychophys. 48, 259–270 (1990).
    [CrossRef] [PubMed]
  68. M. K. Albert and D. D. Hoffman, “Genericity in spatial vision,” in Geometric Representations of Perceptual Phenomena: Articles in Honour of Tarow Indow’s 70th Birthday, D.Luce, K.Romney, D.Hoffman, and M.D’Zmura, eds. (Erlbaum, 1995).
    [PubMed]
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    [CrossRef]
  71. A. Michotte, G. Thinés, and G. Crabbé, “Les compléments amodaux des structures perceptives,” Studia Psychologica (Institut de Psychologie de l’Université de Louvain, 1964).
  72. H. G. Barrow and J. M. Tenenbaum, “Interpreting line drawings as three-dimensional surfaces,” Artif. Intell. 17, 75–116 (1981).
    [CrossRef]

2010 (1)

G. Ben-Yosef and O. Ben-Shahar, “Minimum length in the tangent bundle as a model for curve completion,” in Proceedings of the 23rd Conference on Computer Vision and Pattern Recognition (CVPR) (IEEE, 2010).

2009 (1)

M. Okamoto, T. Naito, O. Sadakane, H. Osaki, and H. Sato, “Surround suppression sharpens orientation tuning in the cat primary visual cortex,” Eur. J. Neurosci. 29, 1035–1046 (2009).
[CrossRef] [PubMed]

2005 (2)

H. Spitzer and Y. Barkan, “Computational adaptation model and its predictions for color induction of first and second orders,” Vision Res. 45, 3323–3342 (2005).
[CrossRef] [PubMed]

S. Grossberg and A. Yazdanbakhsh, “Laminar cortical dynamics of 3D surface perception: Stratification, transparency, and neon color spreading,” Vision Res. 45, 1725–1743 (2005).
[CrossRef] [PubMed]

2003 (3)

R. Shapley, M. Hawken, and D. L. Ringach, “Dynamics of orientation selectivity in the primary visual cortex and the importance of cortical inhibition,” Neuron 38, 689–699 (2003).
[CrossRef] [PubMed]

H. Spitzer, Y. Karasik, and S. Einav, “Biological gain control for high dynamic range compression,” in Proceedings of IS&T/SID Eleventh Color Imaging Conference (IS&T/SID, 2003).

P. J. Kellman, “Interpolation processes in the visual perception of objects,” Neural Netw. 16, 915–923 (2003).
[CrossRef] [PubMed]

2002 (3)

J. R. Cavanaugh, W. Bair, and J. A. Movshon, “Nature and interaction of signals from the receptive field center and surround in macaque V1 neurons,” J. Neurophysiol. 88, 2530–2546 (2002).
[CrossRef] [PubMed]

H. Spitzer and S. Semo, “Color constancy: a biological model and its application for still and video images,” Pattern Recogn. 35, 1645–1659 (2002).
[CrossRef]

H. Spitzer and A. Rosenbluth, “Color constancy: The role of low-level mechanisms,” Spatial Vis. 15, 277–302 (2002).
[CrossRef]

2001 (2)

N. Rubin, “The role of junctions in surface completion and contour matching,” Perception 30, 339–366 (2001).
[CrossRef] [PubMed]

E. Peterhans and F. Heitger, “Simulation of neuronal responses defining depth order and contrast polarity at illusory contours in monkey area V2,” J. Comput. Neurosci. 10, 195–211 (2001).
[CrossRef] [PubMed]

2000 (5)

A. Sarti, R. Malladi, and J. A. Sethian, “Subjective surfaces: A method for completing missing boundaries,” Proc. Natl. Acad. Sci. USA 97, 6258–6263 (2000).
[CrossRef] [PubMed]

W. D. Ross, S. Grossberg, and E. Mingolla, “Visual cortical mechanisms of perceptual grouping: interacting layers, networks, columns, and maps,” J. Neural Netw. 13, 571–588 (2000).
[CrossRef]

Victor and M. Conte, “Illusory contour strength does not depend on the dynamics or relative phase of the inducers,” Vision Res. 40, 3475–3483 (2000).
[CrossRef] [PubMed]

E. R. Kandel, J. H. Schwartz, and T. M. Jessell, Principles of Neural Science, 4th ed. (McGraw-Hill, 2000).

B. Spehar, “Degraded illusory contour formation with non-uniform inducers in Kanizsa configurations: the role of contrast polarity,” Vision Res. 40, 2653–2659 (2000).
[CrossRef] [PubMed]

1999 (3)

M. Singh, D. D. Hoffman, and M. K. Albert, “Contour completion and relative depth: Petter’s rule and support ratio,” Psychon. Sci. 10, 423–428 (1999).

E. Saund, “Perceptual organization of occluding contours of opaque surfaces,” Comput. Vision Image Underst. 76, 70–82(1999).
[CrossRef]

F. Heitger, “A computational model of neural contour processing : Figure–ground segregation and illusory contours,” in Proceedings of the Seventh IEEE International Conference on Computer Vision (IEEE, 1999).

1998 (4)

D. Geiger, H. Pao, and N. Rubin, “Salient and multiple illusory surfaces,” in IEEE Computer Society Conference on Computer Vision and Pattern Recognition (IEEE, 1998).

F. Heitger, R. Von der Heydt, E. Peterhans, L. Rosenthaler, and O. Kübler, “Simulation of neural contour mechanisms: representing anomalous contours,” Image Vision Comput. 16, 407–421 (1998).
[CrossRef]

J. D. Victor and M. M. Conte, “Quantitative study of effects of inducer asynchrony on illusory contour strength,” Invest. Ophthalmol. Visual Sci. 39, S206 (1998).

P. J. Kellman, C. Yin, and T. F. Shipley, “A common mechanism for illusory and occluded object completion,” J. Exp. Psychol. 24, 859–869 (1998).
[CrossRef]

1997 (3)

L. R. Williams and D. W. Jacobs, “Local parallel computation of stochastic completion fields,” Neural Comput. 9, 859–881(1997).
[CrossRef]

S. Grossberg, E. Mingolla, and W. D. Ross, “How does the cortex do perceptual grouping?” Trends Neurosci. 20, 106–111 (1997).
[CrossRef] [PubMed]

S. Grossberg, “Cortical dynamics of three-dimensional figure-ground Perception of two-dimensional pictures,” Psychol. Rev. 104, 618–658 (1997).
[CrossRef] [PubMed]

1996 (4)

D. L. Ringach and R. Shapeley, “Spatial and temporal properties of illusory contours and amodal boundary completion,” Vision Res. 36, 3037–3050 (1996).
[CrossRef] [PubMed]

K. Kumaran, D. Geiger, and L. Gurvits, “Illusory surface perception and visual organization,” Network: Comput. Neural Syst. 7, 33–60 (1996).
[CrossRef]

G. Guy and G. Medioni, “Inferring global perceptual contours from local features,” Int. J. Comput. Vision 20, 113–133 (1996).
[CrossRef]

R. Dahari and H. Spitzer, “Spatiotemporal adaptation model for retinal ganglion cells,” J. Opt. Soc. Am. A 13, 419–435 (1996).
[CrossRef]

1995 (3)

M. K. Albert and D. D. Hoffman, “Genericity in spatial vision,” in Geometric Representations of Perceptual Phenomena: Articles in Honour of Tarow Indow’s 70th Birthday, D.Luce, K.Romney, D.Hoffman, and M.D’Zmura, eds. (Erlbaum, 1995).
[PubMed]

R. Benyishai, R. L. Baror, and H. Sompolinsky, “Theory of orientation tuning in visual-cortex,” Proc. Natl. Acad. Sci. USA 92, 3844–3848 (1995).
[CrossRef]

A. Gove, S. Grossberg, and E. Mingolla, “Brightness perception, illusory contours, and corticogeniculate feedback,” Visual Neurosci. 12, 1027–1052 (1995).
[CrossRef]

1993 (3)

I. Kojo, M. Liinasuo, and J. Rovamo, “Spatial and temporal properties of illusory figures,” Vision Res. 33, 897–901 (1993).
[CrossRef] [PubMed]

Z. F. Kisvarday and U. T. Eysel, “Functional and structural topography of horizontal inhibitory connections in cat visual cortex,” Eur. J. Neurosci. 5, 1558–1572 (1993).
[CrossRef] [PubMed]

G. W. Lesher and E. Mingolla, “The role of edges and line-ends in illusory contour formation,” Vision Res. 33, 2253–2270 (1993).
[CrossRef] [PubMed]

1992 (5)

M. F. Wesner and S. K. Shevell, “Color-perception within a chromatic context—changes in red green equilibria caused by noncontiguous light,” Vision Res. 32, 1623–1634 (1992).
[CrossRef] [PubMed]

T. Banton and D. Levy, “The perceived strength of illusory contours,” Percept. Psychophys. 52, 676–684 (1992).
[CrossRef] [PubMed]

B. Dresp, “Local brightness mechanisms sketch out surfaces but do not fill them: Psychophysical evidence in the Kanizsa square,” Percept. Psychophys. 52, 562–570 (1992).
[CrossRef] [PubMed]

T. F. Shipley and P. J. Kellman, “Strength of visual interpolation depends on the ratio of physically specified to total edge length,” Percept. Psychophys. 52, 97–106 (1992).
[CrossRef] [PubMed]

F. Heitger, L. Rosenthaler, R. Vonderheydt, E. Peterhans, and O. Kübler, “Simulation of neural Contour mechanisms—from simple to end-stopped cells,” Vision Res. 32, 963–981 (1992).
[CrossRef] [PubMed]

1991 (1)

P. J. Kellman and T. F. Shipley, “A theory of visual interpolation and object perception,” Cogn. Psychol. 23, 141–221 (1991).
[CrossRef] [PubMed]

1990 (1)

T. F. Shipley and P. J. Kellman, “The role of discontinuities in the perception of subjective figures,” Percept. Psychophys. 48, 259–270 (1990).
[CrossRef] [PubMed]

1989 (3)

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

S. Zucker, “Two stages of curve detection suggest two styles of visual computation,” Neural Comput. 1, 68–81 (1989).
[CrossRef]

L. H. Finkel and G. M. Edelman, “Integration of distributed cortical systems by reentry—a computer-simulation of interactive functionally segregated visual areas,” J. Neurosci. 9, 3188–3208 (1989).
[PubMed]

1988 (2)

M. Kass, A. Witkin, and D. Terzopoulos, “Snakes—active contour models,” Int. J. Comput. Vision 1, 321–331 (1988).
[CrossRef]

D. H. Hubel, Eye, Brain, and Vision, Scientific American Library Series (Scientific American Library, 1988).

1987 (1)

B. C. Skottun, A. Bradley, G. Sclar, I. Ohzawa, and R. D. Freeman, “The effects of contrast on visual orientation and spatial-frequency discrimination—a comparison of single cells and behavior,” J. Neurophysiol. 57, 773–786 (1987).
[PubMed]

1985 (4)

H. Spitzer and S. Hochstein, “A complex-cell receptive-field model,” J. Neurophysiol. 53, 1266–1286 (1985).
[PubMed]

R. Shapley and J. Gordon, “Nonlinearity in the perception of form,” Percept. Psychophys. 37, 84–88 (1985).
[CrossRef] [PubMed]

S. Grossberg and E. Mingolla, “Neural dynamics of form perception: Boundary completion, illusory figures, and neon spreading,” Psychol. Rev. 92, 173–211 (1985).
[CrossRef] [PubMed]

S. Grossberg and E. Mingolla, “Neural dynamics of perceptual grouping: textures, boundaries, and emergent segmentations,” Percept. Psyhophys. 38, 141–171 (1985).
[CrossRef]

1984 (2)

R. Von der Heydt, E. Peterhans, and G. Baumgartner, “Illusory contours and cortical neural responses,” Science 224 (4654), 1260–1262 (1984).
[CrossRef] [PubMed]

R. Shapley and C. Enroth-Cugell, “Visual adaptation and retinal gain controls,” Progr. Retin. Res. 3, 263–346 (1984).
[CrossRef]

1983 (2)

K. Prazdny, “Illusory contours are not caused by simultaneous brightness contrast,” Percept. Psychophys. 34, 403–404 (1983).
[CrossRef] [PubMed]

S. Petry, A. Harbeck, J. Conway, and J. Levey, “Stimulus determinants of brightness and distinctness of subjective contours,” Percept. Psychophys. 34, 169–174 (1983).
[CrossRef] [PubMed]

1982 (1)

D. Marr, Vision (Freeman, 1982).

1981 (1)

H. G. Barrow and J. M. Tenenbaum, “Interpreting line drawings as three-dimensional surfaces,” Artif. Intell. 17, 75–116 (1981).
[CrossRef]

1979 (2)

G. Kanizsa, Organization in Vision (Praeger, 1979).

M. Jory and R. Day, “The relationship between brightness contrast and illusory contours,” Perception 8, 3–9 (1979).
[CrossRef] [PubMed]

1976 (2)

G. Kanizsa, “Subjective contours,” Sci. Am. 234, 48–52 (1976).
[CrossRef] [PubMed]

S. Ullman, “Filling-in gaps—shape of subjective contours and a model for their generation,” Biol. Cybern. 25, 1–6 (1976).

1972 (1)

E. Heinemann, “Simultaneous brightness induction,” in Handbook of Sensory Physiology, D.Jameson and L.M.Hurvich, eds. (Springer, 1972).

1969 (1)

B. Sakmann and O. D. Creutzfeldt, “Scotopic and mesopic light adaptation in cats retina,” Pflugers Arch. 313, 168–185 (1969).
[CrossRef] [PubMed]

1968 (1)

D. Hubel and T. Wiesel, “Receptive fields and functional architecture of monkey striate cortex,” J. Physiol. 195, 215–243(1968).
[PubMed]

1964 (1)

A. Michotte, G. Thinés, and G. Crabbé, “Les compléments amodaux des structures perceptives,” Studia Psychologica (Institut de Psychologie de l’Université de Louvain, 1964).

1962 (1)

J. Kinney, “Factors affecting induced color,” Vision Res. 2, 503–525 (1962).
[CrossRef]

1956 (1)

G. Petter, “Nuove ricerche sperimentali sulla totalizzazione percettiva,” Riv. Psicologia 50, 213–227 (1956).

1955 (1)

E. Heinemann, “Simultaneous brightness induction as a function of inducing-and test-field luminances,” J. Exp. Psychol. 50, 89–96 (1955).
[CrossRef] [PubMed]

Albert, M. K.

M. Singh, D. D. Hoffman, and M. K. Albert, “Contour completion and relative depth: Petter’s rule and support ratio,” Psychon. Sci. 10, 423–428 (1999).

M. K. Albert and D. D. Hoffman, “Genericity in spatial vision,” in Geometric Representations of Perceptual Phenomena: Articles in Honour of Tarow Indow’s 70th Birthday, D.Luce, K.Romney, D.Hoffman, and M.D’Zmura, eds. (Erlbaum, 1995).
[PubMed]

Bair, W.

J. R. Cavanaugh, W. Bair, and J. A. Movshon, “Nature and interaction of signals from the receptive field center and surround in macaque V1 neurons,” J. Neurophysiol. 88, 2530–2546 (2002).
[CrossRef] [PubMed]

Banton, T.

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Figures (5)

Fig. 1
Fig. 1

Schematic flowchart of the model. The main computational stages (a)–(c) of the model are indicated. Arrows indicate the direction of information flow between the model’s different components. The white rectangles represent components of the on-center channel, the black rectangles represent components of the off-center channel, and the gray rectangles are mutual components of both channels. The weight coefficients of orientation enhancement ( w l m ) and cross-orientation inhibition ( α l m ) are schematically shown.

Fig. 2
Fig. 2

Predictions of the classical illusory contour effects: (A) Kanizsa square with white inducers and its characteristics: proximity, P; inducer diameter, D; and inner distance, d. (B) Kanizsa square with B & W inducers. (C) Illusory bar with black inducers. The input images are shown in the left column, the model predictions of the perceived contour strength in the middle column, and the demonstration of these predictions at the horizontal cross sections of the vertical illusory contours in the right column (solid black lines). Output pixel values higher than those of the corresponding input pixel (above 0.5 on ordinate) indicate the increase in the contour brightness compared to that of its background. The following horizontal cross-sections are shown in the third column: (A)  Y = 50 (B)  Y = 48 (C)  Y = 45 . The structural organization of this figure repeats itself in Figs. 3, 4.

Fig. 3
Fig. 3

Predictions of the known necessary conditions to perceive illusory contours and the known characteristics of contour strength: (A) “relatability”. (B) contrast. (C, D) lack of spatial interference. The following horizontal cross-sections are shown in the third column: (A, B, D) Y = 50 (C) Y = 45 .

Fig. 4
Fig. 4

Predictions of the suppressive stimulus conditions: (A) Kanizsa square with nonuniform and balanced inducers, compared to the classical stimulus B. (B) Kanizsa square with empty inducers. (C) Kanizsa square with inducers of decreased area. (D) Kanizsa square with nonuniform and unbalanced inducers, compared to the case of balanced inducers (A). The following horizontal cross sections are shown in the third column: (A,D) Y = 48 (B,C) Y = 50 , compared to the classical stimulus A.

Fig. 5
Fig. 5

Additional properties of illusory contours: (A,B) The importance of ”tangent discontinuities”. Compare discontinuity-present (DP), and discontinuity-absent (DA) displays. Adapted by permission from [67] (Figure 6). (C) An exception for the necessity of “tangent discontinuities” to the Kanizsa effect. The luminance of each inducer falls off in space as a Gaussian function. Adapted by permission from [65] (Figure 15). (D,E) Comparison of ”spatial interference” (D) and ”transparency” (E) conditions. (F) An example of Kanizsa illusion with eliminated inner corners (”L-junctions”).

Equations (40)

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I on ( x , y ) = I in ( x , y ) ,
I off ( x , y ) = L max I in ( x , y ) ,
I c , on / off ( x , y ) = visual field I on / off ( x x , y y ) · f c ( x , y ) d x d y ,
I s , on / off ( x , y ) = visual field I on / off ( x x , y y ) · f s ( x , y ) d x d y .
f c ( x , y ) = ( π ρ cen 2 ) 1 · exp { ( x 2 + y 2 ) / ρ cen 2 } ,
f s ( x , y ) = ( π ρ sur 2 ) 1 · exp { ( x 2 + y 2 ) / ρ sur 2 } .
I r , on / off ( x , y ) = visual field I on / off ( x x , y y ) · f r ( x , y ) d x d y ,
f r ( x , y ) = { ( π ρ rem 2 ) 1 · exp { ( x 2 + y 2 ρ sur 2 ) / ρ rem 2 } x 2 + y 2 > ρ sur 2 0 elsewhere .
C on / off ( x , y ) = I c , on / off ( x , y ) I c , on / off ( x , y ) + σ c , on / off ( x , y ) ,
S on / off ( x , y ) = I s , on / off ( x , y ) I s , on / off ( x , y ) + σ s , on / off ( x , y ) .
σ c / s , on / off ( x , y ) = σ c / s , on / off , local ( x , y ) + σ c / s , on / off , remote ( x , y ) ,
σ c / s , on / off , local ( x , y ) = α c / s · I c / s , on / off ( x , y ) + β c / s ,
σ c , on / off , remote ( x , y ) = c c · I r , on / off ( x , y ) ,
σ s , on / off , remote ( x , y ) = c s · I r , off / on ( x , y ) .
R con on / off ( x , y ) = ξ C S · C on / off ( x , y ) S on / off ( x , y ) ,
E on / off ( x , y , l ) = visual field R con on / off ( x x , y y ) · f bar ( x , y , l ) d x d y ,
f bar ( x , y , l ) = { N 1 ( ν bar , L bar ) · cos 2 ν bar [ Ω l θ ( x , y ) ] x 2 + y 2 < L bar 2 0 elsewhere .
E r , on / off ( x , y , l ) = visual field f r ( x , y ) · m = 1 N Ω w l m · | E on / off ( x x , y y , m ) | d x d y ,
w l m = cos 2 ν short ( Ω m Ω l ) m = 1 N Ω cos 2 ν short ( Ω m ) .
E on / off enh ( x , y , l ) = E on / off ( x , y , l ) · ϵ + | E on / off ( x , y , l ) | ϵ + E r , on / off ( x , y , l ) .
E ^ on / off ( x , y , l ) = { E on / off enh ( x , y , l ) E on / off enh ( x , y , l ) > E on / off min · ( 1 τ or ) + E on / off max · τ or 0 elsewhere ,
E on / off min = min x , y , l E on / off enh ( x , y , l ) ; E on / off max = max x , y , l E on / off enh ( x , y , l )
L on / off ( x , y , l ) = visual field E ^ on / off ( x x , y y , l ) · f con ( x , y , l ) d x d y .
f con ( x , y , l ) = N 2 ( ν con , L con ) · cos 2 ν con [ Ω l θ ( x , y ) ] · exp { ( x 2 + y 2 ) / L con 2 } .
E ^ on / off X O ( x , y , l ) = [ m = 1 N Ω α l m · E ^ on / off ( x , y , m ) E ^ on / off ( x , y , l ) ] + ,
α l m = 1 cos 2 ν short ( Ω m Ω l ) N Ω m = 1 N Ω cos 2 ν short ( Ω m ) ,
L inh ( x , y , l ) = visual field [ g X O ( x x , y y , l ) ] P inh · f con ( x , y , l ) d x d y ,
g X O ( x , y , l ) = M ^ ( x , y , l ) · [ E ^ on X O ( x , y , l ) + E ^ off X O ( x , y , l ) ] .
G on / off inh ( x , y , l ) = { 1 L on / off enh ( x , y , l ) > a inh · [ L inh ( x , y , l ) ] 1 / P inh 0 elsewhere ,
L r , on / off ( x , y , l ) = visual field f r ( x , y ) · m = 1 N Ω W l m · | L on / off ( x x , y y , m ) | d x d y ,
W l m = cos 2 ν long ( Ω m Ω l ) m = 1 N Ω cos 2 ν long ( Ω m ) .
L on / off enh ( x , y , l ) = L on / off ( x , y , l ) · ϵ + | L on / off ( x , y , l ) | ϵ + L r , on / off ( x , y , l ) ,
M ( x , y , l ) = L on enh ( x , y , l ) + L off enh ( x , y , l ) .
M ^ ( x , y , l ) = { 1 M ( x , y , l ) > M min · ( 1 τ ill ) + M max · τ ill 0 elsewhere ,
M min = min x , y , l M ( x , y , l ) , M max = max x , y , l M ( x , y , l ) ,
L ^ ( x , y , l ) = { max [ L on enh ( x , y , l ) , L off enh ( x , y , l ) ] M ^ ( x , y , l ) = 1 0 elsewhere .
I con on / off ( x , y ) = l = 1 N Ω I con on / off l ( x , y )
I con on / off l ( x , y ) = { L on / off enh ( x , y , l ) · G on / off inh ( x , y , l ) L on / off k , enh ( x , y , l ) = L ^ k ( x , y , l ) 0 elsewhere .
I on / off out ( x , y ) = R con on / off ( x , y ) + g ill · I con on / off ( x , y ) ,
I out ( x , y ) = L max 2 + g out 2 · [ I on out ( x , y ) I off out ( x , y ) ] ,

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